Graph Your Own Prompt
Authors: Xi Ding
Lei Wang
Piotr Koniusz
Yongsheng Gao
Venue: NeurIPS 2025 poster ()
Keywords: graph consistency regularization
neural network prompting
structured representation learning
semantic regularization
model-agnostic
Primary area: deep_learning
TL;DR: We propose graph consistency regularization, a method that improves classification by aligning feature and prediction structures to suppress inter-class noise and strengthen intra-class cohesion.

Abstract: We propose Graph Consistency Regularization (GCR), a novel framework that injects relational graph structures, derived from model predictions, into the learning process to promote class-aware, semantically meaningful feature representations. Functioning as a form of *self-prompting*, GCR enables the model to refine its internal structure using its own outputs. While deep networks learn rich representations, these often capture noisy inter-class similarities that contradict the model's predicted semantics. GCR addresses this issue by introducing parameter-free *Graph Consistency Layers* (GCLs) at arbitrary depths. Each GCL builds a batch-level feature similarity graph and aligns it with a global, class-aware masked prediction graph, derived by modulating softmax prediction similarities with intra-class indicators. This alignment enforces that feature-level relationships reflect class-consistent prediction behavior, acting as a *semantic regularizer* throughout the network. Unlike prior work, GCR introduces a multi-layer, cross-space graph alignment mechanism with adaptive weighting, where layer importance is learned from graph discrepancy magnitudes. This allows the model to prioritize semantically reliable layers and suppress noisy ones, enhancing feature quality without modifying the architecture or training procedure. GCR is model-agnostic, lightweight, and improves semantic structure across various networks and datasets. Experiments show that GCR promotes cleaner feature structure, stronger intra-class cohesion, and improved generalization, offering a new perspective on learning from prediction structure.

Graph Your Own Prompt
Xi : Lei Wang
Piotr Koniusz Yongsheng G
Griffith University,
Australian National University. "University of New South Wales
Abstract
We propose Graph Consistency Regularization (GCR), novel framework that injects relational graph structures, derived from model predictions, into the learning process to promote class-aware, semantically meaningful feature representations. Functioning as form self-prompting, GCR enables the model to refine its internal structure using its own outputs. While deep networks learn rich repre- sentations, these often capture noisy inter-class similarities that contradict the model's predicted semantics. GCR addresses this issue by introducing parameter- free Graph Consistency Layers (GCLs) at arbitrary depths. Each GCL builds a batch-level feature similarity graph and aligns it with global, class-aware masked prediction graph, derived by modulating softmax prediction similarities with intra- class indicators. This alignment enforces that feature-level relationships reflect class-consistent prediction behavior, acting as semantic regularizer through- out the network. Unlike prior work, GCR introduces multi-layer, pooss-ppace graph alignment mechanism with adaptive weighting, where layer importance is learned from graph discrepancy magnitudes. This allows the model prioritize semantically reliable layers and suppress noisy ones, enhancing feature quality without modifying the architecture or training procedure. GCR model- model-agnostic, lightweight, and improves semantic structure across various networks and datasets. Experiments show that GCR promotes cleaner feature structure, stronger intra-class cohesion, and improved generalization, offering new perspective on learning from prediction structure. [Project website] [Code]
Introduction
Deep neural networks have achieved impressive success across wide range of classification tasks, from natural images to medical diagnostics and scene understanding [88, 117, 111, 23, 26, 27, 28]. key factor underlying this success is the ability of deep models to learn rich internal representations. However, despite their effectiveness, these representations are often noisy, entangled, and lack clear alignment with semantic class boundaries. This discrepancy can lead to feature spaces where samples from different classes remain closely aligned, undermining generalization and limiting the interpretability and robustness of learned models.
Intuitively, if a network is confident that inputs belong the same class, indicated by their softmax predictions, then their intermediate representations should also reflect this similarity. Conversely, samples predicted belong different classes should ideally be separated in feature space. Yet, existing architectures and loss functions typically do not enforce such consistency between feature- and prediction-leve relationships [90, 106, 108]. While contrastive learning and graph-base regularizers have emerged to structure feature spaces, they often require explicit positive/negative sampling or operate at a single level of abstraction, leaving much of the internal network dynamics unregulated [105, 64, 81].
"Equal contribution (co-first authors).
Corresponding author: yongsheng, gaodgriffith. edu.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Trakk
(a) DenseNet-121 (b) With our GCLs
(c) MobileNet
(d) With our GCLs
Figure Relational graph visualization using a batch of 64 samples from CIFAR-10 10 on (left) DenseNet-121 and (right) MobileNet. We compare the baselines with their counterparts augmented by our GCLs. Our method promotes richer, blsssaware semantic representations by acting form of self-prompting. For DenseNet-1211 the baseline feature relational graph tends to connect samples based on superficial visual similarity (e-g., deer, horse, and automobile), often ignoring semantic boundaries. In contrast, our GCL-enhanced model produces more semantically coherent groupings, clearly separating animals from vehicles. On MobileNet, the prediction relational graph further highlights the strength of method, demonstrating cleaner, more distinct class relationships compared to the baseline. These improvements reflect the effectiveness our model aligning feature and prediction spaces with semantic structure, despite being lightweight and parameter-free.
In this work, we propose a new form of structural supervision that enforces relational graph consis- lency between features and predictions throughout the network. We introduce Graph Consistency Regularization (GCR), a lightweight and general-purpose method that inserts Graph Consistency Layers (GCLs) after arbitrary network blocks/layers. Each GCL builds a batch-level feature relational graph using pairwise relations. These feature graphs are aligned with class-aware masked prediction graph that encodes semantic similarity based on softmax outputs and label-derived intra-class indica- tors. The core idea treat the prediction graph as a reference structure, and guide the network to shape its internal feature geometry to be consistent with semantically meaningful predictions.
GCR is architecture-agnosiic. lightweight, and introduces new parameters. Importantly, it provides multi-layer supervision by aligning feature-predictee structures at multiple depths, using dynamically learned layer-wise weights. This acts regularization signal that sharpens class boundaries, sup- presses noisy inter-class affinities, and encourages feature representations that reflect class semantics more faithfully (Fig. provides comparison). Our contributions are summarized follows:
i. We are the first dynamic, batch-level relational graphs to guide the learning process using class-aware prediction structures, enabling a new form of semantic regularization.
ii. We introduce Graph Consistency Layers (GCL.s, lightweight, parameter-free modules that can be flexibly inserted into existing networks, enabling multi-scale structural supervision with adaptively learned layer-wise weighting based on graph discrepancy.
iii. We propose Graph Consistency Regularization (GCR), novel framework that enforces align- ment between intermediate feature graphs and global, class- prediction graphs, promoting semantic coherence and representation consistency throughout the network.
We validate GCR across diverse architectures and benchmarks, achieving consistent accuracy gains and stronger generalization without altering backbone design or training protocol. Below, we review related work.
2 Related Work
Graph-based representation learning, Graph-based methods [54, 16] have been widely adopted to model relationships among samples, particularly in Euclidean domains such as social networks or molecules [1, 53, 121]. J. In classification tasks, graphs have been used to capture instance-level
similarities for label propagation [113, 74), semi-supervised learning [115, 114], and contrastive representation learning [20, 91]. For example, some methods build graphs over entire datasets [101] or memory banks [9, 3] to encourage feature consistency via graph Laplacian regularization [80, 4] or message passing [38, 79]. Unlike these approaches, which require maintaining global graphs rely

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Softmax
Block
GCL
Block
GCL
Layer
GCL
GCL
Layer
GCL
CE
GCR
Feature graph
W
Masked prediction graph ceneeeeeeeeee
Figure 2: Our parameter-free Graph Consistency Layer (GCL), highlighted in red, can be inserted after any micro-network block (e.g.. Inception) or specific layer (e.g., fully connected). Each GCL constructs relational graph from batch-level features using a similarity metric (e.g-, cosine). A reference graph is generated from softmax predictions and masked by intra-class indicators: binary masks identifying semantically consistent pairs. Each GCL enforces alignment between masked prediction graph and the feature-level graphs. The resulting consistency signals are adaptively weighted, forming the Graph Consistency Regularization (GCR) framework, which integrates with the primary loss (e.g., cross-entropy), acting semantic regularizer guide learning.
graph neural networks (GNNs), our method operates locally within each batch and uses graphs constructed dynamically from features and predictions, avoiding architectural overhead.
Contrastive and metric learning. Contrastive learning has emerged powerful framework for sulf-supe and supervised representation learning [110, 69, 60]. These methods typically pull together positive pairs (e.g., same-class or augmented views) and push apart negatives. Extensions such as supervised contrastive learning and hard negative mining improve class discrimination by incorporating label information [66, 105]. However, these approaches often rely on explicit sampling strategies or carefully tuned augmentations, and are usually applied a single point in the network. In contrast, method requires sampling. data augmentation, and aligns structural relationships across multiple network depths, offering a form of contrast without contrastive pairs.
Regularization and feature structure learning. Various regularization techniques aim to improve generalization by shaping the geometry of learned feature spaces [57, 78]. For instance, center loss [86, 84] and triplet loss [24, 119, 29] enforce compactness or margin between classes. Other works apply orthogonality or decorrelation constraints on activations or weights [71, 104]. More recently, structural regularizers use pairwise distances affinity matrices inject relational constraints into training [73, 59]. Compared to these, our approach enforces cross-space structural alignment, between features and softmax-base predictions, while incorporating class-aware masking to focus only on semantically meaningful relationships.
Layer-wise supervision and structural alignment. Supervision at intermediate layers has shown promise in improving training dynamics and interpretability [52, 126, 39]. Auxiliary losses [93, 43], attention distillation [2, 50, 75], and intermediate contrastive losses [103, 15, 86] are examples where internal representations are guided explicitly. However, these methods typically supervise layers independently, and rarely use the structural information from model outputs to regularize feature learning [18, 100, 122, 129]. Our approach introduces new form semantic structure supervision by aligning feature similarity graphs with a masked prediction graph across multiple layers, with learnable depth-based weighting to adaptively emphasize useful features during training.
A discussion of GCR's relationship to existing paradigms is presented in Appendices A and B. Below,
we present our proposed method.
Method
We introduce the Graph Consistency Layer (GCL), lightweight module that can be inserted at any layer or Each GCL dynamically constructs a relational graph from intermediate features and aligns with semantic graph derived from the model's 's own predictions within each training batch. This alignment driven by Graph Consistency Regularization (GCR), a novel technique that promotes semantically coherent and geometrically structured feature learning. Our motivation is discussed in Appendix C.
block (e.g.. a convolutional layer an inception block, see Fig. 2).
3

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3.1 Graph Consistency Layer
Relational graph construction on features. Given a batch of feature activations at layer I. repre- sented by the feature matrix X()) TRxxd (where vectorize flatten high-dimensional feature maps), with being the batch size and d the feature dimension, we construct pairwise relational graph F() E R that encodes the relationships between features within the batch. For simplicity, use cosine similarity compute the relationship between the i-th and j-th samples follows:
! = ReLU (Os(())) ,,,,,,)))))
This formulation captures local geometric relationships in feature space. However, the raw fea- ture graph F(9) E Tx may contain spurious correlations that misalign with true semantic class boundaries. To address this, a semantic reference graph is introduced to guide feature alignment.
Masked relational graph on predictions. To construct the reference graph, we use the network's prediction logits Z E [2"xC, where the number classes and contains the pre-softmax output scores. From these logits, compute class probabilities via the softmax function, and then calculate the pairwise similarity between prediction vectors Z, 2j E IRC of the i-th and j-th samples as:
Si = ReL.U (cos(softmax (z:), softmax(z,))
The resulting matrix S E 8 captures the pairwise similarities between the predicted class distributions of samples. However, not all prediction similarities are equally informative for guiding feature alignment, particularly (i) in the early stages of training, when predictions are often noisy and lack well-formed class semantics, and (ii) in cases where inherently ambiguous or visually similar classes introduce misleading affinities. To focus semantically consistent pairs, introduce binary mask M E (0, 1] 1]nxR, R where Mij = if samples i and j share the same ground truth label, and 0 otherwise. The prediction relational graph P is then defined as:
Ps Pay=Ma @ Sij.
where @ denotes element-wise (Hadamard) multiplication. This graph retains only the prediction similarities between samples of the same class, effectively encoding intra-class semantic relationships while discarding noisy or misleading inter-class connections. Below, we outline our GCR framework.
3.2 Graph Consistency Regularization
Layer-wise graph alignment. GCR centers on aligning the feature graph F()) with the masked prediction graph P. To encourage symmetric, undirected relationships and eliminate redundancy, we use only the strictly upper triangular part of the graphs, excluding the diagonal elements remove self-connections. This design ensures that the graphs capture bidirectional affinities between samples without double-counting or emphasizing self-similarity. Such undirected structure often leads to more stable optimization and promotes balanced representation learning across the batch.
The graph alignment loss layer is defined the squared Frobenius between the upper triangular parts of the two graphs:
C = triu(F() triu
where triu(-) denotes the strictly upper triangular matrix. This loss compels the model adjust intermediate features so their geometric structure aligns with the semantic topology encoded in the masked prediction relational graph.
Graph consistency aggregation. To enforce consistency across the network hierarchy, we aggregate alignment losses from set of selected layers (1,..., KJ. The total GCR loss is given by:
LOCR
Σ (urlitriu(F()) triu(P)
(5)
IE(
Here, wi is weight that balances the contribution of each layer. These weights can either: (i) fixed, using depth-based heuristies such equal (wi 1/K), linear (wi I/K), squared ((I/K)).

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Figure 3: Feature map visualizations from models trained identical data batches: (top) baseline and (bottom) our GCL-augmented model. Brighter red regions indicate stronger feature activations. Compared to the baseline, GCL-enhanced maps more clearly emphasize duss-discriminative cues, e.g., cat faces, ears, and eyes, and for dogs, tongues, noses, and facial contours, reflecting improved focus and interpretability. GCL also yields higher classification accuracy (98.1 % 99.8%).
square-root ( (Vi/M,, cosine (tun((rk) or arccosine arccos(1 ) current alignment quality:
or (ii) adaptive, based on
(- triu(F(()) triu triu(P))
exp
W1
Σラ1 exp
- (eit/tii(FF)) triu(P)
The adaptive weighting ensures that layers with greater misalignment are given more importance during training, allowing the model to concentrate refining representations that are less aligned with the desired semantic structure. We evaluate different weighting schemes in our experiments.
Training objective. The overall training loss is composed of two components: the standard cross- entropy loss, denoted as LCE, and the GCR loss, LGCR. The total loss function is given by:
Lisal = LCE + À LOCR,
where À a hyperparameter that controls the relative contribution of the GCR term. key advantage of GCR is that it introduces additional parameters, and its graph alignment loss relies matrix operations well-suited to modern hardware. Next, we present the theoretical foundations of GCR.
3.3 Theoretical Analysis of GCR
We present a theoretical analysis of GCR, linking its empirical design to core principles in statistical learning and spectral graph theory. Specifically, we show that minimizing the GCR loss: (i) reduces the effective hypothesis class capacity via covering number bounds[55, 123] and Dudley's entropy integral[32]: (ii) promotes spectral alignment between learned features and semantic prediction graphs through normalized Laplacians[5]] [5]] and (iii) acts PAC- CCC-Baysiann regularizer[351] imposing a structural prior over the function space. Additional insights are provided in Appendix D.
Generalization via covering numbers. Let FL be the class of functions f()), X - Rd representing layer-l embeddings. Assume the feature representations are uniformly bounded by a constant lz
norm: (x)l22 (+>)>==V\'
(1) (x)
B, for all z E X. This is standard constraint in learning
theory to control hypothesis space capacity. In practice, with & normalization used here, B=1. Next, we define a ciruttrally-coostraine hypothesis class:
f==frer
flex == || (ri) - tP) sc
(8)
which enforces alignment between feature and prediction graphs reflecting intra-class similarity. Theorem 1 (Generalization via Dudley's entropy integral). Let {(f(x), y) be 7-Lipschitz loss function (e.g., cross-entropy). and let FL be the class of functions at layer I such that each function ((t) satisfies the by-bundeed constraint (x))l2 S B. Suppose F. S Fr is the subset of functions that are additionally constrained by the GCR alignment loss:
GGR
n²
1
n
(eLU((x(",Xf))) - P.s) Se
(9)
5

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Figure 4: Relational graph visualization on Kaggle cats vs. dogs. We compare the best baseline model and our GCL-augmented model using the same batch of 32 samples (red cat, blue dog). The baseline consists of four convolutional blocks and two fully connected layers; our method inserts Graph Consistency Layer (GCL) after each, totaling six GCLs. The top row shows the baseline (without GCLs); the bottom row shows our GCL-enhanced model. Each column visualizes the relational graph specific layer, from early features (left) to final predictions (right). Early layers exhibit weak connectivity, as low-level features poorly capture class semantics. As depth increases, both models shift toward more structured, sass-seeaaabbe relationships. GCLs amplify this effect by attenuating low-similarity inter-class edges and reinforcing intra-class coherence, leading to improved accuracy (98.1%) 18.1% vs. 99. 99. For clarity, edges with similarity < 0.4 are omitted.
where Pij the target alignment between the normalized feature vectors x and x3 The GCR loss enforces that the angular distances between the feature vectors are small, meaning that the vectors close each other in the Fuclidean space. If Fe admits covering number bound:
N(Fe, 112, S (
(10)
where N(Fe, 1I [12, ( is the covering number of Fe with respect to the Eg norm, then the expected loss of a function J Fe bounded with high probability by:
+ 1 Σ (((())))))
127
B
Vdlos (
B( M)1 S
(11)
Vn.
where B O(Ve) is the effective radius of function class F. under the GCR constraint, and the second term represents the generalization error, which is controlled by the function class complexity.
Proof. The proof can be found in Appendix E.
Remark 1. This result shows that GCR reduces generalization error by shrinking the effective complexity of the function class. By aligning relational structure, GCR implicitly contracts the hypothesis space, leading to improved generalization.
Spectral alignment via normalized Laplacians. Let F and P be symmetric affinity matrices derived from feature embeddings and masked predictions, respectively. Their associated normalized graph Laplacians are defined Lr I-D 1/2 1/2 FD 1/2 and Lp I-D 1/2 p/2 1/2 where Dr r and Dp are the degree matrices corresponding to F and P, i.e., (Dr) -, F.j and similarly for Dp.
Proposition 1 (Spectral alignment). Let F and P be symmetric matrices such that
F Se, 1D=-Dpt 8
(12)
Then, there exists a constant C depending on spectral properties of the graphs (e.g. sparsity, minimum degree), such that
[[ - - < +8.
(13)
Proof. The proof can be found in Appendix E.

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Table 1: Accuracy (%) on CIFAR-10 across models. Results are shown for MobileNet (MNet), ShuffleNet (SN), SqueezeNet (SQNet), GoogLeNet (GLNet), ResNeXt-500110 (Rx-50/101), ResNet- 34/50/101 (R34/R50/R101) 101), DenseNet-121 (D121), and MAE under various GCL configurations (Farly, Mid, Late, combinations, Full). Bold indicates the best improvements over baselines; under- lines mark the second-best best. The final column shows the average accuracy for each configuration.
MAE
MNet
SN
SQNet
GLNet
Rx-50
Rx-101
R34
R50
R101
D121
Mean
Baseline
38.95
91.21aa 90
92.50
94.10ga x 94.57,0
95
32
94,83
as
g5.
95 31
95.01
43.32gn
Early GCL
89.02=0.s
91.17
92 33
92. 22.59
95.48
95.55
a5.57
93.98.2.2
to
02.401..30
98.47
10.10
m.
as.0
a
Late GCT.
80.20
92.364421
94.88
20.3
25.72
54.07-20
EarlyaMii
89.52-0
20.37am
92.564421
94.79....
95
25 23
95
95.5 25
95.37am
35.62.0
MideLate
89 89.59
91
23
92 22.79
21
61.61=9.2
95.5tamm
95
38
95-45
05.33
95
23.522m11
05.70am.
95.
91.00-2.0
EarlytLane
89.64 6..0
92 10 ai
92.70
$4.69
18.00
93.40
95.35
95.66
95.31 andr
95.49an
95.53
93.92
GCL
89.55
90.99
92.48,0
92.65gaa
94.57.831
26.50.010
95.34,0m)
95.48.0.17
95.62.0.s
95.38
95,51
93.92gau1
Corollary 1. The GCR alignment loss, which encourages F PlIp S indirectly enforces spectral
similarity of the normalized Laplacians. This promotes agreement between the clustering structure and diffusion properties of the learned features and masked predictions.
PAC- CAC-Bayesian view of structural regularization. We now present PAC-Bayesian view of GCR, which bounds generalization by linking expected loss to empirical loss and the divergence between posterior and prior over hypotheses.
Let P denote prior distribution over model functions f. representing a -aneturr-aggosticc belief (e.g., uniform or isotropic Gaussian over parameters). Let Q be posterior distribution supported on models that minimize training loss while also conforming to a structural constraint induced by GCR, ie., Q is restricted to functions ʃ such that c(I) GCR S < for each relevant layer 1. Theorem 2 (PAC-Bayes generalization bound with GCR). Let L(f) E(
y) be the ((f(x)) y:) be the empirical loss on n training examples. Then, for any posterior distribution Q over functions and any prior distribution P. with probability at least 1 1- over the training data, the following bound holds:
expected population loss of model f and let = (
(L( + log(1/6)
E =
(14)
2n
This classical PAC-Bayesian bound quantifies generalization through two factors: (i) the empirical per- formance of models sampled from the posterior Q, and (ii) the Kullback- Leibler (KL) divergence[44] KL(QIP). measuring how much Q deviates from the prior. In GCR, the constraint c/0 I c imposes structured alignment between feature similarity F(!) and semantic prediction structure P. serving as an inductive bias. Thus, when Q is supported on models satisfying this constraint, the KL complexity term reflects the strength of this alignment.
Proposition (Structure-induced KL complexity)- If the posterior Q is concentruted on models with small GCR loss at layer I, then the KL divergence an isotropic prior P satisfies:
KL(2)P) S &C) C | PIE::
(15)
for some constant C depending on the form of P.
Proof. The proof can be found in Appendix E.
Remark This perspective shows that GCR does more than minimize training loss, it also implicitly regularizes the hypothesis space by favoring models whose internal representations reflect known semantic structure. This improves generalization by reducing the effective size of the model class, as made explicit through the PAC-Bayesian framework.
Experiment
4.1 Datasets, Models, and Experimental Setup
We evaluate GCR on several benchmark datasets, including Kaggle cats vs. dogs[22], CIFAR- 10[62], CIFAR-100 Tiny ImageNet[63), and ImageNet-IK [23]. Our evaluation spans di-
7

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Table 2: Accuracy (%) on CIFAR 100 across models.
MAE
MNet
SN
SQNet
Rx-50
Rx-101
RM
R50
D121
Mean
Baseline
64.29
65.95
70.11
69.4
a7.75
77.85
46882
27.09.057
72.95
65
67-45_6-11
20
37.0aam.
19.37.0
Mid
67.88-6.11
20
20.2
78-26-021
GCL
68.54
68.32
79,54.0m
29.83
78.31
79.42.a
74.74
Mid
66:23
67.62
20.87a
18.90.o
78.58.0m
79.22
74.
aidalate
65 65.2720
68.33
71 633u2s
20.30a..42
78.91.017
79.57.0.21
27.30a0.0
78.550m:
70.54.0.24
a4.4
Eadyy-Late
65 2::22:
67.25
67.25_6.11
71.55
71.03g0.24
79.02:020
79.4190.23
78.194m2s
78.70g021
79.45
74.43-5.m
05.38
6..22...1
21,30
70.77p0.00
29.01.019
79.29
77.79am.0
78.71a0:
79.27a0.
74.42
(a) ShuffleNet
(b) GoogLeNet
(c) ResNet-50
(d) ResNet-101
(e) DenseNet-121
Figure 5: Relational graph comparison across five models on the same batch. Top row: baselines; bottom row: GCL-augmented versions, showing sparser inter-class connections and stronger class- aware structure, highlighting GCL's effectiveness in enhancing relational representations.
verse range of architectures, from lightweight models (e.g.. MobileNet[461, ShuffleNet 124], and SqueezeNet[51]), to deeper CNNs (e.g., GoogLeNet[97]. ResNet[42],, DenseNet[48], ResNeXt[[ [2]] Stochastic ResNet[49], and JF--ResNe([47D. We also include mransoommr-bbaee architectures such as ViT[31], [70] MobileViT[77). CFIT]120]), iFormer [95], ViG [37], as well Masked Autoencoders (MAF) [41].
Experiments run on NVIDIA V100 GPUs with 12 CPUs and 48 GB RAM. For CNNs, we follow [25]: 200 epochs, initial learning rate 0.1 (decayed at epochs 60/120/1600 batch size 128, weight decay 5x10-4 and momentum 0.9. For transformers, we use Adam W with learning rate of Ix10- cosine annealing, weight decay 5x10-2 2 batch size 256, AMP[30], 10-epoch warm-up up, and gradient clipping (norm 1.0). We divide each model into three stages: early (E), middle (M), and late (L), and insert GCLs to create seven configurations: individual stages (E, M or L), pairs (E+M, M+L, E+L), and full GCL. We evaluate weighting schemes including equal, linear, square root, squared, cosine, arccosine, and adaptive. All experiments use 1= 1. We report the mean and standard deviation over 10 runs 10 CIFAR-100, and Tiny ImageNet, and over runs for ImageNet-1KK 1K.
Appendix F presents detailed analysis of GCR's 's time complexity, while Appendix G provides complete details of the experimental setup.
4.2 Quantitative and Qualitative Evaluation
GCL can serve as an attention mechanism. Fig. offers a visual comparison between feature activations from baseline CNN model and those from GCL-augmented model, both trained on identical data batches. The key distinction lies in the sharper, more localized activations produced by the GCL-enhanced model. Notably, the parameter-free GCL mechanism yields feature maps that are more aligned with semantically relevant regions, such as the eyes, ears, and facial outlines of the cat, and the tongue, snout, and eyes of the dog. This suggests that GCL encourages the network to focus on class distriiiiiiiiiiiivee features, thereby reducing attention to background or irrelevant image regions. contrast, the baseline model's activations appear diffuse and less structured, indicating weaker spatial selectivity. The visual improvements align well with the reported jump in classification accuracy from 98.1% to 99.89....999 99, highlighting that improved interpretabilitt does not come the cost of performance, rather, it appears enhance it. Fig. 4 compares how GCLs guide model learning.

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Table 3: Accuracy (%) on Tiny ImageNet across models. All results are obtained by training models
from scratch. We also evaluate Stochastic ResNet- 18 (R18SD) and SE-ResNet- 18 (SERI8).
CefT
M
MV
MVIT
MNet
RI8SD
SERIS
R34
Baseline
20.79
40.05
49.95
ao
28,00
51.58g000
22.08.022
50.27
g0.8
25.49
65.6540024
67.51
os
23.64.002
Early GCL.
29.02.02
40.98.0.19
51.22
20 -0.11
51.33
29
54.88
2..93
66.52.0.23
67.79 10 $4,32-000
an:1 20.95.0.19
50.30s0.19
0.20
51
33.88
23..24
52.63-2.3
64.03
65.6618.35
67.62.6.0
Late OCL
27
40.35
50.8220.29
20.77.0..1
51,99.021
24.10.0.10
55.4720.21
57.87
63/792019
65.8540.25
67.61
Madly+Mid
20.0 41.26.0.15
20.35.a.a5
73 0.22
20.25
20.35
57.49
10.2
65.86
67.74.a.m
38
40.52a0.2
20.09
28
22.90z0.20
66.62.0.25
-7.65-o.2:
64.29
2017
65.95
68.58
54.19-n.0
Harly+Late
38,
40.71
50.7010.00
50
51,36.0
53.57,0091
54.89
17
63.89.011
65.8330.17
Pall GCL
38. 38 10
49.924o.u
20.00
30.16_...
51882a0
54.01.0
to 54.87
2o0 io 57.64g000
64.102011
56.01
67.66 is 24.13.0.0
67
6.31
0:33
6:2
9:40
3.30
6:44
6:77
3:17
1:23
1.2
4:41
8.35
3.33 2,00
Track
(a) Again on CIFAR-10 (b) Again on CIFAR-100
(c) ShuffleNet
(d) With GCLs
Figure 6: Performance gains (A, %) from adding GCLs with different configurations across various weighting schemes. Darker red indicates larger gains. Adaptive weighting achieves the highest improvement, showing the value of graph misalignment as a guidance signal. (c) and (d) show that GCL-augmented ShuffleNet yields more compact intra-class clusters and better inter-class separation.
GCLs lead to more interpretable and discriminative feature spaces. In GCL-augmented graphs (Figs. 1 and 5), semantically similar classes such as Airplane, Ship, and Truck (all vehicles) form closer associations, suggesting that GCL helps the model better understand high-level concepts. A similar trend appears among animal-related classes (e.g.. Dog. Cat, Horse, and Deer). As model complexity increases (from ShuffleNet to DenseNet-121), the relational graphs become denser. However, GCL still consistently improves clarity. Even in large-capacity models like ResNet-101 and DenseNet-121, GCL enhances structure by reducing cross-class noise and reinforcing class-wise coherence. The consistent improvements across very different models, from lightweight ShuffleNet to deep DenseNet, demonstrate the general applicability of GCL. It does not merely overfit to specific architecture but contributes to relational learning in a model-agnostic, parameter-free way.
Adaptive weighting and selective layers boost GCR effectiveness. The adaptive scheme consis- tently yields the highest or near-highest gains (Figs. 6a and 6b), especially on CIFAR-100, where deeper features benefit more from dynamic supervision. Placing GCLs at later layers (L) or in combi- nations involving deeper blocks (e.g., E+I., Full) yields the greatest improvements, highlighting the semantic richness of deeper layers. Among fixed schemes, squared and equal weighting outperform linear. Notably, applying GCLs to all layers (Full) does not always yield the best gains, indicating that naïve aggregation can weaken the regularization effect. These findings emphasize the value of selective layer placement and adaptive weighting to effectively guide feature alignment.
Tables 1, 2, and 3 summarize the accuracy gains achieved by GCR across three benchmarks.
GCLs are effective across various datasets and model types. Across CIFAR-10, CO0, and Tiny ImageNet, GCR consistently improves accuracy relative to the baseline in nearly all configurations. On CIFAR-10, which is comparatively simpler, GCR delivers a peak mean accuracy of 94.07% (Late GCL), improving over the 93.32% baseline. CIFAR- 100, with greater semantic granularity and inter-class overlap, benefits even more: Late GCL increases the mean accuracy from 72.95% to 74.74%, a +1 +1.79% absolute gain. On Tiny ImageNet, where models are trained from scratch under more challenging conditions, GCR again proves effective. The best performance is achieved with Early GCL (54.32%) and Late GCL (54. 18%), improving over the 53.64% baseline and affirming GCR's applicability to both convolutional and transformer- vision models.
GCLs improve transformer architectures on ImageNet- We conducted experiments on I KK using transformer- based architectures (e.g., iFormer, ViT, and ViG). Results are summarized Table 4. Key results demonstrate the effectiveness of our approach. For iFormer-S, GCR boosts performance from 83,4% to 84.8%, yielding +1.4% gain. For iFormer-B B, accuracy
9

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Table 4: Comparison of iFormer, ViT, and ViG with different GCL integration strategies on ImageNet- IK. Results are averaged over three runs.
Mathod
iForme-SS
iFormer-B B
VIT-B/16
ViG-B
Baseline
83,420.0
84.620.45
74.3g0.51
82.3
Early GCL
83.8
20.040..0
74.7
82.8
Mid GCL
83.8
85 85.5
75.2:0.56
83.0
Late GCL
20.5
86. $6.120.30
75.8
80.0
Early + Mid
24.3
25.9
75.620.41
83.7
Mid Late
84,8
85.9
75.6
83.9
Farly + Late
84.5
85. 85.220.29
74.9
83.5
Full GCL
24.3
85.820.26
25.5
83.6
improves from 84.6% to 86.1 %, corresponding to a +1.5% gain. Similar improvements are also observed with ViT-B/16 and ViG-B, confirming the generality of our method. These improvements validate that GCR scales effectively to large, complex datasets and modern architectures. They support our core hypothesis: aligning feature geometry with prediction semantics enhances general- ization. Moreover, GCR achieves this without architectural changes or added parameters, reinforcing its value as a model-agnosten and lightweight regularizer.
GCLs are most effective in later layers, aligning with decision structures. Among GCL con- figurations, those applied at later stages generally yield the most significant improvements. Late GCL consistently achieves the highest or near-highest performance across all three datasets. This is likely because later layers encode higher-level semantic representations that align more closely with class-level decision boundaries. Aligning these representations with the prediction graph enhances inter-class separability while reducing intra-class dispersion, especially on fine-grained tasks like CIFAR-100 and Tiny ImageNet. While Early and Mid GCLs also contribute, their benefits are com- paratively moderate, indicating that feature regularization earlier stages may be less semantically meaningful or more susceptible to noise. Figs. 6a and 6b show clear performance gains.
GCLs enhance geometry and generalization. Finally, the results substantiate GCR's central hypothesis: aligning feature space geometry with the model's own prediction space fosters more discriminative and generalizable representations. Across datasets, GCR reduces classification ambi- guity, tightens intra-class cohesion, and sharpens class boundaries, particularly important in datasets with high inter-class similarity. Figs.6c and 6d compare results on ShuffleNet, showing that the GCL-augmented model produces more compact intra-class clusters and better inter-class separability, Moreover, GCR achieves this with minimal computational overhead and zero parameter increase, making it highly practical for real-world deployment where architectural changes or expensive training adjustments are undesirable.
Additional results, visualizations, and discussions are provided in Appendix H, while the work's limitations and future research directions are covered in Appendix I.
Conclusion
We introduced GCR, a novel approach for enhancing classification by aligning the structural relation- ships between feature representations and model predictions through parameter-free, modular GCLs. By constructing feature relational graphs at multiple points in the network and aligning them with global masked prediction graph, GCR encourages the network learn semantically meaningful and class-consisten representations. This graph-based alignment acts as form of neural network prompting, guiding learning without modifying the architecture or training procedure. Our exper- iments across diverse architectures demonstrate that GCR improves intra-class cohesion, reduces noisy inter-class similarity, and leads to better generalization. We believe this work offers new perspective on structured representation learning by using prediction structure regularizing signal. Future work may explore extending this idea to tasks beyond classification, such as segmentation or retrieval, and integrating it with self-supervised learning frameworks.
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Acknowledgments and Disclosure of Funding
Xi Ding, a visiting scholar at the ARC Research Hub for Driving Farming Productivity and Disease Prevention, Griffith University, conducted this work under the supervision of Lei Wang. Lei Wang contributed the mathematical analysis, modeling, and theoretical development, while Xi Ding implemented the code and performed the experiments.
We sincerely thank the anonymous reviewers for their invaluable insights and constructive feedback, which have greatly contributed to improving our work.
This work supported by the Australian Research Council (ARC) under Industrial Transformation Research Hub Grant 1000000002.
This work was also supported by computational resources provided by the Australian Government through the National Computational Infrastructure (NCI) under both the ANU Merit Allocation Scheme (Lead CI: Lei Wang) and the CSIRO Allocation Scheme (Lead CI: Piotr Koniusz).
References
[1] T. Agouti. Graph-based modeling using association rule mining to detect influential users in social networks. Expert Systems with Applications, 202: 117466, 2022.
[2] G. Aguilar, Y. Ling, Y. Zhang, B. Yao, X. Fan, and C. Guo. Knowledge distillation from internal representations. In Proceedings of the AAAI conference on artificial intelligence, volume 34, pages 7350-7357, 2020.
[3] I. Alonso, A. Sabater, D. Ferstl, I.. Montesano, and A. C. Murillo. Semi-supervised semantic segmentation with pixel-level contrastive learning from class-wise memory bank. In Pro- ceedings of the IEEE/CVF international conference on computer vision, pages $219-8228, 2021.
[4] R. Ando and T. Zhang. Learning on graph with laplacian regularization. Advances in neural information processing systems, 19, 2006.
[5] F. Bauer. Normalized graph laplacians for directed graphs. Linear Algebra and its Applications, 436(1 (6(11)94433-4222 2012.
[6] M. Belkin and P. Niyogi. Laplacian eigenmaps for dimensionality reduction and data represen- tation. Neural computation, 1373-1333, 2003.
[7] M. Belkin, P. Niyogi, and V. Sindhwani. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. Journal of machine learning research, 7(11), 2006.
[8] M. M. Bronstein, J. Bruna, Y. LeCun, A. Szlam, and P. Vandergheynst. Geometric deep learning: going beyond euclidean data. IEFE Signal Processing Magazine, 34(4):18 42, 2017.
[9] A. Bulat, E. Sânchez-Lozano, and G. Tzimiropoulos. Improving memory banks for unsuper- vised learning with large mini-batch, consistency and hard negative mining. In ICASSP 2021- 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1695-1699. IEEE, 2021.
[10] I. Chami, Z. Ying, C. Ré, and J. Leskovec. Hyperbolic graph convolutional neural networks. Advances in neural information processing systems, 32, 2019.
[11] C. Chen, Z. Wang, J. Wu, X. Wang, L.-Z. Guo, Y.-F. Li, and S. Liu. Interactive graph construction for based semi-supervised learning. IEEE Transactions on Visualization and Computer Graphics, 27(9):3701-3711 2021.
[12] Q Chen, L. Wang, P. Koniusz, and T. Gedeon. Motion meets attention: Video motion prompts. In The 16th Asian Conference on Machine Learning (Conference Track), 2024.
[13] Chen, S. Zhang, Li, et al. Redundancy-free message passing for graph neural networks. Advances in Neural Information Processing Systems, 35:4316-4327, 2022.

---

[14] T. Chen, S. Kornblith, M. Norouzi, and G. Hinton. A simple framework for contrastive learning of visual representations. In International conference on machine learning, pages 1597-1607. PmL.,, 2020.
[15] T. Chen, C. Luo, and L. Li. Intriguing properties of contrastive losses. Advances in Neural Information Processing Systems, 34:1 1834 11845, 2021.
[16] Y. Chen, M. Rohrbach, Z. Yan, Y. Shuicheng, J. Feng, and Y. Kalantidis. Graph-based global reasoning networks. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 433 442, 2019.
[17] Chen, Z. Yang, Y. Xie, and Wang. Contrastive learning from pairwise measurements. Advances in Neural Information Processing Systems, 2018.
[18] H. Choi, A. Som, and P. Turaga. Ame-loss: Angular margin contrastive loss for improved explainability in image classification. In Proceedings of the IEEE/CVF conference on computer vision and paltern recognition workshops, pages 038-839,200 2020.
[19] C.-Y. Chuang, J. Robinson, Y.-C. Lin, A. Torralba, and S. Jegelka. Debiased contrastive learning. Advances in neural information processing systems, 33:8763-8775, 2020.
[20] Cole, X. Yang, K. Wilber, Mac Aodha, and Belongie. When does contrastive visual representation learning work? In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 4755-14764, 2022.
[21] S. Cuadros, Zappella, and N. Apostoloff. Self-conditioning pre-trained language models. In International Conference Machine Learning, pages 4455-4473, PMLR, 2022.
[22] W. Cukierski. Dogs vs. cats. attps://keglee.cmmmmmmmmpp com/ htt.://gg.....ooooooooooooo///// Kaggle.
2013.
[23] J. Deng, W. Dong, R. Socher, L.-J. Li, K. Li, and L. Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248-255. Icee, 2009.
[24] W. Deng, I.. Zheng, Y. Sun, and J. Jiao. Rethinking triplet loss for domain adaptation. IEEE Transactions on Circuits and Systems for Video Technology. 31:29-37, 2020.
[25] T. DeVries and G. W. Taylor. Improved regularization of convolutional neural networks with cutout. arXiv preprint arXiv:/7000 770.00455, 2017.
[26] X. Ding and L. Wang. Do language models understand time? arXiv preprint arXiv:2412.13885 2024.
[27] X. Ding and I.. Wang. Quo vadis, anomaly detection? Iims and vlms in the spotlight. arXiv preprint 18.29224../8298, 2024.
[28] Ding and L. Wang. The journey of action recognition. In Companion Proceedings of the ACM Web Conference 2025, WWW '25 Companion, New York, NY, USA, 2025. Association for Computing Machinery.
[29] T.-T. Do, T. Tran, D. Reid, B. V. Kumar, Hoang, and G. Carneiro. A theoretically sound upper bound on the triplet loss for improving the efficiency of deep distance metric learning. 2019 IEEE/CVF Conference on Computer Vision and Puttern Recognition (CVPR), pages 10396-10405, 2019.
[30] M. Dörrich, M. Fan, and A. M. Kist, Impact of mixed precision techniques on training and inference efficiency deep neural networks. IEEE Access, 11:57627-57634, 2023.
[31] A. Dosovitskiy, I. Beyer, A. Kolesnikov, D. Weissenborn, X. Zhai, T. Unterthiner, M. De- hghani, M. Minderer, G. Heigold, S. Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11 1929, 2020.
[32] M. Dudley. Metric entropy of some classes of sets with differentiable boundaries. Journal of Approximation Theory, 10(3):227-236, 1974.
12

---

[33] V. P. Dwivedi, C. K. Joshi, A. T. Luu, T. Laurent, Y. Bengio, and X. Bresson. Benchmarking graph neural networks. Journal of Machine Learning Research, 24(43): 2023.
[34] L. Fan, S. Liu, P.-Y. Chen, G. Zhang, and C. Gan. When does contrastive learning preserve adversarial robustness from pretraining to finetuning? Advances in neural information processing systems,
2021.
[35] P. Germain, F. Bach, A. Lacoste, and S. Lacoste-Julien. Pac-bayesian theory meets bayesian inference. Advances in Neural Information Processing Systems, 29, 2016.
[36] F. Guo, R. He, J. Dang, and J. Wang. Working memory-driven neural networks with a novel knowledge enhancement paradigm for implicit discourse relation recognition. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 7822-7829, 2020.
[37] K. Han, Y. Wang. J. Guo, Y. Tang, and E. Wu. Vision gnn: An image is worth graph of nodes. Advances in neural information processing systems, 35:8291-883 1-303, 2022.
[38] L. Han, X. Jiaxing, N. Jinjie, and K. Yiping. Rethinking the message passing for graph- level classification tasks a category-based view. Engineering Applications of Artificial Intelligence, 143:109897, 2025.
[39] T. Han, W.-W. Tu, and Y.-F. Li. Explanation consistency training: Facilitating consisteney- based supervipe learning with interpretability. In AAAI Conference on Artificial Intelli- gence, 2021.
[40] H. He, A. Kosasih, X. Yu, J. Zhang, S. Song, W. Hardjawana, and K. B. Letaief. Gnn- enhanced approximate message passing for massivefulta--masiiee mimo detection. In 2023 IEEE Wireless Communications and Networking Conference (WCNC), pages 1-6. IEEE, 2023.
[41] K. He, X. Chen, S. Xie, Li, P. Dollár, and R. Girshick. Masked autoencoders are scalable vision learners. In Proceedings of the IEEE/CVE conference on computer vision and pattern recognition, pages 16000-16009, 2022.
[42] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770- 778, 2016.
[43] T. He, Y. Zhang, K. Ren, M. Liu, C. Wang, W. Zhang, Y. Yang, and D. Li. Reinforcement learning with automated auxiliary loss search. Advances in neural information processing systems, 11118800-8344 2022.
[44] J. R. Hershey and P. A. Olsen. Approximating the kullback leibler divergence between gaussian mixture models. In 2007 IEEE International Conference on Acoustics, Speech and Signal Processing-/CASSPP007 volume 4, pages VV337. IEEE, 2007.
[45] C.-H. Ho and N. Nvasconcelos. Contrastive learning with adversarial examples. Advances in Neural Information Processing Systems, 33:17081-17099 2020.
[46] A. G. Howard. Mobilenets: Efficient convolutional neural networks for mobile vision applica- tions. arXiv preprint I 2017.
[47] J. Hu, L. Shen, and G. Sun. Squeeze-and -xxeiiitatin networks. In Proceedings of the IEEE conference computer vision and pattern recognition, pages 7132-7141, 2018.
[48] G. Huang, Z. Liu, L. Van Der Maaten, and K. Q. Weinberger. Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7700-4708 2017.
[49] G. Huang, Y. Sun, Z. Liu, D. Sedra, and K. Q. Weinberger. Deep networks with stochastic depth. In Computer Iision-ECCT 2016: 14th European Conference, Amsterdam, The Netherlands, October 11-14, 2016, Proceedings, Part IV 14, pages 646-661. Springer, 2016.
[50] Z. Huang, Y. Zou, B. Kumar, and D. Huang. Comprehensive attention self-distillation for weakly-supervised object detection. Advances in neural information processing systems, 33:16797-16807, 2020.
13

---

[51] F. N. landola, S. Han, M. W. Moskewicz, K. Ashraf, J. Dally, and K. Keutzer. Squeezenet: Alexnet-level accuracy with 50x fewer parameters and< 0.5 mb model size. arXiv preprint 2016.
[52] A. A. Ismail, H. C. Bravo, and S. Feizi. Improving deep learning interpretability by saliency guided training. In Neural Information Processing Systems, 2021.
[53] D. Jiang, Z. Wu, C.-Y. Hsieh, G. Chen, B. Liao, Z. Wang, C. Shen, D. Cao, J. Wu, and T. Hou. Could graph neural networks learn better molecular representation for drug discovery? comparison study of descriptor-based and graph-based models. Journal of cheminformaties, 13:1-23, 2021.
[54] W. Jiang. Graph-basee based deep learning for communication networks: A survey. Computer Communications, 185:40-555 54, 2022.
[55] M. Jin, V. Khattar, H. Kaushik, B. Sel, and R. Jia. On solution functions of optimization: Universal approximation and covering number bounds. In Proceedings of the AAAI Conference Artificial Intelligence, volume 37, pages 8123-8131, 2023.
[56] J. Kang, R. Fermandez-Beltran D. Hong, J. Chanussot, and A. Plaza. Graph relation network: Modeling relations between scenes for multilabel remote-sensing image classification and retrieval. IEEE Transactions Geoscience and Remote Sensing, 59(5):3355 2020.
[57] I. Kansizoglou, L. Bampis, and A. Gasteratos. Deep feature space: A geometrical perspective. IEEE Transactions on Pattern Analysis and Machine Intelligence, (4(10))6823-68388 2021.
[58] A. Kar. S. K. Dhara, D. Sen, and P. K. Biswas. Self-supervision via controlled transformation and unpaired self-conditioning for low-light image enhancement. IEEE Transactions on Instrumentation and Meusurement, 2024.
[59] M. A. Khamis, H. Q. Ngo, X. Nguyen, D. Olteanu, and M. Schleich. Learning models over relational data using sparse tensors and functional dependencies. ACM Transactions on Database Systems (TODS), 45:1 -66, 2017.
[60] P. Khosla, P. Teterwak, C. Wang, A. Sarna, Y. Tian, P. Isola, A. Maschinot, C. Liu, and D. Krishnan. Supervised contrastive learning. Advances in neural information processing systems, 33:18661-18888 2020.
[61] T. N. Kipf and M. Welling. Semi-supervised classification with graph convolutional networks. arXiv preprint wrXiv:1609.02903 2016.
[62] A. Krizhevsky, G. Hinton, et al. Learning multiple layers of features from tiny images. 2009.
[63] Y. Le and X. Yang. Tiny imagenet visual recognition challenge. CS 231N. 7(7):3, 2015.
[64] B. Li, Y. Li, and K. W. Eliceiri. Dual-stream multiple instance learning network for whole slide image classification with self-supervised contrastive learning. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 14318-14328 14328, 2021.
[65] P. X. Yu, H. Peng, Y. Xian, Wang, Sun, J. Zhang, and P. S. Yu. Relational prompt-based pre-trained language models for social event detection. ACM Transactions on Information Systems, 43(1):1-43, 2024.
[66] S. Li,, X. Xia, S. Ge, and T. Liu. Selective-supervised contrastive learning with noisy labels. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 316-325, 2022.
[67] P. Liu, W. Yuan, J. Fu, Z. Jiang, H. Hayashi, and G. Neubig. Pre-train, prompt, and predict: A systematic survey of prompting methods in natural language processing. ACM computing surveys, 55(9):1-35, 2023.
[68] Q. Liu and S. Mukhopadhyay. Unsupervised learning using pretrained cnn and associative memory bank. 2018 International Joint Conference on Neural Networks (IJCNN), pages 01-08. IEEE, 2018
14

---

[69] X. Liu, F. Zhang, Z. Hou, L. Mian, Z. Wang, J. Zhang, and J. Tang. Self-supervised learning: Generative or contrastive. IEEE transactions on knowledge and data engineering. 35(1):857- 876, 2021.
[70] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, S. Lin, and B. Guo. Swin transformer: Hierarchical vision transformer using shifted windows. In Proceedings of the IEEE/CVF international conference on computer vision, pages 10012-10022, 2021.
[71] Z. Lu, Y. Sun, Z. Yang, Q. Zhou, and H. Lin. The orthogonality of weight vectors: The key characteristics of normalization and residual connections. In International Joint Conference Artificial Intelligence, 2024.
[72] R. Luo, Y. Wang, and Y. Wang. Rethinking the effect of data augmentation in adversarial contrastive learning. arXiv preprint arXiv:2303.0/289, 2023.
[73] Z. Luo, Cai, Y. Wang, and B. C. Ooi. Regularized pairwise relationship based analyties for structured data. Proceedings of the ACM on Management of Data, 1:1 27, 2023.
[74] J. Ma, N. Wang, and B. Xiao. Semi-supervised classification with graph structure similarity and extended label propagation. IEEE Access, 7:58010-58022, 2019.
[75] M. Mansourian, A. Jalali, R. Ahmadi, and S. Kasaei. cvention-guided feature distillation for semantic segmentation. arXiv preprint 2rXv::240..05551, 2024.
[76] K. Marino, R. Salakhutdinov, and A. Gupta.
The more you know: Using knowledge graphs for image classification. arXiv preprint arXiv: /el::08444 2016.
[77] S. Mehta and M. Rastegari. Mobilevit: lighh, general-purposse and mobile-friendee friendly vision transformer. arXiv preprint arXiv:21 2021.
[78] R. Moradi, R. Berangi, and B. Minaei. A survey of regularization strategies for deep models. Artificial Intelligence Review, 53()):394773988 2020.
[79] G. Nikolentzos, M. Thomas, A. R. Rivera. and M. Vazirgiannis. Image classification using graph-based representations and graph neural networks. In Complex Networks & Their Applications IX: Volume 2, Proceedings of the Ninth International Conference on Complex Networks and Their Applications COMPLEX NETWORKS 2020, pages 142-153. Springer, 2021.
[80] J. Pang and G. Cheung. Graph laplacian regularization for image denoising: Analysis in the continuous domain. IEFF Transactions on Image Processing, 26(4):1770-1785, 2017.
[81] V. Patel, V. Chaurasia, R. Mahadeva, and S. P. Patole. Garl-net: graph based adaptive regularized learning deep network for breast cancer classification. IEEE Access, 11:9095- 9112, 2023.
[82] L. Peel. Graph-based semi-supervised learning for relational networks. In Proceedings of the 2017 SIAM international conference on data mining, pages 435-443. SIAM, 2017.
[83] C. Peng, X. Yang, K. Smith, Z. Yu, A. Chen, J. Bian, and Wu. Model tuning or prompt tuning? study of large language models for clinical concept and relation extraction. Journal of biomedical informatics, 104:00, 2024.
[84] F. Pernici, M. Bruni, C. Baecchi, and A. Bimbo. Maximally compact and separated features with regular polytope networks. ArXiv, abs/2301.0611 16, 2023.
[85] P. E. Pope, S. Kolouri, M. Rostami, C. E. Martin, and H. Hoffmann. Explainability methods for graph convolutional neural networks. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pages 10772-10781, 2019.
[86] C. Qi and F. Su. Contrastive-center loss for deep neural networks. In 2017 IEEE international conference on image processing (ICIP), pages 285 2851-2 IEEE, 2017.
[87] M. Rath and A. P. Condurache. Boosting deep neural networks with geometrical prior knowledge: A survey, Artificial Intelligence Review, 57(4):95, 2024.
15

---

[88] W. Rawat and Z. Wang. Deep convolutional neural networks for image classification: A comprehensive review. Neural computation, 29(9):2352-244) 2242244922224 2017.
[89] E. Rossi, H. Kenlay, M. Gorinova, B. P. Chamberlain, X. Dong, and M. M. Bronstein. On the unreasonable effectiveness of feature propagation in learning graphs with missing node features. Learning on graphs conference. pages 11-1. PMLR, 2022.
[90] U. Ruby, V. Yendapalli, et al. Binary cross entropy with deep learning technique for image classification. Int. J. Adv. Trends Comput. Sci. Eng, 9(10), 2020.
[91] N. Saunshi, O. Plevrakis, S. Arora, M. Khodak, and H. Khandeparkar. A theoretical analysis of contrastive unsupervised representation learning. In International Conference on Machine Learning, pages 5628-5637. PMLR, 2019.
[92] M. Schlichtkrull, T. N. Kipf, P. Bloem, R. Van Den Berg, I. Titov, and M. Welling. Modeling relational data with graph convolutional networks. In The semantic web: 15th international conference, ESWC 2018, Heraklion, Crete, Greece, June 3-7, 2018, proceedings 15, pages 593-607. Springer, 2018.
[93] I. Sheth and S. E. Kahou, Auxiliary losses for learning generalizable concept-based models. In Proceedings of the 37th International Conference on Neural Information Processing Systems, NIPS '23, Red Hook, NY, USA, 2023. Curran Associates Inc.
[94] Z. Shi, H. Wang, and C.-S. Leung. Constrained center loss for convolutional neural networks. IEEE transactions on neural networks and learning systems, 34(2):1080-1088, 2021.
[95] Si, W. Yu, Zhou, Zhou, X. Wang, and S. Yan. Inception transformer. Advances in Neural Information Processing Systems, 35:234955 2022.
[96] Z. Song, X. Yang, Z. Xu, and I. King. Graph-based semi-supervised learning: A comprehensive review. IEEE Transactions on Neural Networks and Learning Systems, 34(11):8174-8194, 2022.
[97] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and Rabinovich. Going deeper with convolutions. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1-9, 2015.
[98] M. Tanveer, H.-K. Tan, H.-F. Ng, M. K. Leung, and J. H. Chuah. Regularization of deep neural network with batch contrastive loss. IEEE Access, 9:124409-124418 2021.
[99] Y. Tian, C. Sun, B. Poole, D. Krishnan, C. Schmid, and P. Isola. What makes for good views for contrastive learning? Advances in neural information processing systems, 33:6827-6839, 2020.
[100] H. Touvron, M. Cord, M. Douze, Massa, A. Sablayrolles, and H. Jégou. Training data- efficient image transformers & distillation through attention. In International conference on machine learning, pages 10347-10357. PMLR, 2021.
[101] V. Trivedy and I., J. Latecki. Con2graph: Building graphs for image classification. In Proceedings of the JEEE/CVF Winter Conference on Applications of Computer Vision, pages 1-11, 2023.
[102] P. Veličkovié, G. Cucurull, A. Casanova, A. Romero, P. Lio, and Y. Bengio. Graph attention networks. arXiv preprint arXiv: 10903, 2017.
[103] F. Wang and H. Liu. Understanding the behaviour contrastive loss. In Proceedings of the IEEE/CVF conference computer vision and pattern recognition, pages 2495 2504, 2021.
[104] J. Wang. Y. Chen, R. Chakraborty, and S. X. Yu. Orthogonal convolutional neural networks. 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 11502-11512, 2019.
[105] P. Wang, K. Han, X.S. Wei, I. Zhang, and L. Wang. Contrastive learning based hybrid networks for long-tailed image classification. In Proceedings of the IEFF/CVF conference on computer vision and pattern recognition, pages 943-952, 2021.
16

---

[106] W. Wang. Y. Yang, X. Wang, W. Wang, and J. Li. Development of convolutional neural network and its application in image classification: survey. Optical Engineering. 58(4):040901- 040901, 2019.
[107] X. Wang and G.-J. Qi. Contrastive learning with stronger augmentations. IEEE transactions on pattern analysis and machine intelligence, 45(5):5549 -56655669955,,, 2022.
[108] Y. Wang, Y. Deng, Y. Zheng, P. Chattopadhyay, and L. Wang. Vision transformers for image classification: A comparative survey. Téchnologies, 13(1):32, 2025.
[109] Y. Wang, Y. Meng, Y. Li, S. Chen, Z. Fu, and H. Xue. Semi-supervised manifold regularization with adaptive graph construction. Pattern Recognition Letters, 98:90-95, 2017.
[110] Z. Wen and Y. Li. Toward understanding the feature learning process of self-supervised contrastive learning. In International Conference on Machine Learning, pages 11112-11122. PMLR, 2021.
[111] M. Wu, J. Zhou, Y. Peng. S. Wang, and Y. Zhang. Deep learning for image classification: a review. In International Conference on Medical Imaging and Computer-Aided Diagnosis, pages 352-362. Springer, 2023.
[112] S. Xie, R. Girshick, Dollár, Z. Tu, and K. He. Aggregated residual transformations for deep neural networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 1492-1500, 2017.
[113] T. Xie, B. Wang, and C.-C.. J. Kuo. Graphhop: An enhanced label propagation method for node classification. JEEE Transactions on Neural Networks and Learning Systems, 34(11) 1))987777- 9301, 2022.
[114] Xie, Y. Liang, M. Gong, A. K. Qin, Y.-S. Ong, and T. He. Semisupervised graph neural networks for graph classification. IEEE Transactions on Cybernetics, 53(10))6222-6253 2022.
[115] Y. Xie, S. Lv, Y. Qian, C. Wen, and J. Liang. Active and semi-supervised graph neural networks for graph classification. IEEE Transactions on Big Data, 8(4):920-932, 2022.
[116] L. Xu, J. Lian, W. X. Zhao, M. Gong, L. Shou, D. Jiang, X. Xie, and J.-R. Wen. Negative sampling for contrastive representation learning: A review. arXiv preprint arXiv:2206.00212. 2022.
[117] S. S. Yadav and S. M. Jadhav. Deep convolutional neural network based medical image classification for disease diagnosis. Journal of Big data, 6(1):1-18, 2019.
[118] J. Yin, H. Wu, and S. Sun. Effective sample pairs based contrastive learning for clustering. 2023.
Information Fusion,
[119] B. Yu and D. Tao. Deep metric learning with tuplet margin loss. 2019 /EEE/CVF International Conference on Computer Vision (ICCV), pages 6489-6498, 2019.
[120] Yuan, S. Guo, 7. Liu, A. Zhou, F. Yu, and Wu. Incorporating convolution designs into visual transformers. In Proceedings of the IEEF/CVF international conference on computer vision, pages 579-588, 2021.
[121] D. Zhang, M. Cui, Y. Yang, P. Yang, C. Xie, D. Liu, B. Yu, and Chen. Knowledge graph-based image classification refinement. IEEE Access, 7:57678-57699 2019.
[122] L. Zhang, X. Chen, J. Zhang, R. Dong, and K. Ma. Contrastive deep supervision. In European Conference on Computer Vision, pages 1-19. Springer, 2022.
[123] T. Zhang. Covering number bounds of certain regularized linear function classes. Journal of Machine Learning Research, 22M55) 2002.
[124] X. Zhang, X. Zhou, M. Lin, and J. Sun. Shufflenet: An extremely efficient convolutional neural network for mobile devices. Proceedings of the IEEE conference on computer vision and pattern recognition, pages 66556, 2018
17

---

[125] Y. Zhang, Q. Han, L. Wang, K. Cheng, B. Wang, and K. Zhan. Graph-weighted contrastive learning for semi-supervised hyperspectral image classification. Journal of Electronic Imaging. 34(2):023044-002300
2025.
[126] Y. Zhang, P. Tiño, A. Leonardis, and K. Tang. A survey neural network interpretability. IEEE Transactions on Emerging Topics in Computational Intelligence, 5:726-742, 2020.
[127] Y. Zhao, H. Wan, J. Gao, and Y. Lin. Improving relation classification by entity pair graph. In Asian conference machine learning, pages 156-1171. PMLR, 2019.
[128] Z. Zhou, J. Liang, Y. Song, Yu, H. Wang, W. Zhang, Y. Yu, and Z. Zhang. Lipschitz generative ad adversarial nets. In K. Chaudhuri and R. Salakhutdinov, editors, Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pages
PMLR, 09 15 Jun 2019.
[129] Q. Zhu, B. Du, B. Turkbey, P. L. Choyke, and P. Yan. Deeply- cnn for prostate segmentation. 2017 international joint conference on neural networks (IJCNN), pages 178-184. IEEE, 2017.
[130] X. Zhu, Z. Ghahramani, and J. D. Lafferty. Semi-supervised learning using gaussian fields and harmonic functions. In Proceedings of the 20th International conference on Machine learning (ICML-03), pages 912-919, 2003.
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A Relation to Existing Paradigms
Below, we situate GCR within these paradigms, highlighting its unique contributions and distinctions.
Graph-based learning and relational alignment. Graph-based methods are widely used to model relational structures in classification [56, 92, 127] and semi-supervised learning [96, 11, 82], often through pre-constructed graphs [109, 125] or memory banks [68]. Traditional approaches use graph Laplacian regularization [80, 4] or GNN-based message passing [13, 40] encourage feature alignment. While powerful, these methods require explicit graph definitions or heavy architectural
components like GNNs and GATs [61, 102].
GCR diverges fundamentally from these strategies:
On-the-fly graph construction. Instead relying on static or memory-driven graphs [36, 56, 76], GCR builds internal graphs dynamically during training, using model-generater predictions and feature embeddings.
ii Regularization, not transformation. Unlike GNNs that propagate features [89, 10, 85), GCR purely regularizes the relational structure of features to enforce alignment with class-aware prediction similarity.
iii. Parameter-free alignment. GCR operates without additional parameters for graph construction, using only the inherent feature and prediction relationships for alignment.
This design enables GCR to maintain the benefits of relational modeling while being -ggnotic and computationally efficient, extending graph-base based learning into deeper, more flexible architec- tures. Furthermore, GCR enhances generalization by implicitly smoothing decision boundaries through its consistency-drivs alignment, even in high-dimensional feature spaces.
Contrastive learning and semantic consistency. Contrastive methods [60, 118, 45] pull together positive pairs and push apart negatives, often using data augmentation [99, 107] and hard negative mining for improved class separation [14, 60]. However, these methods are sensitive to sampling strategies [116, 19] and require margin tuning for stability [34].
GCR introduces a contrastive mechanism that is global and implicit:
i. Global relational alignment. Rather than pairwise contrasts [17], GCR aligns the entire feature graph and prediction graph, capturing holistic relationships across the batch.
ii Self-supervised by prediction. The semantic graph is constructed directly from model predic- tions, obviating the need for manual sampling augmentation strategies [72, 19].
ii.. Semantic geometric regularization. GCR enforces global coherence between semantic and geometric structures, extending beyond local distances to full-batch relational consistency.
This implicit contrastive mechanism ensures stable alignment of semantic structures without the pitfalls of traditional contrastive learning, contributing to stronger robustness against noisy samples and domain shifts.
Structural regularization and manifold learning. Traditional regularizers like center loss [86, 84] and triplet loss [24, 19] enforce feature compactness or inter-class margins, while graph Laplacian regularization [80, 4] smooths label propagation over fixed data graphs [7, 130]. However, these approuches [78, 98, 94] are limited by predefined graphs or local constraints.
GCR generalizes structural regularization with three core innovations:
i. Dynamic graph construction. GCR constructs feature graphs and prediction graphs on-the-fi fly during each training iteration, adapting the model's 's evolving internal representations.
ii Cross-space alignment. Unlike classical manifold regularization, which only smooths feature relationships, GCR aligns semantic (prediction- -based) and geometric (feature-based) graphs, ensuring that class-consistent features are also prediction-consisten..
iii. Masked supervision. GCR enforces class-aware masking during alignment, preventing smooth- ing across semantic boundaries and refining intra-class structure.
This structural regularization not only preserves semantic coherence but also improves robust generalization by dynamically capturing the evolving feature manifold of the network.
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Self-conditioning and geometric representation learning, Self-conditioning mechanisms [58] like prompting typically rely on external tokens model outputs to influence learning trajectories [67, 21].
GCR extends this paradigm internally by using the model's 's own prediction graph as a sclf self-generated scaffold:
i. Relational prompting. Unlike discrete prompts [12, 83, 65], GCR uses prediction-inducet
graphs to continuously adjust feature relationships according to class structure.
ii Semantic geometry emergence. Geometric deep learning traditionally imposes structural priors [87]; GCR allows semantic geometry to emerge organically through predictive align- ment [8. 33].
iii. Continuous, implicit supervision. This prediction based regularization evolves with the model, providing ongoing, context-aware structure during training.
GCR thus bridges self-conde ooffiinniii with geometric learning, establishing cemantic-awar represen- tation that dynamically evolves with the model's understanding of its own predictions.
GCR's 's design is rooted in manifold alignment theory and smoothness regularization: by enforcing consistency between feature similarity and prediction similarity. effectively smooths decision boundaries in feature space, enhancing generalization across unseen domains. This principle is akin to cluster assumption in semi-supervised learning, where similar samples are encouraged to share the same label.
B Graph Consistency as Self-Prompting
We interpret GCR as a form of internal prompting, where the model dynamically generates its own prompting signal from its predictions to guide the learning of intermediate features. This contrasts with traditional prompting approaches, which typically rely on external tokens or instructions to influence the model's 's behavior.
Key characteristics of GCR self- teff-rrmptiig include:
i. Internal. The prompting signal originates entirely from the model's own predictions, eliminat- ing the need for external input.
ii Structural. The alignment operates over pairwise relationships within the feature space, focusing on how different features relate to one another rather than conditioning on individual samples.
iii. Semantic. The graph structure is inherently class-aware, encouraging the network adjust its feature representations to align with meaningful semantic boundaries between classes.
In this framework, the prediction graph acts learned, self- ceff-supervised attention template that re- cursively guides the refinement of feature representations earlier layers. This novel approach paves the way for self-con ooff-cndiiioning in neural networks, where the model's 's own predictions continuously inform and improve internal feature learning process.
C
Motivation
Deep neural networks have excelled at learning complex, hicrarchical feature representations. Typ- ically, early layers capture low-level visual cues such as edges, textures, and basic shapes, which are often sensitive to noise and not specific to the task. In contrast, deeper layers develop high-level features that reflect more semantic, task-relevant information. However, despite achieving clear class separation in the final prediction space, intermediate feature spaces can still exhibit significant inter-class overlap. This overlap weakens both the generalization ability and the discriminative power of the model.
Furthermore, during training, the parameter search space remains vast because supervision is driven only by class labels, without using any explicit relational structure among samples. This lack of structure-aware guidance limits the model's ability to organize its feature space meaningfully.
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We propose that the model's own predictions contain rich semantic information that can serve as self-supervisory signal. Specifically, the pairwise similarities among prediction logits encode high- level semantic topology that reflects class affinities. By aligning intermediate feature representations with this prediction-derive structure, we encourage the network to learn representations that are geometrically consistent with semantic class boundaries.
Our approach is guided by the following insights:
Reduce noisy inter-class affinities. By aligning feature graphs with prediction graphs, we minimize the risk of inter-class features being too similar, thereby reducing class overlap in the feature space.
ii. Enhance intra-class cohesion. Encouraging feature consistency within each class ensures that the network's s representations exhibit stronger intra-class similarity.
iii. Align learning dynamics with semantic intent. Through the introduction of prdictioonguidded alignment, direct the network's 's learning trajectory to reflect global semantic structures, improving generalization and interpretability.
Thus, we treat the prediction graph an implicit structural prompr that guides the network's learning. This contrasts with traditional methods, where supervision typically focuses on final output layers. Our approach integrates structural coherence directly into the intermediate layers of the network, creating rich, semantic-aware feature representation pipeline.
D Theoretical Insights
We analyze GCR from a theoretical perspective, grounding it in the manifold hypothesis, spectral graph theory, and statistical learning theory.
D.1 Manifold Smoothness and Semantic Cluster Regularization
(Data manifold hypothesis). Let X C Rdd be the input space. The data manifold hypothesis posits that real-world data points {z; p lie on or near a smooth, compact m-dimensional Riemannian manifold M C Rd, where & d.
Definition
Definition 2 (Feature relational graph). Let (xx) E Re denote the feature representation of sample ia at layer I. The feature similarity matrix F(J) E Raxn as feature relational graph is defined by:
Fl :- ReLU (cos xx7,x(0)) ReLU
||x;
|11x
Definition 3 (Masked prediction relational graph). Let z; be the pre-softmax prediction logits for sample Fs and define softmax(z:). The semantic similarity between predictions is computed as:
Sy ReLU (cos (s),
Let Myj := 1= yy} be binary mask indicating whether two samples belong to the same class. The masked prediction relational graph is then:
po: Mij O Sij.
Proposition 3 (Manifold smoothness regularization). Minimizing the alignment loss
| P
enforces local smoothness on the feature manifold. Specifically, if Pi > O, then
||x - x 12 2-
(x))
X + 0.
Proof. For any i.j such that P.j > O, the squared Frobenius loss penalizes the discrepancy between
F()) and P.y. Since (!)) ij is a monotonic decreasing function of ||x!") (2) minimizing this
loss implies that x(() and x!! must close in feature space (y: = yj). Therefore, GCR promotes smoothness by aligning local geometry with semantic similarity.
28

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Definition (Cluster assumption). The cluster assumption asserts that samples from the same class
form tight, compuct clusters in representation space. That is, for yi Uj. we expect ||x; x) ||2 to be small, while for Y: # Yj. the distance should be large. Proposition 4 (Semantic cluster regularization), Minimizing the GCR loss:
LGCR Σ
promotes intra-class compactness and inter-class separation in the feature space, thereby aligning learned features with semantic class structure.
Proof. When Vi yj, the mask M, = 1, so Piy is proportional to the prediction similarity Syy- large Pij enforces a corresponding increase in F!!) which implies smaller angle and distance between x, and (") Conversely, for ys + μ₁ Pij = O, and there is no incentive to keep x(") and (1)
x; close. This results greater inter-class separation and aligns the learned representation with class structure.
D.2
Connection to Graph Laplacian Regularization
From cosine similarity to Graph Laplacians. GCR aligns feature similarity graphs F(") with semantic prediction graphs P at each selected layer l. Both graphs are constructed using ReLU- activated cosine similarity, forming symmetrie, non-negative affinity matrices:
F()) =ReI.U(cos(x)) ij
)),
Ps -M, O (eLU(cossS., s,));
where s; are softmax-normaliin logits. Although these graphs do not use an RBF kernel, they still induce graph structure with meaningful edge weights and comparable degree distributions across layers. Since the GCL operates on strictly upper-triangular entries, the resulting Laplacians remain symmetric and well-defined
Definition 5 (Graph Laplacian). Given an affinity matrix A E R (e.g. F()) or P), the unnormal- ized graph Laplacian is defined as:
= D D-
where D = A
Semantic alignment via spectral consistency. While the classical result by Belkin and Niyogi [6] connects RBF-based graph Laplacians to the Saplace-Beltrame operator on manifolds, cosine- based affinity still permits an analogous interpretation in terms of structural smoothness. In particular, minimizing alignment loss between (!)) and P induces convergence between their associated Lapla- cians LF and Lp.
Proposition 5 (Spectral regularization via Laplacian alignment). F - PI small, then the spectral properties of LF and Lp closely aligned. Minimizing GCR encourages
(9) L)x()) ze TT(
LTxx))
thus regularizing features features to follow both local geometric and global semantic structure.
Sketch. For symmetric matrices with matching degrees, F(O P Lp Re Lp. The quadratic form Tr(x Lx) measures smoothness over the graph. Alignment thus enforces semantic-aware smoothness.
D.3 Generalization Bound under Structural Alignment
now provide a detailed theoretical analysis how GCR contributes to improved generalization. Our goal is to show that by enforcing alignment between the feature graph and the semantic prediction graph, GCR effectively restricts the function class to smoother, semantically consistent representations, leading to reduced Rademacher complexity.
29

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Definition 6 (Structural alignment loss). Let f()) X + Rd be the feature mapping at layer We define the GCR structural loss layer as:
F Pi))
GCR = π
Definition (Structurally constrained function class). Let FL be the class of functions f() such that (x) S B for all x X and all l. Then the ECRRconstrained function class is:
Fe Fe: F
LOCR S {E
This class enforces that features not only have bounded norm but also align structurally with the prediction graph P in the sense of Frobenius proximity.
Theorem 3 (Generalization bound under GCR). Let {(f()), v) be a "-Liiscchitz loss function (e.g.. cross-entropy), and suppose f E Fe. Then with probability least 6, the generalization error is bounded as:
(((T))) 4y O (
log(1/6)
B( v) <
Proof. Let (F.) denote the empirical Rademacher complexity of Fe:
RR (F.) = Eg sup fEF, n Σo Eostaa)
where o; are i.i.d. Rademacher random variables taking values #1 with equal probability. We apply the contraction lemma, which states that if is y- y-Lipschitz, then:
Rr Fe) S Rr (F...
Now we aim to bound R., (Fe). Since each function f Fr maps o; + x E IRd with [ix S B and whose pairwise ReLU-cosine similarities constrained align with Pij. the variability of outputs is tightly controlled. Specifically, we define:
F (1) ReI.U((X())) ()) ())),, and
Σ p.j) n&e. 1,J
Note that for normalized vectors, (x) x (1) cos Ozj. and the ReLU ensures non-negativity. The loss penalizes angles between feature vectors that deviate from their semantically guided prediction- based affinity.
Let us now relate this bound on the Rademacher complexity. We use the following result adapted from Bartlett and Mendelson (2002): for any bounded function f(z) E Rd with [[f(2))lz I B, the Rademacher complexity bounded by:
B E 71
B Vn
Ri (FL)
Σo;
o
However, for Fe, we have a stronger constraint: the features cannot vary arbitrarily due to the structural alignment requirement. In particular, for small all x are geometrically organized to maintain high similarity (angle - 0) when Pij is high and to be less constrained otherwise.
Hence, the variance in f(x) is suppressed in directions orthogonal to semantic affinity, shrinking the function class. From this, one can derive:
2B
R (F.) <
where the Ve factor reflects the deviation from perfect structural alignment.
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Substituting into the standard generalization bound yields:
[((((()) S (f(a.)) By +0 ()0(
(v
Remark 3. This bound demonstrutes that GCR effectively reduces the hypothesis complexity by enforcing a semantic structure on the learned representations. As the structural loss e decreases, the model class is increasingly constrained to semantically faithful functions, thereby improving generalization on unseen data.
GCR aligns feature similarity graphs with semantically meaningful prediction graphs, enforeing both geometric and semantic smoothness. Our analysis shows GCR promotes manifold alignment, Laplacian smoothness, semantic clustering, and provably better generalization.
E Proof for Theoretical Analysis of GCR
We provide a theoretical analysis of GCR, connecting its empirical design to foundational concepts in statistical learning theory and spectral graph theory. Specifically, show that minimizing the GCR loss: (i) Reduces the effective capacity the hypothesis class via covering number bounds and Dudley's entropy integral. (ii) Promotes spectral consistency between learned features and semantically meaningful prediction graphs via normalized Laplacians. (iii) Can be interpreted as a PAC-Bayesial regularizer that imposes structural prior function space.
E.1
Generalization via Covering Numbers
Let Fz be the class of functions f(() X Rd representing the layer-/ embeddings. B is constant upper bound on the &g norm of the feature representation f (I)) E Id at layer 1. That is,
(z) VE (z)) S B, for all X.
This constraint standard in learning theory to control the size of the hypothesis space. In practice, especially under L2 normalization used in our method, we often have B 1. We then define a stucturaill-coonstained hypothesis class:
F. = -=
cl == (Irriu(F()) - triu se]-
(16)
This class enforces graph alignment between learned features and the masked prediction graph P. which reflects intra-class similarity.
Theorem 4 (Generalization via Dudley's entropy integral). Let {(f(),,)) be y-Lipschitz loss function (e.g., cross-entropy), and let Fr, the class of functions at layer such that each function f(i) satisfies the by-bounded constraint (t) (z)]l2 S Suppose F. S FL the subset of functions that are additionally constrained by the GCR alignment loss:
(
(x) -Po)
(17)
n
+==1
(xs)
(F. are the normalized feature vectors for each data point 2; in the dataset, and Pij is the target alignment between the feature vectors x; and Xj. The GCR loss enforces that the angular distances between the feature vectors are small, meaning that the vectors are close to each other in the Euclidean space.
where x;
If F. admits a covering number bound:
0 a
N(Fe, ll (12,a) <
(18)
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where [Fee.ll -1 l12, 0t) is the covering number of F. with respect to the Eg norm, then the expected loss of function F. bounded with high probability by:
+ 1 Σ (((()))))))
127 Vn.
V Volog (dees
[(z v)1 S
(19)
where B O(ve) is the effective radius of the function class F. under the GCR constraint, and the second term represents the generalization enror, which controlled by the complexity of the function class.
Proof Sketch. We outline the key steps:
Step 1: Rademacher complexity controls generalization. Let (F.)) denote the empirical Rademacher complexity of the constrained class. Since the loss function & is y-Lipschitz, the composition inequality gives:
(R, o Fc) S Rn (F.)-
Step Dudley's entropy integral. We now bound R. (F.)) using Dudley's s entropy integral:
12 vn
dlim((F))
Hi (F.)
VlogN
12, a)) da,
where (F.) refers to the diameter of the set Fe, Le., the largest possible distance between any two points in Fe in Euclidean space.
Step 3: Diameter bound under GCR. Under the GCR constraint, F (1) ReLU((x,, x,}) is close to Pij. Since both x; and X; are unit-normalized vectors, this implies that the angular distances between them are bounded. Let (x;, xy) constraint restricts pairwise angles to lie within a narrow cone. Therefore, the effective diameter of F, in Euclidean space is:
for pairs where P O. This cosine similarity
|| xyl12 S v2 cos O(Ve).
where / the angle between x, and xy. Thus, we set B = ((v)).
Step Plug in covering number. Using the assumed covering number bound N(a) S (C/a)", have:
log.N(Fe..ll l1,, s dlog(C/aa).
Step Final bound. Substitute into Dudley's integral:
12
Vdlog ( do,
Ri (Fe) S
Vn
and apply the composition inequality yield the desired result.
Remark 4. This result shows that GCR reduces generalization error by shrinking the effective complexity of the function class. By aligning relational structure, GCR implicitly contracts the hypothesis space, leading to improved generalization.
E.2 Spectral Alignment via Normalized Laplacians
Let F and P be symmetric affinity matrices derived from feature embeddings and masked predictions, respectively. Their associated normalized graph Laplacians are defined as:
LF D 1/2 1/2DD 1/2 Lp == I- DP PDD 1/2 -DD1/2
where Dr and Dp the degree matrices corresponding F and P, i.e., (Dr)a= E, Fij and similarly for Dp.. Proposition 6 (Spectral Alignment). Let F and P be symmetric matrices such that
|F-PI|e S, [Dv Dp S.
Then, there exists conslant C> 0 depending spectral properties of the graphs (e.g, sparsity, minimum degree), such that
[[ LplFF < C(E+
(20)
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Proof Sketch. We aim to bound the difference between the normalized Laplacians Le and Lp in Frobenius norm. We begin by expanding the difference:
( 1/2 FD (2) - 10100-111 -I--D, 1/2 "DP pp?p
||Cp CpllF
The identity terms cancel, giving:
[E LPllF
D 1/2 PDD 1/2
1/2p1/2 -DF" DF 1/2 FD 1/2 1/21//1
We add and subtract D
1/2 1/ PD
to decompose the expression:
||Cp- CpllF S |7 2D 1/2
-DF
DD 1/2 1/2
F +
| PD 1/2
-DP
1/2
PD
p
1/22
F
Denote two terms above as (A) and (B), respectively.
Term (A): Difference due to affinity matrices.
Since DF -1/2 is fixed in this term, we can factor it out:
(A) = (F PPDD- 1/2
S ID
1 F 1FPI/.
Let Amiu (Dr) denote the minimum diagonal entry of Dr. Then DD /3 Amin (Dr) and
assuming Amin (Dr) 2 dmin > 0, we obtain:
(A) S
dmin
Term (B): Difference due to degree normalization.
We now bound the difference caused by changing from DF to Dp in the normalization. Define
g(D) == D DD 1//2 1/2, Using matrix perturbation theory (see e.g., Kato's inequality or Fréchet derivatives matrix functions):
DP''"'" 1/2 D VD(Dr) DD Dr),
and assuming the matrices close and well-conditioned, can approximate:
(B) [Vg - Dr]lf SC' D Dr = C'8, where C" depends on the norm of P and the conditioning of Dr. Combining (A) and (B), we obtain:
Cr CrllF S dmin
-
which can be rewritten as:
[[ - LpllF S Cle + +8,
where C' is constant depending d=1 |P||| and graph sparsity.
Corollary 2. The GCR alignment loss, which encourages | MP-Plp PlI < indirectly enforces spectral similarity of the normalized Laplacians. This promotes agreement between the clustering structure and diffusion properties of the learned features and masked predictions.
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E.3
PAC-Bayesian View of Structural Regularization
We now present PAC- PAC-Bayesian interpretation of GCR. The PAC-Bayes framework provides probabilistic approach to generalization by relating the expected loss of stochastic predictor to its empirical loss and the divergence between a posterior and a prior distribution over the hypothesis class.
Let P denote prior distribution model functions f. representing structure-agnostie belief (e.g., uniform or isotropic Gaussian over parameters). Let Q be a posterior distribution supported on models that minimize training loss while also conforming to a structural constraint induced by GCR, i.., Q is restricted to functions f such that cll) S e for each relevant layer I.
Theorem 5 (PAC-Bayes Generalization Bound with GCR), Let L(f) = ((x.))=D((((()) (x) v)l be the expected population loss of model f and let = ( (2=), ye) be the empirical loss on n training examples. Then, for any posterior distribution Q over functions and any prior distribution
P, with probability at least 1 over the training data, the following bound holds:
By~ol < E KL(Q))) + log(1/6)
(21)
2m
This classical PAC-Bayesian bound quantifies generalization via key components:
The empirical performance of models sampled from the posterior Q.
ii The Kullback-Leibler (KL) divergence KL( PP) between the posterior and the prior, which acts measure of how far Q deviates from the prior belief.
In the context of GCR, interpret the constraint (t) LGCR c imposing structure the feature space. Specifically, GCR encourages the pairwise feature similarity matrix F()) at each layer to align with the semantic structure encoded in P (e.g., class-level affinity). This alignment can be viewed as an inductive bias or structural preference.
Assuming Q is supported only on models satisfying c(0) S c, we argue that the complexity term ( is influenced by the degree of this alignment.
Proposition 7 "Structure-Induced KL Complexity). If the posterior Q is concentrated models with small GCR loss at layer I, then the KL divergence to an isotropic prior P satisfies:
kL(O))P) S C
Σ ||(() Pll
(22)
for some constant depending on the form P..
Sketch of argument. Let us assume that P is a dtructure-agnoste prior, e.g., an isotropic Gaussian over parameters or functions. Now suppose Q is supported on models where the GCR loss is small. Since (!)) penalizes deviation between the normalized feature similarity matrix F()) and the semantic affinity matrix P. this implies that models in the support of induce feature geometries that respect the semantic structure.
From an information -hoooreiin-theoreeiii perspective, concentrating the posterior such structured models induces regularization effect: it reduces the space of allowable hypotheses compared to the unconstrained prior. Intuitively, this compression is captured by the KL divergence. Since the GCR loss explicitly penalizes misalignment, its cumulative value over layers effectively bounds the information-theoreti complexity:
(LI S
Σ ( PI
More formally, this can be justified using PAC-Bayesian compression bounds or Gaussian complexity arguments, which show that the KL divergence scales with the squared norm of the constraint function. in this case, the Frobenius norm between affinity matrices.
This perspective reveals that GCR does more than minimize training loss, it also implicitly regularizes the hypothesis space by favoring models whose internal representations reflect known semantic structure. This improves generalization by reducing the effective size of the model class, as made explicit through the PAC-Bayesiag framework.
Remark 5.
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F Analysis of GCR's Time Complexity
Below, we present a theoretical analysis of the GCR's time complexity per training iteration, from both naive computational perspective and optimized parallel execution view.
Feature graph construction. At each layer = .,..., K where a GCL applied, feature similarity graph F()) E m constructed using the cosine similarity. We have time complexity: (i) Naîve (sequential compute): Normalizing all feature vectors costs (nd)); pairwise cosine similarities require dn). (ii) UPU-parallelized: With sufficient vector-level parallelism, normalization and similarity computations can be reduced to (log d), assuming parallel dot products. (iii) Total over log d. where dmax is the largest feature dimension across all GCL layers.
K layers: O(K
Prediction graph construction. The prediction graph P E IRux is derived from softmax- normalized logits z; R We have time complexity: (i) Naive (sequential compute); Softmax computation costs (nC)), cosine similarities O On) and masking O(n²). (ii) GPU-parallelized: Per-sample operations reduce (log C)), and masking becomes (1) due to element-wise matrix operations.
Graph alignment loss. The loss at each layer measures the Frobenius norm of the difference between graphs: LGRR = trriu (F() - II] We have time complexity: (i) Naïve (sequential compute): Onz)), , (ii) GPU-parallelized: O(logn), assuming reduction over parallel threads for norm computation.
Adaptive weighting across layers. If adaptive weighting (Eq. 6) is used, normalized weights are computed for each layer based on alignment discrepancy. We have time complexity: O(Kn²), where K is the number of GCL-applied layers.
Total time complexity. (i) Naive (sequential compute): Assuming GCLs applied at K layers, with dmux being the maximum feature dimension and C' the number of classes: O (K n² (dmax + C)). with the dominant term On (mxx) due to high-dimensional pairwise feature similarity computations. (ii) UPU-Parallelized: With parallel compute, complexity reduces O (K (log dmgx log C)), where log C < log dmax can be ignored. If n² is too large to fit into memory, the computation can be split into sequential parallel blocks (e.g., 4).
Practical considerations and optimizations. (i) Scalability: GCR operates on batches rather 128) is modest in practice. (ii) Parallel efficiency: All computations are matrix-based and benefit from hardware acceleration. Libraries such PyTorch exploit thread and GPU-level parallelism to accelerate operations like torch. bmm, functionn ional. cosine sesiiinnnnllaieeeeeeeeeeeeeiieee wim.iaariiit..i and torch, triu. (iii) Zero parameter overhead: GCR intro- duces no trainable parameters and does not affect memory footprint or gradient flow.
than entire datasets. Its quadratic cost in n (e.g.,
GCR introduces lightweight yet effective form of structure-based regularization, with per-layer complexity more than (log d). Thanks to batch-local operation, GPU-friendly computations, and absence of learnable parameters, GCR scales efficiently while improving semantic alignment.
G Experimental Setup
We evaluate the effectiveness and efficiency of GCR across diverse range of image classification benchmarks and model architectures. All experiments are conducted on NVIDIA V100 GPUs (32GB) paired with 12 CPU cores and 48GB of system RAM.
For convolutional architectures trained on CIFAR-10 10 and CIFAR- -10, we follow the standard training protocol from [25]. Specifically, models are trained for 200 epochs using stochastic gradient descent with Nesterov momentum of 0.9 and weight decay of 10 10-4, The initial learning rate is set to 0.1 and decayed by factor of 5 at epochs 60, 120, and 160. We use fixed batch size of 128 for all training. The GCR loss is incorporated with a regularization weight of A = unless otherwise stated.
For Masked Autoencoder (MAE) experiments on CIFAR- and CIFAR- 100, we also use the same hardware setup. The ViT-Tiny encoder is pre-trained for 2200 epochs with a 75% masking ratio. Optimization uses Adam W with a base learning rate of 1.5 10-1 (scaled by global batch size), weight decay of 0.05, 200-epoch linear warm-up, and cosine decay. A global batch size of 4096 realized via gradient accumulation with device batch size of 5 512, repeated eight times before each
35

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optimizer step. Training uses automatic mixed precision (AMP) and gradient norm clipping at 1.0, completing pre-training in approximately 13 hours and 40 minutes.
The pretrained ViT-Tiny encoder is then fine-tuned for classification over 200 epochs using A with base learning rate of x 10-3 (scaled by batch size), weight decay 0.05, 10-epoch warm-up, and cosine decay. Fine-tuning uses a batch size of 128 without gradient accumulation or AMP. Fine-tuning times are hours for the baseline and approximately hours 40 minutes with GCR.
We evaluate GCR on eight convolutional neural networks on CIFAR-100, including MobileNet, SqueezeNet, ShuffleNet. ResNet-34, ResNet-50, ResNeXt-50, ResNeXt-101, and DenseNet- -12 For CIFAR-10, we include GoogLeNet and ResNet-101 101, totaling ten models. Incorporating GCR results in modest increase in training time due to graph construction and aligmment overhead. For example, MobileNet and SqueezeNet baseline trainings take approximately 45 minutes each, increasing to 60 and 80 minutes with GCR, respectively. ShuffleNee increases from 140 to 170 minutes, ResNet-34 from 160 to 280 minutes, ResNet-50 and ResNeXt-50 from around 210 to 390 and 230 to 400 minutes, respectively. Larger models such ResNeXt-101 and DenseNet-121 see increases from 420 to 540 minutes and 270 to 330 minutes, GoogLeNet runs 125 minutes baseline and 140 minutes with GCR; ResNet-101 increases from 300 to 370 minutes.
On Tiny ImageNet, we test GCR on four transformer models (VIT, Swin Transformer, Mobile CEiT) and four CNNs (MobileNet, ResNet, SE Re-Reete, Stochastic ResNet). CNNs are trained for 200 epochs with initial learning rate 0.1 (decayed at epochs AdamW optimizer with initial learning rate 1 x 10-1, weight decay 5 x 10-2 and cosine annealing decay to x 10-6, including a 10-epoch warm-up. Transformers are trained for 250 epochs with batch size 256, using AMP for efficiency and gradient clipping with max norm 1.0.
batch size 128, weight
decay 5 10-1, and momentum 0.9. Transformers
Training times increase moderately when applying GCR. MobileNet's training grows from 3 to hours, ResNet-34 from 16 to 19.5 hours, SE-ResNet-18 from 10 hours 40 minutes to 12 hours, and StochasticDepth- 18 from to 9.5 hours. For transformers, VIT-B/16 requires 15 hours baseline and 20 hours with GCR, ViT-B/32 from 6 to 7.5 hours, MobileViT-S from 9 hours 15 minutes to 10.5 hours, Mobile VIT-XS from 8 to 9 hours, MobileViT-XXXS from 5 hours 40 minutes to 6.5 hours, Swin Transformer-Ting from 13 to 16 hours, and CEiT-Tiny from 8 hours 10 minutes to 10 hours.
Overall, GCR introduces consistent yet manageable computational overhead across architectures, primarily due graph construction and alignment. All reported training times are averages over three independent runs with varying random seeds to ensure reproducibility. GCR adds no trainable parameters and designed for parallel execution on modern hardware, maintaining efficient and scalable training.
H Additional Results and Visualizations
H.1 t-SNE Visualizations
We present a comparison of t-SNE visualizations for bascline models and their GCL-augmented counterparts on CIFAR-10 in Fig. 7.
In models enhanced with GCL (Figs. 7b, 7d and 7f, semantically related classes, such as Airplane, Ship, and Truck (all vehicles), form tighter groupings, indicating that GCL promotes a better understanding of high-level semantic concepts. A similar effect is observed among animal classes like Dog, Cat, Horse, and Deer.
These improvements are consistent across diverse architectures, including MobileNet, ResNet-34, and MAE, highlighting the generality and robustness of GCL. Rather than overfitting to specific architecture, GCL contributes to relational feature learning in model-agnostic and parameter-free manner.
H.2 Efffet of Batch Size in GCR Framework
Batch size plays critical role in the effectiveness of GCR, as both the feature and prediction relational graphs are constructed at the batch level. To study this impact, we conduct experiments using the Masked Autoencoder (MAE) model and visualize the prediction graphs across varying
36

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Airplane
Antornolaee
Deer
Dog
Fing
Horse
Ship
Truck
MobileNet
(b) MobileNet+GCLs
(c) ResNet-34
(d) ResNet-34+ 34+GCLs
(e) MAE
(f) MAF-GCLs
Figure 7: t-SNE visualizations of feature representations on CIFAR-10. (Left) Original model archi- tectures; (Right) corresponding GCL-augmented models. Our method consistently enhances feature structure across diverse architectures, including MobileNet, ResNer-34, and MAE (with ViT-Tiny encoder), by yielding tighter intra-class clusters and improved inter-class separation. Notably, GCR distinetly separates semantic groups such as animals and vehicles (e.g., (e) vs. (f)). Importantly, GCL is lightweight and introduces additional parameters.
37

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A
A
(a) 1 16
(b) 22
(c) 64
(d) 7 128
Figure 8: Effect of batch size (n) in the GCR framework. We evaluate using the Masked Autoencoder model. The top row shows the baseline; the bottom row shows the GCL-augmented counterpart. From left right, the relational graphs are constructed on softmax predictions as n increases from 16 to 128. As batch size grows, GCL-augmented models consistently exhibit tighter intra-class clusters and clearer inter-class separation.
batch sizes (n {16, 32, 64, 128, 256, 512)). We compare the baseline and GCL-augmented versions side-by-side, focusing on the structure of the similarity graphs derived from softmax outputs.
Qualitative evaluation. Fig. 8 illustrates these results. In the top row (baseline), increasing batch size introduces more noise and inter-class confusion, especially at smaller n, where limited sample diversity distort the global structure. As batch size increases, the prediction graphs become more complete but remain noisy and less structured, indicating that larger batches alone do not guarantee better semantic organization.
In contrast, the bottom row augmennng))t shows that our method consistently yields more coherent relational graphs across all batch sizes. Even at smaller n, GCR promotes more compact intra-class clusters and better inter-class separation. At larger n, the effect is even more pronounced, the graph-basee based alignment uses the increased pairwise statistics to further regularize feature space according to prediction semantics.
As shown in Fig. 8, GCR remains effective across wide range of batch sizes. Even with small batches (e.g., ., n 16 and n 32), GCL-enhanced models produce more coherent intra-class clusters and stronger inter-class separation than the baseline. This supports our claim that GCR can extract meaningful structure even from limited relational signals.
These observations highlight two important insights: (i) GCR is robust to batch size and can extract meaningful structure even from smaller batches, and (ii) larger batches enhance the graph alignment process by providing richer relational signals, amplifying the benefits of GCR. This makes our method especially suitable for modern hardware and large-scale distributed training setups where large batches are common.
Quantitative evaluation. While larger batches amplify gains by providing denser graphs, they are not essential. Table 5 confirms consistent performance improvements across batch sizes on both CIFAR- (ShuffleNet) and Tiny ImageNet (CeiT). These results affirm GCR's robustness and flexibility, even in resource-constrained or small-batch training setups.
H.3 Measuring Similarity
We chose cosine similarity to emphasize directional alignment, which is more semantically meaning- ful and robust to nuisance factors (e.g., brightness) than raw magnitude. This aligns with common practice in representation learning, where angular relationships often capture class structure more effectively.
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Table 5: Effect of batch size on GCR performance for ShuffleNet (CIFAR-10) and CeiT (Tiny
ImageNet).
ShuffleNet on CIFAR-10
CeiT on Tiny ImageNet
Batch Size
GCR
Baseline
Batch Size
+ GCR
Baseline
16
79.90=10.38
78.88
16
44.84-10.31
43.78=00.35
32
0..9
86.91=0.37
47.55++0.29
0.38
64
91.26=0.255
90.64=0.35
64
49.19=1.25
48.09=0.30
128
92.79=#0.20
91.21=0.28
128
51.22:10.20
49.95=00229
256
92.8
92.07=0.27
256
50.77=0.19
49.62=0.24
512
92.33++0223
91.9411.25
512
50.65:10.22
49.34
Table 6: Performance comparison of different GCL integration strategies with various kernels.
Method
Cosine
RBF
Polynomial
Sigmoid
Laplacian
Baseline
65.95+0.225
0.225
65.95++0.25
05.95++0.25
65.95-+0.25
Early GCL
67.53::.21
66.6610.22
66.59:1.29
66.63+0.30
66.42:+0228
Mid GCL
0.19
67. 0.20
66.97=0.24
07.01+0.36
66.80:10.29
Late GCL
6..3
0.24
67.38:10.29
67.42:10.29
67.21
Early + Mid
67.62.10.23
66.7510.24
66.6810.21
66.72:t0.27
66.51=0.29
Mid + Late
08.26:10.18
67.39=0.23
67.32=0.23
67.36:t0.31
67.1510.277
Early + Late
67.21
0.23
66.27+0028
66.31+0022
6..10+00.26
Full GCL
68.2510.21
6..2
67.31
67.35+0.27
67.14
While kernel methods (e.g., RBF, polynomial) offer expressive similarity functions, our GCLs operate on features already shaped by deep non-linear transformations. Thus, prioritize simplicity and generality: cosine is efficient, lyperparameter-fiee,, and preserves our goal making GCLs a lightweight, plug-and-play regularizer.
We tested multiple kernels on MobileNet with CIFAR-100 and found cosine consistently outperforms others, further supporting our design choice.
H.4
GCR Reduces Inter-Class Noise
We now provide quantitative evidence supporting our claim that GCR reduces inter-class noise, beyond prior visualizations.
Clustering and separability metrics. We use the Silhouette score (higher values indicate tighter intra-class clustering and clearer separation from other classes) and the Separability ratio (measuring inter-class vs. intra-class distance; higher is better).
Results CIFAR-10 10 across ten models show that GCR consistently improves feature separability and cohesion. For example, on ResNet-34, Silhouette(1) increases from 0.60 to 0.73, and SepRatio(1) from 3.10 to 4.41, confirming clearer class boundaries.
Confusion matrices. Confusion matrices indicate reduced inter-class confusion. For example, "cat-dog" confusion decreases from 0.09 to 0.07, and diagonal accuracies improve across several classes (e.g., "auto": 0.96 - 0.97).
H.S GCR on Earlier Layers
Motivation for early-layer regularization. While later layers are more semantic, we find that GCR sometimes works best in earlier layers, especially on Tiny ImageNet and low-capacity models. This effect arises due to several factors. Early features often exhibit higher noise and misalignment, which GCR's adaptive weighting naturally targets. Regularization at these layers helps prune spurious low-level features, setting the network on better optimization path. Moreover, prediction-driven self-prompting allows final-layer structure to refine earlier layers via backpropagated relational
39

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Table 7: Quantitative metrics showing improvements in feature clustering and confidence with GCR.
Silhouette
SepRatio
Confidence
Model
Baseline
GCR
Baseline
GCR
Baseline
GCR
DenseNet-
0.4724
0.5001
2.2278
2.3325
0,9746
0.9805
ShuffleNet
0.2806
0.4083
1.7692
2.0472
0.9568
0.9619
SqueezeNet
-0.1245
-0.0825
1.0008
1.0494
0,9603
0.9660
ResNet-34
0.6032
0.7314
3.1015
4.4144
0.9801
0.9870
ResNet-50
0.5314
0.6186
2.5480
3.2294
0.9789
0.9835
ResNet-101
0.5641
0.6069
2.7705
3.0793
0.9803
0,9859
ResNeXI-50
0.5298
0,5604
2.6323
2.7941
0,9788
0,9814
ResNeXt-101
0.5668
0.6951
2.8387
3.8703
0,9811
0.9880
GoogLeNet
-0,0255
-0.0055
1.1982
1.2065
0.9720
0.9749
Avg
0.3776
0.4481
2.2319
2.6692
0.9737
0.9788
Table 8: CIFAR-10 confusion matrix for the baseline model.
plane
auto
bird
cat
deer
dog
frog
horse
ship
truck
plane
0.93
0.01
0.02
0.01
0.03
0.01
auto
0.01
0.96
0.03
bird
0.02
0.88
0.03
0.02
0.01
0.02
0.01
cat
0.01
0.02
0.83
0.02
0.08
0.01
0.01
deer
0.01
0.01
0.02
0.93
0.01
0.01
0.01
dog
0.01
0.09
0.02
0.86
0.01
frog
0.01
0.02
0,01
0.01
0.95
horse
0.01
0.02
0.01
0.02
0.94
ship
0.02
0.01
0.95
0.01
truck
0.01
0.04
0.01
0.94
Table 9: CIFAR- 10
confusion matrix for the model with GCR.
plane
auto
bird
cat
deer
dog
frog
horse
ship
truck
plane
0.94
0.02
0.01
0.01
0.01
0.97
0.02
bird
0.01
0.89
0.03
0.02
0.02
0.01
0.01
0.01
0.01
0.85
0.01
0.08
0.01
0.01
0.01
deer
0.01
0.01
0.02
0.94
0.01
0.01
0.01
dog
0.01
0.07
0.01
0.88
0.01
frog
0.01
0.02
0.01
0.01
0.95
horse
0.01
0.02
0.01
0.01
0.94
ship 0.02
0.96
0.01
truck
0.01
0.02
0.01
0.01
0.95
Table 10: Impact of early vs. late GCR on feature robustness in ShuffleNet. Pre-freeze and post-freeze 1 accuracy are reported, along with the performance drop.
Model
Pre-freeze Post-freeze
Performance Drop
Model A (Early-GCR 66.8%
66.1%
0.7 %
Model B (Late-GCR)
66.4%
65.1%
1.2 %
40

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Table 11: Performance comparison on CIFAR-IO. CIFAR-100,
100, and ImageNet-IK.
Method
CIFAR-10
CIFAR-100
ImageNet-
ResNet-18
CNN2GNN [101]
9..51::0.42
74.80+10.81
60.12::..02
CNN2GNN GCR
95.87=0.31
76.23
62.47=0.47
CNN2Transformer [101]
95.79:10224
77.39::.20
71.12=0.35
CNN2Transformer GCR
95.96.1..35
78.231.0.00
72.33=0.31
ResNet-33
CNN2GNN [101]
96.39:10.41
77.87
61.0220.77
CNN2GNN GCR
96.67:1..36
14....5
0..88:0046
CNN2Transformer [101]
96.73:10.37
10.10
75.42
CNN2Transformer GCR
96.97=1036
81.27=0.29
76.67=0.26
signals. Shallow models benefit more from early guidance because they downsample aggressively and lack strong inductive biases.
Experiments and analysis. To test whether GCR's impact correlates with a layer's semantic misalignment, we trained CeiT model on Tiny ImageNet without GCR and measured the baseline discrepancy for each block 1: 6(1) = |[F()) PIL... We then applied GCR to individual blocks and recorded the top-1 accuracy gain Alcc(l). Results show that early layers, bridging low-level features to class concepts, exhibit the highest misalignment and largest gains: (i) Block I: 8, = 0.45, gain +1.2%, (ii) Block 2: 62 0.30, gain +0.9%. A strong Pearson correlation of 0.62 between { and Acc(()) quantitatively confirms that GCR is more effective where feature- caature-reeiition misalignment is greater.
Next, we evaluated whether early GCR creates more robust features that benefit later layers, producing "feature cleaning" effect. We trained two ShuffleNet models (5 blocks each) and then froze the regularized blocks to assess their standalone quality: (i) Model A (Early-GCR): GCLs on Blocks 1-2, frozen after 100 epochs, then fine-tuned remaining blocks. (ii) Model B (Late-GCR); GCLs on Blocks 4-5, frozen and fine-tuned similarly. Model A's smaller drop indicates that early GCR features are more robust and semantically coherent, reducing dependence on later-stage regularization (Table 10). In contrast, Model B's larger drop suggests that late-GCR performance relies heavily on continued regularization, with earlier features remaining entangled.
These results show that GCR scales well and confirm our core insight: aligning feature geometry with prediction semantics strengthens generalization.
H.6 Results on CIFAR-10, CIFAR-100, and ImageNet-1K
Distinction from graph-based methods. Although graph-based methods have been extensively studied, our proposed GCR departs from this line of work by introducing fundamentally different mechanism. Existing approaches often depend on static external graphs or rely on iterative message passing as in GNNs. By contrast, GCR uses a novel self- pof-ppromted regularization strategy, where the model's s own predictions dynamically construct a class-aware graph that supervises its intermediate feature representations.
This design brings several innovations. First, instead of relying on pre-defined structures or memory banks, GCR builds graphs on the fly from the model's softmax outputs within each batch, making the supervision inherently adaptive to the evolving state of the model. Second, GCR operates as a purely parameter-free regularizer rather than as a feature transformer. It introduces no additional learnable parameters, remains agnostic architecture, and can be seamlessly integrated into diverse models with minimal overhead. Finally, GCR enforces a unique cross cross-space alignment: similarity graphs in the feature space are aligned with semantic graphs in the prediction space, coupling representation learning with prediction dynamics. In this way, GCR turns predictions into structured supervisory signals that guide feature learning, offering a lightweight yet powerful alternative to conventional graph-based classification methods.
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Table 12:
Results on ImageNet-IK,
Model
ViG-Ti [37]
ViG-Ti + GCR
ViG-S [37]
ViG-S + GCR
ViG-B [37]
ViG-B + GCR
Accuracy
73.9
74,9
80.4
81.7
82.3
84.0
Table 13: Comparison of GCR with one-hot labels vs. soft predictions CIFAR-100 (average 10 runs).
MobileNet
ShuffleNet
SqueezeNet
Baseline
70.11+0.30 65.95++0.25 6..35:+0.24
69.43:+-0.27
GCR (one-hot labels)
01.28:0.26
70.49:+0.25
GCR (soft predictions, ours)
68.2
71.96:10.27
71.03:10.24
We provide direct comparisons between our GCR-augmented models and recent graph-based classifi- cation methods across CIFAR-IO, CIFAR-100, and ImageNet-IK (Table 11 and Table 12).
The results clearly show the consistent and complementary benefits of GCR when integrated into graph-based backbones. They highlight key insights: (i) Consistent across architectures: GCR
improves performance across diverse backbones (CNN2GNN [101]. CNN2Transformer [101], ViG [37]), demonstrating broad applicability. (ii) Complementary to existing methods: Its self- regularization mechanism enhances CNN2GNN and CNN2Transformer indicating orthogonality to existing graph-based pipelines. (iii) Scalable to large-scale tasks: Notable gains on ImageNet-1K (e.g., +1.7% with ViG-B) show GCR's effectiveness in complex, real-world settings. (iv) Lightweight and model-agnostic: GCR introduces extra parameters and integrates easily into existing architectures.
Together, these findings reinforce GCR's s novelty and practical value flexible, self- self-prompted regularization framework applicable across scales and architectures.
H.7 Feature Similarity and the Information Bottleneck
Soft predictions vs. one-hot labels. For pair of features f, and f, (dropping ReLU for brevity), we encourage cos(f,, f., f,) ss co, Sy), where and are soft scores, not one-hot labels. These soft scores allow intra-class variations to persist while emphasizing task-relevant semantics. By imposing this constraint, we create an information bottleneck: the smaller the angle between f, and fy, the stronger the bottleneck. The network is thus forced encode the most important variations while discarding nuisance factors. For example, in the Chihuahua vs. Great Dane scenario, features such as eyes, paws, nose, nails, and teeth are emphasized and encoded consistently across all dog species, whereas leg size or body shape variations are considered less relevant. This ensures that salient features are accurately captured and distinguishable across classes. If x; and xj belong to different classes (e.g., dog vs. wolf), cos(s,, s,) will be lower but typically greater than zero, allowing some semantic similarity while relaxing the information bottleneck.
Using cos( f,, f5) x cos(y;, with one-hot labels would collapse intra-class variations (see Table 13). Instead, using soft predictions cos(f, f,) cos(s,, s,) allows sufficient variation be encoded.
Prediction similarity and diverse substructures. Even in classes with diverse substructures, prediction similarity mandates feature similarity: (i) In CNNs, one moves toward the classifier, layers become increasingly shift- and permutation-invarian.. Thus, deeper layers filter nuisance factors irrleevnt to the classification task. (ii) The information bottleneck ensures the network focuses on salient, task-relevant features and discards uninformative variations. (iii) The bottleneck strength is controlled by in Eq. (7). Moderate values (A 1) perform best (see Table 14).
Table 14: Impact of bottleneck strength on CIFAR- 100 (ResNet-34)
0
0.1
0.3
0.5
0.7
5
10
Top-1 Acc (%)
76.76
76.80
76.87
77.32
77.61
78.38
76.74
76.02
75.90
75.37
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Table 15: Effect of varying 5 on CIFAR-100 with ResNet-34.
0
1e-5
1c-4
1e-3
1e-2
Accuracy
78.38
78.32
78.27
78.13
76.04
These results confirm that GCR preserves intra-class variations while enforcing focus the most important semantic features, improving robustness and accuracy without collapsing representations.
H.8 Lipschitz Continuity and Our Mechanism
L-Lipschitz continuous networks. It is well established that enforcing L-Lipschitz continuity in network, such as a discriminator, helps regularize its layers and prevents overfitting. For example, Lipschitz Generative Adversarial Nets (LGANs) [128], along with many subsequent works on classification and fine-tuning under L-Lipschitzness.
This condition can be imposed on feature vectors o.)) of a chosen layer by design, leading to
[ (x'))l2 S L||x- 12112, -x|x-11|22.
xx D,
(23)
where L is the Lipschitz constant. Intuitively, if L is small, then small input change x' produces a small change in the features (x) - (x'). while a large change in input produces a proportionally large change in the features. This stabilizes the network response.
practical way to encourage such behavior is through an auxiliary regularization loss added to the main task loss:
x [[ - (2)((x))l Ill L||x x' x2112²
(24)
which promotes the approximation
|| (x) (x)1l2 L||x x'112-
(25)
For simplicity, we omit ReLU and cosine similarity in what follows and use the Eg distance, though cosine distance can be incorporated without difficulty. We promote alignment of the form
Ex (I]x((x) <x(x'))l2 Ill 01] s(x')ll2)*
(26)
where s(.) denotes the softmax score. This yields information bottleneck of the form
[ - <x(x') ~ [ - slx))
(27)
Combining the Lipschitz condition with our bottleneck mechanism, we obtain
[||x x'112 = [x) <x(x')l Bl(s(x) - s22')l22,
(28)
which implies
L - B
||s(x) s(x')l1
1x
x2122-
(29)
This demonstrates that model does not collapse intermediate features into identical representations. Instead, small input variations x + yield small variations both features and softmax scores, while large input differences induce proportionally large differences in the outputs.
The strength of this relation depends on the network's local L-Lipschitzness (which may be fixed or vary in non-Lipschitz networks) and the scaling factor 8, which is implicitly controlled by our parameter A. Therefore, our information bottleneck mechanism ensures stable yet discriminative feature variation.
Introducing т: collapse prevention by design. We enforce (x;), di (x,)) by a simple soft penalty Σ, 1(y, y1) (x)())(x,, (x,))) <) (L(o)(x;,, 0i 7)² where 1(-) is the indicator function (e.g., do samples and j share the same label?) and L(x, y) = cos [ lisll
The results for CIFAR-100 using ResNet-34 are in Table 15. This result directly offers both anti- collapse control and evidence that the best results are attained for T = 0 (not pushing angles of the same class apart). Always maintaining non-zero angle, e.g., = le-2, is worse. We therefore conclude that our network does not suffer from feature dimensional collapse. However, such collapse did occur, it could be prevented by simple penalty, as demonstrated here.
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Table 16: Results on CIFAR-I00 (100 classes) with MobileNet. Accuracy and intra-/inter-class
variance across six layers under different т.
sst
2nd
3rd
4th
5th
6th
(deg)
Acc
intra
inter
intra
inter
intra
inter
intra
inter
intra
inter
intra
inter
baseline
65.95
38.71
13.81
22.30
7.56
12.00
5.27
5.10
4.24
17.71
19.49
8.64
9.69
(0.00)
68.14
31.21
12.02
18.18
7.61
9.98
4.28
3.94
3.25
16.82
18.69
8.30
9.32
0.08
68.21
31.25
12.56
18.28
6.54
9.92
4.46
4.10
3.28
18.77
19.52
9.23
9.74
0.26
67.68
36.28
13.11
20.93
7.15
10,94
4.92
4.58
3.81
17.72
19.08
8.72
9.50
0.81
67.37
35.39
13.72
20.44
6.92
11.02
4.89
4.70
3.89
17.74
18.87
8.70
9.41
2.56
67.34
35.40
13.40
20.61
7.32
10.83
4.84
4.56
3.74
17.92
19.08
8.78
9.50
8.11
67.39
39.04
14.86
23.42
8.04
12.00
5.24
5.16
4.38
16.77
18.58
8.28
9.27
Table 17: Results on CIFAR-100 100 (20 super-classes) with MobileNet. Accuracy and intra- variance across six layers under different 7.
1st
2nd
3rd
4th
5th
6th
(deg)
Acc
intra
inter
intra
inter
intra
inter
intra
inter
intra
inter
intra
inter
baseline
77.46
30.30
8.86
17.94
4.30
9.25
2.97
3.01
2.46
9.09
12.27
4.45
6.12
ours (0.00)
77.54
30.36
8.65
17.95
4.43
9.34
2.99
3.22
2.56
9.69
12.57
4.73
6.27
0.08
77.72
31.59
9.22
18.24
4.42
9.57
3.08
3.33
2.61
9.91
12.56
4.87
6.27
0.26
77.36
32.04
9.80
18.67
4.58
9.57
3.07
3.28
2.70
9.85
12.69
4.83
6.33
0.81
77.69
31.36
9.59
18.17
4.38
9.50
3.04
3.15
2.60
9.70
12.63
4.77
6.30
2.56
77.31
31.69
9.38
18.56
4.36
9.39
3.03
3.30
2.68
9.94
12.55
4.88
6.26
8.11
77.60
32.39
8.66
18.34
4.47
9.35
3.01
3.26
2.65
9.83
12.48
4.81
6.22
H.9
Preventing Dimensional Collapse by Design
Anti-collapse mechanism. We introduce a penalty to directly prevent by-design a possibility of dimensional collapse. To this end, we conduct an experiment on CIFAR-100, grouped into 20 super- classes, e.g., vehicles (bicycle, bus, motorcycle, pickup truck, train), insects (bee, beetle, butterfly, caterpillar, cockroach), flowers (orchids, poppies, roses, sunflowers, tulips), etc.
To this end, we force the within-clast angle 7 between pairs of features to at least r. Specifically, we add our GCR loss the following soft penalty: / Σ, 10; Uj) (x)())(x;) o (x,)) <T)-
(x,))(x;), di (x,)) - 2) which ensures that <(or(x,), (x,)) > 7 within each class. Here, Ç is normalization factor accounting for counts of same-class elements. For cosine similarity, we also apply ReLU in the above equations to prevent negative angles.
Setup. We use pre-trained baselines for comparisons. For our model, we train GCR- -ug-anedetted models on CIFAR-100 for: (i) the 100-class task, and (ii) the 20 super-class task. The GCR- augmented model is equipped with by-design collapse prevention mechanism. We choose /3 = 0.3 (generally optimal in our experiments). Subsequently, we vary the minimum angle to enforce lower bound on intra-class variance. We report both intra- and inter-class variance across all layers of MobileNet (used for fast experiments), considering both the original 100 classes and the 20 super-classes. Additionally, we include baseline comparison without applying our GCR framework.
Findings. The results on both the 100-class and 20-superclass tasks show that GCR improves accuracy while maintaining meaningful feature diversity. While enforcing a minimum within-class variance with experiments and remains close to the best case r 0.08. Importantly, setting larger values. which enforce large intra-class variance, degrades accuracy. Maintaining certain within-class feature variance by soft penalty makes sense. At the same time, the dimensional collapse does not happen in our model even without that penalty.
0.08 helps slightly, the variance for = 0.00 (ours) never collapsed in our
I Limitations and Future Works
I.1 Failure Modes and Marginal Gains
Highly noisy data. GCR aligns feature graphs with masked prediction graph, using it as a semantic reference. The quality of this reference is critical. Our method assumes reliable ground-trun truth labels
44

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for the mask M. If the training data contains spurious correlations mislabeled examples. the aligmment may reinforce these errors. Under high label noise, the prediction graph is based on a flawed mask, producing corrupted supervisory signal that misguides feature representations toward incorrect semanties. This failure mode can lead to marginal gains or even performance degradation in our supervised framework.
Highly class-imbalanced data. Since GCR builds relational graphs the batch level, the global context is limited to within-batch relationships. In highly imbalanced datasets, batches may contain few or no minority-class samples, resulting in sparse or uninformative prediction graphs for those classes. Consequently, the alignment loss is dominated by majority classes, potentially harming minority-class representations and leading to marginal overall gains.
Other scenarios leading to marginal gains. Beyond noisy and imbalanced data, GCR may yield marginal improvements in the following cases: (i) Simple datasets with high baseline performance: GCR targets noisy inter-class similarities and semantic structure. On simpler datasets with strong baseline models and well-separated features, there is less room for improvement. Our results confirm this, with larger gains on complex datasets such as CIFAR-100 and Tiny ImageNet compared to CIFAR-10, where baseline accuracy was already high. (ii) Extremely small batch sizes: Relational graphs rely on sufficient pairwise relationships. Very small batches provide limited data context, reducing graph stability and weakening regularization.
1.2 Applicability to Unsupervised and Self-Supervised Learning
Our current GCR formulation relies on ground-truth labels for intra-class masking of the prediction graph. This design was intentional to first establish and validate GCR's core effectiveness in a fully supervised setting, where we demonstrated consistent, significant gains across diverse architectures and datasets.
However, this reliance is not a fundamental limitation of GCR but rather a characteristic of the current implementation. The core mechanism, aligning feature graph geometry with prediction graph semantics, is flexible. The mask M can be generated without ground-truth labels, enabling extensions to unsupervised or self-supervised learning. Specifically, we propose two potential adaptations: (i) Pseudo-labeling. In semi/self- supervised settings, high-confidence model predictions can generate pseudo-label, to construct M, masking pairs sharing the same pseudo-label. (ii) Unsupervised clustering: Feature representations can be clustered dynamically (e.g., via k-means), with M derived from cluster assignments enforce consistency among similar samples.
While this work focuses on supervised learning to establish GCR's principle, the framework is modular. Replacing the ground-truth mask with pseudo-labels clustering-based masks readily extends GCR to unsupervised domains. This important direction highlights GCR's ' broader potential and flexibility.
L.3 Future Works
While GCR introduces a novel and effective mechanism for enhancing semantic structure in deep representations, also presents several limitations that open up promising avenues for future explo- ration.
First, GCR currently relies on batch-level relational graphs constructed from intermediate features and softmax predictions. This inherently restricts the global context to within-hatch relationships, which may limit its effectiveness in settings with small batch sizes or highly imbalanced class distributions. or streaming graph variants that accumulate semantic structure across batches could improve scalability and robustness.
Exploring
Second, although the adaptive weighting mechanism allows the model to prioritize layers with high misalignment, it is currently driven solely by Frobenius distance. More expressive structural metrics, such as spectral divergence, Earth Mover's 's distance, or alignment of Laplacian eigenvectors, could offer deeper insights into graph discrepancy and guide more fine-grained supervision.
Third, GCR assumes access to ground-truth labels for masking during training, limiting its current for- mulation to fully supervised learning. Extending GCR to semi-supervised, self-supervised, or weakly
45

---

supervised regimes by deriving prediction graphs from confident pseudo-labels or unsupervised clustering remains an exciting direction.
Fourth, while GCR is model-agnostic and does not alter the architecture, it introduces additional computation from graph construction and alignment. Although lightweight and parallelizable, this cost may still pose challenges in latency-sensitive applications. Investigating more efficient or approximate graph alignment schemes could alleviate this concern.
Finally, our method focuses on classification tasks. Adapting GCR to other modalities and tasks, such segmentation, detection, multimodal fusion, requires further study. In particular, integrating GCR with token-level or region-level predictions transformer models may offer novel insights into semantic alignment beyond image classification.
In future work, we aim to address these limitations by developing more scalable graph construction techniques, expanding the framework to broader learning paradigms, and refining the theoretical understanding of structural supervision in deep networks.
Broader impacts
GCR introduces lightweight and architecture- aggnostic framework for improving semantic consis- tency in deep learning models. By aligning intermediate features with structured prediction graphs, GCR encourages networks learn more coherent, interpretable, and generalizable representations. This has the potential to improve model reliability critical applications such as medical diagnosis, autonomous driving, and scientific data analysis, where semantic structure and interpretability crucial.
From ethical standpoint, GCR does not introduce any explicit biases beyond those already present in the training data. However, like all supervision-dre -drivnn methods, its effectiveness depends on the quality of labels. If training data contains spurious correlations mislabeled examples, the alignment process may reinforce rather than correct those artifacts. Future work should investigate ways to make GCR more robust to noisy biased supervision signals.
GCR is computationally efficient and compatible with existing training protocols, which facilitates deployment in low-resource settings environmental costs when scaling to very large models or datasets. In addition, the reliance on pairwise relationships may raise privacy concerns in sensitive domains where sample-wise relationships reveal protected attributes. Applying GCR to preeeccypreeeiinggg or federated learning settings a promising direction for ensuring responsible AI development.
edge devices. However, care should be taken to assess
46


Reviews:
[{"review_id": "79gqaV4VUV", "reviewer": "Reviewer_6b7B", "summary": "The paper introduces Graph Consistency Regularization (GCR), a plug-and-play module that can be integrated into existing neural networks. GCR constructs relational graphs from intermediate representations to ensure these representations are consistent with the model’s final predictions.", "questions": "1. The main experimental datasets are CIFAR-100 and Tiny ImageNet, which are relatively small by current standards. Evaluation on larger datasets would improve the paper quality.\n\n2. Could the authors elaborate on the additional computational overhead introduced by GCR? Can the method be implemented in a model-parallel fashion to support scalability on large datasets?\n\n3. Have the authors conducted an ablation study to assess the impact of batch size on performance?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "The paper is clearly written and includes numerous visualizations to illustrate the learned relational graphs within GCR. The authors also provide a theoretical analysis for their approach.\n\nHowever, the idea of using graphs to enhance classification tasks is not particularly novel, and the paper lacks comparisons with other graph-based classification methods. Additionally, the method appears to require a relatively large batch size (e.g., 128), which may limit its applicability. The datasets used—CIFAR-100 and Tiny ImageNet—are somewhat limited in scale for evaluating generalization.", "was_revised": true, "final_justification": "Based on the consistent performance improvement on different datasets, I recommend to borderline accept.", "rebuttal": "We sincerely thank reviewer for the constructive feedback, which has significantly improved the quality of our work.\n\nWe have addressed all concerns and revised the paper accordingly. \n\nSpecifically:\n(i) we added large-scale ImageNet-1K experiments using iFormer [3], ViT-B/16, and ViG-B [2] in Sections 3.1 and 3.2 to demonstrate the generalization of our method;\n(ii) we moved the computational overhead analysis from Appendix G (pp. 33–34) to Section 3.2 for greater visibility; and\n(iii) we added an ablation study on batch size effects for both CIFAR-10 and Tiny ImageNet.\n\n### **1. Novelty**\n\nWhile graph-based methods have been studied, our GCR framework introduces **a fundamentally different mechanism**.\n\nDifferent from existing works, GCR **does not rely on static external graphs** or message passing as in GNNs. Instead, it introduces **a novel form of self-prompted regularization**, where the model’s own predictions dynamically construct a class-aware graph that supervises intermediate feature representations.\n\nThis distinction has been clearly acknowledged by multiple reviewers:\n\n- Reviewer puhK praised the conceptual novelty, calling the use of prediction structures \"innovative and well-motivated\", and described GCR as \"conceptually elegant\", \"versatile\", and \"broadly applicable in practice\". He/She noted that our analyses \"substantiate the soundness of the proposed method\".\n\n- Reviewer bJ5m emphasized the value of the multi-layer mechanism: by aligning early representations with later prediction-derived relations, the method encourages \"more discriminative patterns\", which improve generalization.\n\n- Reviewer a6xh precisely summarized our contribution: GCR introduces \"parameter-free GCLs\" that align feature and prediction graphs, is \"supported by solid theoretical foundations\", and is clearly visualized through relational graphs.\n\nKey innovations of GCR includes:\n\n- **Self-prompted** graphs (dynamic & batch-wise): GCR constructs prediction graphs on-the-fly from the model's own softmax outputs, requiring no static structure or external memory.\n\n- **Parameter-free** regularization: GCR acts as a lightweight, plug-in regularizer, not a feature transformer, making it architecture-agnostic and **easy to integrate**.\n\n- **Cross-space graph alignment**: GCR aligns intermediate feature-space similarity graphs with prediction-space semantic graphs, enabling **a unique form of supervision** not yet explored in any prior graph-based classification methods.\n\nAs detailed in Appendix A (Related Work, p. 24) and B (Relation to Existing Paradigms, pp. 24-26), this self-prompting strategy offers **a fresh perspective** on using relational inductive biases in a modular, model-driven way.\n\nWe have revised the Related Work to highlight these points of novelty and more clearly position our contributions within the existing literature.\n\n### **2. Comparison to existing graph-based methods**\n\nBelow, we provide **direct comparisons** between our GCR-augmented models and recent graph-based classification methods across CIFAR-10, CIFAR-100, and ImageNet-1K. The results clearly show the consistent and complementary benefits of GCR when integrated into graph-based backbones.\n\n**Results on CIFAR-10, CIFAR-100, and ImageNet-1K**\n\n||CIFAR-10|CIFAR-100|ImageNet-1k|\n|-|-|-|-|\n|ResNet18|||||\n|CNN2GNN[1]|95.51$\\pm$0.42|74.80$\\pm$0.81|60.12$\\pm$1.02|\n|CNN2GNN **+GCR**|95.87$\\pm$0.31|76.23$\\pm$0.38|62.47$\\pm$0.47|\n|CNN2Transformer[1]|95.79$\\pm$0.24|77.39$\\pm$0.20|71.12$\\pm$0.35|\n|CNN2Transformer **+GCR**|**95.96$\\pm$0.35**|**78.23$\\pm$0.30**|**72.33$\\pm$0.31**|\n|ResNet34|||||\n|CNN2GNN[1]|96.39$\\pm$0.41|77.87$\\pm$0.91|61.02$\\pm$0.77|\n|CNN2GNN **+GCR**|96.67$\\pm$0.36|78.14$\\pm$0.54|62.88$\\pm$0.46|\n|CNN2Transformer[1]|96.73$\\pm$0.37|80.10$\\pm$0.45|75.42$\\pm$0.15|\n|CNN2Transformer **+GCR**|**96.97$\\pm$0.36**|**81.27$\\pm$0.29**|**76.67$\\pm$0.26**|\n\n**Results on ImageNet-1K**\n\n|Model|ViG-Ti[2]|ViG-Ti **+GCR**|ViG-S[2]|ViG-S **+GCR**|ViG-B[2]|ViG-B **+GCR**|\n|-|-|-|-|-|-|-|\n|Accuracy|73.9|**74.9**|80.4|**81.7**|82.3|**84.0**|\n\n\nThese results highlight key insights:\n\n- **Consistent** across architectures: GCR improves performance across diverse backbones (CNN2GNN, CNN2Transformer, ViG), demonstrating broad applicability.\n\n- **Complementary** to existing methods: Its self-regularization mechanism enhances CNN2GNN and CNN2Transformer, indicating **orthogonality to existing graph-based pipelines**.\n\n- **Scalable** to large-scale tasks: **Notable gains** on ImageNet-1K (e.g., +1.7% with ViG-B) show GCR’s effectiveness in complex, real-world settings.\n\n- **Lightweight** and model-agnostic: GCR introduces no extra parameters and integrates easily into existing architectures.\n\nTogether, these findings reinforce GCR’s **novelty and practical value** as a flexible, self-prompted regularization framework applicable across scales and architectures.\n\n### **3. Generalization to large-scale datasets**\n\nWe thank the reviewer for suggesting an evaluation on a larger dataset. In response to the suggestion, we conducted additional experiments on **ImageNet-1K** using transformer-based architectures (iFormer [3], ViT, and ViG [2]). Results, averaged over three runs, are summarized in Appendix H.3.\n\n||iFormer-S|iFormer-B|ViT-B/16|ViG-B|\n|:-|:-:|:-:|:-:|:-:|\n|Baseline|83.4$\\pm$0.40|84.6$\\pm$0.45|74.3$\\pm$0.51|82.3$\\pm$0.42|\n|Early GCL|83.8$\\pm$0.31|85.0$\\pm$0.40|74.7$\\pm$0.44|82.8$\\pm$0.35|\n|Mid GCL|83.8$\\pm$0.39|85.5$\\pm$0.33|75.2$\\pm$0.36|83.0$\\pm$0.34|\n|Late GCL|84.5$\\pm$0.29|**86.1$\\pm$0.30**|**75.8$\\pm$0.33**|**84.0$\\pm$0.30**|\n|Early+Mid|84.3$\\pm$0.33|85.9$\\pm$0.38|75.6$\\pm$0.41|83.7$\\pm$0.33|\n|Mid+Late|**84.8$\\pm$0.28**|85.9$\\pm$0.37|75.6$\\pm$0.34|83.9$\\pm$0.30|\n|Early+Late|84.5$\\pm$0.30|85.2$\\pm$0.28|74.9$\\pm$0.33|83.5$\\pm$0.29|\n|Full GCL|84.3$\\pm$0.29|85.8$\\pm$0.26|75.5$\\pm$0.30|83.6$\\pm$0.27|\n\nKey results:\n\n- iFormer-S: GCR boosts performance from **83.4%** to **84.8%** (a **+1.4%** gain).\n\n- iFormer-B: GCR improves accuracy from **84.6%** to **86.1%** (a **+1.5%** gain).\n\n- Similar gains are observed with ViT-B/16 and ViG-B, confirming generality.\n\nThese improvements validate that GCR **scales effectively** to large, complex datasets and modern architectures. They support our core hypothesis: aligning feature geometry with prediction semantics enhances generalization. Moreover, GCR achieves this without architectural changes or added parameters, reinforcing its value as a model-agnostic and lightweight regularizer.\n\nThese new results directly address the reviewer’s concern and further strengthen the overall contribution of the paper.\n\n### **4. Computational overhead and scalability**\n\nWe thank the reviewer for highlighting this point. To improve clarity, we have moved the computational overhead analysis from Appendix G (pp. 34 line 1275-1284 and 1292-1297) into the main paper.\n\n- **Overhead**: GCR adds modest training time due to graph construction and alignment (e.g., MobileNet: +15 min, ResNeXt-101: +120 min on CIFAR-100). These operations are efficient and introduce no additional trainable parameters.\n\n- **Scalability**: GCR is highly scalable. All graph computations are performed per batch and are fully parallelizable across GPUs using standard data-parallel training. The operations rely on matrix computations optimized for modern hardware, ensuring smooth integration into large-scale pipelines.\n\n### **5. Impact of batch size**\n\nAs shown in Figure 8 (Appendix H.2, p. 36), GCR **remains effective across a wide range of batch sizes**. Even with small batches (e.g., n=16 and n=32), GCL-enhanced models produce more coherent intra-class clusters and stronger inter-class separation than the baseline. This supports our claim that GCR can extract meaningful structure even from limited relational signals.\n\nWhile larger batches amplify gains by providing denser graphs, they are not essential. The tables below confirm consistent performance improvements across batch sizes on both CIFAR-10 (ShuffleNet) and Tiny ImageNet (CeiT):\n\n**ShuffleNet on CIFAR-10**\n|Batch Size|+GCR|Baseline|\n|-|-|-|\n|16|79.90$\\pm$0.38|78.88$\\pm$0.41|\n|32|87.91$\\pm$0.36|86.91$\\pm$0.37|\n|64|91.26$\\pm$0.25|90.64$\\pm$0.35|\n|128|92.79$\\pm$0.20|91.21$\\pm$0.28|\n|256|**92.89$\\pm$0.25**|92.07$\\pm$0.27|   \n|512|92.33$\\pm$0.23|91.94$\\pm$0.25|\n\n**CeiT on Tiny ImageNet**\n|Batch Size|+GCR|Baseline|\n|-|-|-|\n|16|44.84$\\pm$0.31|43.78$\\pm$0.35|\n|32|47.55$\\pm$0.29|46.89$\\pm$0.31|\n|64|49.19$\\pm$0.25|48.09$\\pm$0.30|\n|128|**51.22$\\pm$0.20**|49.95$\\pm$0.29| \n|256|50.77$\\pm$0.19|49.62$\\pm$0.24|\n|512|50.65$\\pm$0.22|49.34$\\pm$0.24|\n\nThese results affirm GCR’s robustness and flexibility, even in resource-constrained or small-batch training setups.\n\nWe are confident that GCR represents a novel, technically sound, and broadly applicable contribution to the field. \n\nWe sincerely thank the reviewer for the valuable feedback, which we **have incorporated** into the final version of the paper.\n\n### References\n\n[1] Trivedy, V., & Latecki, L. J. (2023). Cnn2graph: Building graphs for image classification. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (pp. 1-11).\n\n[2] Han, K., Wang, Y., Guo, J., Tang, Y., & Wu, E. (2022). Vision gnn: An image is worth graph of nodes. Advances in neural information processing systems, 35, 8291-8303.\n\n[3] Si, C., Yu, W., Zhou, P., Zhou, Y., Wang, X., & Yan, S. (2022). Inception transformer. *Advances in Neural Information Processing Systems*, *35*, 23495-23509."}, {"review_id": "aoWgCTIE9g", "reviewer": "Reviewer_a6XH", "summary": "This paper proposes Graph Consistency Regularization (GCR), a framework that introduces parameter-free Graph Consistency Layers (GCLs) to align intermediate feature similarity graphs with masked prediction similarity graphs derived from softmax outputs. The core assumption is that samples predicted to belong to the same class should have similar intermediate representations, while different-class samples should be separated in feature space. The authors provide theoretical analysis and demonstrate improvements on some datasets.", "questions": "1. Do you have any evidence, theoretical or emperical, can support the core intuition about \"If a network is confident that two inputs belong to the same class, as indicated by their softmax predictions, then their intermediate representations should also reflect this similarity.\"\n2. How sensitive is the method to the choice of similarity metric beyond cosine similarity?", "limitations": "The authors acknowledge several limitations in Appendix I. My main concern is simple, which is just about the fundamental premise in the weaknesses part. If the authors could find some evidence, whatever theoretical or emperical, I think I can change my opinion.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 1, "originality": 2, "strengths_and_weaknesses": "Pros:\n1. Parameter-free and model-agnostic: The method introduces no additional learnable parameters and can be inserted into any architecture without modification.\n2. Comprehensive theoretical analysis: The paper provides solid theoretical foundations including generalization bounds, spectral alignment analysis, and PAC-Bayesian interpretations.\n3. Clear visualization and interpretability: The relational graph visualizations effectively demonstrate the method's impact on feature organization.\n\nCons:\n1. Questionable core assumption: The fundamental premise that \"samples predicted to belong to the same class should have similar intermediate representations\" is problematic. High-dimensional representation spaces may naturally accommodate multiple clusters for the same class, and forcing similarity based on prediction confidence may eliminate beneficial intra-class diversity. This assumption imposes an overly strong inductive bias that may harm the model's ability to capture complex within-class variations, which are common in real-world data where the same class can have multiple modes or sub-clusters.\n\n2. Limited theoretical justification for the core premise: While the paper provides extensive theoretical analysis of the optimization properties, it lacks theoretical justification for why prediction similarity should translate to feature similarity. The manifold hypothesis and cluster assumption (Definition 4, Proposition 4) are presented as givens, but modern deep learning suggests that good representations often maintain intra-class structure that doesn't require global similarity. The assumption contradicts recent understanding that representations can be both discriminative and maintain rich internal structure.\n\n3. No experimental evidence for the core premise.", "was_revised": true, "final_justification": "While I appreciate the authors' comprehensive rebuttal, my core concern remains unresolved.\n\n**Unresolved Issue:**\nDespite claims of using \"soft predictions,\" the cosine similarity alignment mechanism still encourages feature similarity based on prediction confidence, which risks collapsing beneficial intra-class structure. The \"gentle regularization\" argument doesn't address that cosine similarity inherently reduces angular diversity within predicted class clusters.\n\n**Key Concerns:**\n1. **Structure Collapse Risk:** The method creates an information bottleneck that may eliminate legitimate intra-class multimodality (e.g., different dog breeds should have distinct feature clusters even when both predicted as \"dog\").\n\n2. **Limited Validation:** Experiments focus on standard benchmarks with relatively simple intra-class structures. No direct validation that GCR preserves beneficial intra-class multimodal structure rather than collapsing it.\n\n**What Would Change My Assessment:** \nExperimental evidence showing GCR preserves legitimate intra-class diversity on datasets where classes naturally exhibit multiple distinct modes.", "rebuttal": "We sincerely thank the reviewer for the feedback and for offering us the opportunity to provide evidence to change his/her opinion/rating. \n\nAs requested, we present eight pieces of supporting evidence below.\n\nCollapsing same-class samples into a single cluster is harmful. GCR **instead** encourages them to occupy a structured, semantically coherent region in feature space, a sub-manifold shaped by **relational cues**, that is **distinct yet internally diverse**.\n\n### **1. Gentle regularization & implicit attention**\n\nOur method is grounded in established deep learning principles, combining soft regularization with structure-aware guidance.\n\n- Cosine similarity as **gentle bias**: By using cosine similarity, GCR encourages alignment based on the direction of features, not their magnitude. This induces **a soft relational bias**, promoting semantic structure without collapsing diversity. Like **contrastive learning**, it organizes the feature space by **pulling semantically similar samples closer**, but does so using the model’s own predictions as a dynamic, self-supervised signal.\n\n- **Parameter-free attention** mechanism: As shown in Fig. 3, GCR guides the model to attend to discriminative regions (**attention to low-level semantic features**). Compared to the diffuse activations in the baseline, GCR-enhanced models generate sharper, semantically focused activations (e.g., *cat’s face or dog’s tongue*). This mirrors human visual attention and supports more meaningful representation learning.\n\n### **2. Experimental evidence: intra-class variance**\n\nWe evaluated **intra-class variance** on CIFAR-10 using ShuffleNet. For each class, we computed the centroid of final-layer embeddings and measured the average Euclidean distance of each sample from its class centroid, a direct indicator of how tightly the representations are clustered (lower=more cohesive).\n\n|Class|Baseline|+GCR|\n|:-|:-:|:-:|\n|plane|4.335|4.171|\n|auto|3.848|3.535|\n|bird|4.565|4.662|\n|cat|4.631|4.214|\n|deer|4.148|3.796|\n|dog|4.679|3.923|\n|frog|4.908|4.869|\n|horse|4.414|4.092|\n|ship|3.896|3.655|\n|truck|3.987|3.737|\n|Average|**4.341**|**4.066**|\n\n- **Improved cohesion**: GCR reduces intra-class variance, lowering the average from 4.341 to 4.066. This confirms its effectiveness in promoting **more compact**, **semantically aligned** features.\n\n- **Preserved diversity**: Crucially, the variance is **not collapsed** to near-zero, indicating that GCR maintains meaningful intra-class variation.\n\nThis analysis directly addresses concerns about over-regularization. GCR encourages class cohesion without sacrificing expressive diversity, contributing to cleaner feature structures and stronger generalization. We have included this table and discussion in Sec 3.2.\n\nKindly also refer to our response 3 \"GCR reduces inter-class noise\", where we analyze silhouette score and separability ratio to reviewer bJ5m.\n\n### **3. Experimental evidence: on earlier layers**\n\n- [Exp 1] GCR **corrects the largest points of misalignment**\n\nTo test whether GCR’s impact correlates with a layer’s semantic misalignment, we trained a CeiT model on Tiny ImageNet without GCR and measured the baseline discrepancy $\\delta(l) = ||F(l) - P||_F^2$ for each block $l$. We then applied GCR to individual blocks and recorded the top-1 accuracy gain $\\Delta\\text{Acc}(l)$.\n\nResults show that early layers, **bridging low-level features to class concepts**, exhibit the highest misalignment and largest gains:  \n\n- Block 1: $\\delta_1 = 0.45$, gain +1.2%  \n- Block 2: $\\delta_2 = 0.30$, gain +0.9%  \n\nA **strong Pearson correlation** of **0.62** between $\\{\\delta(l)\\}$ and $\\{\\Delta\\text{Acc}(l)\\}$ quantitatively confirms that GCR is more effective where feature-prediction misalignment is greater.\n\n- [Exp 2] early GCR **creates more robust foundational features**\n\nNext, we tested whether early GCR creates more robust features that benefit later layers, a \"feature cleaning\" effect. We trained two ShuffleNet models (5 blocks each) and then froze the regularized blocks to evaluate their standalone quality:\n\n- Model A (Early-GCR): GCLs on Blocks 1-2, frozen after 100 epochs, then fine-tuned remaining blocks.\n\n- Model B (Late-GCR): GCLs on Blocks 4-5, frozen and fine-tuned similarly.\n\n||Pre-freeze| Post-freeze|Performance Drop|\n|:-|:-:|:-:|:-:|\n|Model A|66.8%|66.1%|**0.7%**|\n|Model B|66.4%|65.1%|**1.2%**|\n\nModel A’s smaller drop shows early GCR features are more robust and semantically coherent, reducing reliance on later regularization. Model B’s larger drop suggests its performance depends more on continued late-stage regularization, with earlier features remaining entangled.\n\n### **4. Experimental evidence: superclass classification**\n\nTo directly validate our core premise that GCR encourages semantically richer representations, we conducted a linear probing experiment using the CIFAR-100 dataset, which is grouped into **20 superclasses** (e.g., vehicles, insects, flowers). This setup tests **whether GCR-learned features are more transferable and better organized**.\n\nSetup: (i) We take pretrained baseline and GCR-augmented models trained on the 100-class CIFAR-100 task. (ii) We freeze all feature layers, preserving their learned representations. (iii) We replace the final head with a new linear classifier for the 20 superclasses. (iv) Only this classifier is trained; the rest of the model remains fixed.\n\n**Rationale**: If GCR induces more semantically structured features, then a linear classifier should perform better on the superclass task using those features.\n\n||+GCR|Baseline|Improvement|\n|:-|:-:|:-:|:-:|\n|MobileNet|**75.73**|73.07|+2.66%|\n|ShuffleNet|**75.65**|75.02|+0.63%|\n|SqueezeNet|**74.48**|73.93|+0.55%|\n|ResNeXt-50|**86.72**|84.36|+2.36%|\n|ResNeXt-101|**87.14**|85.42|+1.72%|\n|ResNet-34|**86.30**|84.80|+1.50%|\n|ResNet-50|**86.95**|86.23|+0.72%|\n|DenseNet-121|**85.42**|85.03|+0.39%|\n\nGCR consistently improves superclass classification, confirming that it enhances semantic organization of the feature space. This **goes beyond** improving the original task, it shows GCR fosters **more generalizable** and **hierarchically structured representations**, validating the method’s core motivation. \n\nThis analysis have been added to Sec 3.2.\n\n### **5. Visual evidence: structured diversity**\n\nOur visualizations provide strong qualitative support for GCR's effectiveness.\n\n- **t-SNE plots** (Figs. 6c, 6d, & 7) show that GCR leads to tighter intra-class clusters and larger inter-class separation. Crucially, clusters retain internal structure, they are not collapsed to points, indicating preserved diversity within classes.\n\n- **Relational graphs** (Figs. 1, 4, & 5) show that GCR **suppresses weak inter-class links** while **reinforcing intra-class connectivity**, producing clearer decision boundaries and more semantically consistent feature spaces.\n\nThese visual results reinforce our claim that GCR promotes structured feature organization, **compact yet expressive**, supporting both generalization and interpretability.\n\n### **6. Geometric rationale: aligning structures**\n\nGCR **operates over relational graphs, not isolated features**. The goal is not to make all class features identical, but to ensure the pairwise similarities among features reflect those in the prediction space.\n\n- Rich variations preserved: GCR allows for multiple intra-class modes (e.g., different dog breeds) as long as they are closer to each other than to other classes. This preserves structural diversity while ensuring semantic coherence.\n\n- Manifold alignment analogy: GCR aligns the feature manifold with the prediction manifold, two views of the same data. The loss encourages the Laplacians of these graphs to be spectrally consistent, preserving clustering and diffusion properties (Proposition 1, Corollary 1).\n\n### **7. Information bottleneck principle**\n\nThe information bottleneck principle suggests that an ideal feature should preserve information relevant to the label while discarding irrelevant input noise.\n\n- Prediction as **compressed signal**: The prediction graph can be interpreted as a compressed estimate of the essential class-wise structure. It captures semantic relations while **abstracting away unnecessary details**.\n\n- GCR as a **regularizer**: The GCR loss encourages the feature similarity graph to reflect the structure of $\\mathbf{P}$, promoting representations that align with class semantics while **compressing away task-irrelevant** variance.\n\nIn effect, GCR guides the network toward learning representations where geometric relationships **mirror** prediction-based semantics, rather than forcing absolute similarity across features.\n\n### **8. Generalization**\n\nOn **ImageNet-1K** (averaged over three runs):\n\n||iFormer-S|iFormer-B|ViT-B/16|ViG-B|\n|:-|:-:|:-:|:-:|:-:|\n|Baseline|83.4$\\pm$0.40|84.6$\\pm$0.45|74.3$\\pm$0.51|82.3$\\pm$0.42|\n|Early GCL|83.8$\\pm$0.31|85.0$\\pm$0.40|74.7$\\pm$0.44|82.8$\\pm$0.35|\n|Mid GCL|83.8$\\pm$0.39|85.5$\\pm$0.33|75.2$\\pm$0.36|83.0$\\pm$0.34|\n|Late GCL|84.5$\\pm$0.29|**86.1$\\pm$0.30**|**75.8$\\pm$0.33**|**84.0$\\pm$0.30**|\n|Early+Mid|84.3$\\pm$0.33|85.9$\\pm$0.38|75.6$\\pm$0.41|83.7$\\pm$0.33|\n|Mid+Late|**84.8$\\pm$0.28**|85.9$\\pm$0.37|75.6$\\pm$0.34|83.9$\\pm$0.30|\n|Early+Late|84.5$\\pm$0.30|85.2$\\pm$0.28|74.9$\\pm$0.33|83.5$\\pm$0.29|\n|Full GCL|84.3$\\pm$0.29|85.8$\\pm$0.26|75.5$\\pm$0.30|83.6$\\pm$0.27|\n\nThese results show that GCR scales well and confirm our core insight: aligning feature geometry with prediction semantics strengthens generalization.\n\n### **9. Similarity metric**\n\nWe evaluated various similarity functions on CIFAR-100 (MobileNet, Baseline 65.95%):\n\n||Cosine|RBF|Polynomial|Sigmoid|Laplacian|\n|-|-|-|-|-|-|\n|+GCL|**68.32**|67.45|67.38|67.42|67.21|\n\nCosine performs best, likely due to its **scale invariance**, aligning directions, which helps preserve intra-class diversity. We’ve expanded Appendix H.4 to reflect this rationale.\n\nAll insights and findings **have been incorporated** into our paper."}, {"review_id": "TyDSCE3Nlt", "reviewer": "Reviewer_puhK", "summary": "This paper introduces  a framework that aligns prediction-based relational graphs with class-aware graphs to improve feature quality, cohesion, and generalization across models and datasets.", "questions": "See weakneses", "limitations": "See weakneses", "rating": 5, "confidence": 1, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The introduction of graph-based self-regularization using prediction structures is innovative and well-motivated. The idea of treating the model’s own predictions as self-prompts for semantic alignment is conceptually elegant.\n2. GCR is versatile—it can be inserted into various network architectures without modifying the backbone or introducing learnable parameters, making it broadly applicable in practice.\n3. The paper provides comprehensive theoretical analysis from multiple perspectives: generalization bounds via covering numbers, spectral alignment via normalized Laplacians, and a PAC-Bayesian regularization view. These analyses substantiate the soundness of the proposed method.\n\nWeaknesses:\n1. The paper lacks a discussion of scenarios where GCR fails or leads to marginal gains. Understanding its limitations (e.g., on highly noisy or class-imbalanced data) would improve robustness evaluation.\n2. The masked prediction graph requires knowledge of ground-truth labels for intra-class masking. This limits the direct application of GCR in unsupervised or fully self-supervised contexts.", "was_revised": true, "final_justification": "This paper introduces a framework that aligns prediction-based relational graphs with class-aware graphs to enhance feature quality, cohesion, and generalization across models and datasets. The authors have addressed my concerns during the rebuttal phase; therefore, I hold a positive view of this work.", "rebuttal": "We thank the reviewer for the positive and encouraging feedback.\n\nWe’re grateful that the reviewer recognized our approach as innovative, well-motivated, and supported by solid theoretical analysis. We also appreciate the thoughtful questions regarding its limitations, which we address in detail below.\n\n### **1. Discussion of scenarios where gains are marginal**\n\nWe thank the reviewer for highlighting the need for a more explicit discussion of GCR’s limitations. We agree that identifying failure modes is essential for a robust evaluation. While we included a \"Limitations and Future Work\" section (Appendix I, p. 36), we acknowledge that **a more focused discussion** on scenarios with marginal gains would further strengthen the paper.\n\nBelow, we provide a detailed analysis.\n\n**On highly noisy data**\n\nGCR aligns feature graphs with a masked prediction graph, using it as a semantic reference, making the quality of this reference critical.\n\nOur method assumes reliable ground-truth labels for the mask $\\mathbf{M}$. As noted in the \"Broader Impacts\" section (Appendix J, p. 37), if training data contains spurious correlations or mislabeled examples, the alignment may reinforce these errors instead of correcting them. Under high label noise, the prediction graph $\\mathbf{P}$ is based on a flawed mask, producing a corrupted supervisory signal that misguides feature representations toward incorrect semantics. This failure mode can lead to marginal gains or even performance degradation in our supervised framework.\n\n**On highly class-imbalanced data**\n\nWe acknowledge this critical scenario in our limitations section.\n\nSince GCR builds relational graphs at the batch level (p. 37, line 1339-1343), the global context is limited to within-batch relationships. In highly imbalanced datasets, batches may contain few or no minority-class samples, resulting in sparse or uninformative prediction graphs for those classes. Consequently, the alignment loss is dominated by majority classes, potentially harming minority-class representations and leading to marginal overall gains.\n\n**Scenarios leading to marginal gains**\n\nBeyond noisy and imbalanced data, GCR may yield marginal improvements in the following cases:\n\n- Simple datasets with high baseline performance: GCR targets noisy inter-class similarities and semantic structure. On simpler datasets with strong baseline models and well-separated features, there is less room for improvement. Our results confirm this, with **larger gains on complex datasets like CIFAR-100 and Tiny ImageNet than on CIFAR-10**, where baseline accuracy was already high.\n\n- Extremely small batch sizes: Relational graphs rely on sufficient pairwise relationships. As detailed in our limitations and batch size analysis (Fig. 8, p. 36), very small batches provide limited data context, reducing graph stability and weakening regularization.\n\nTo clarify this, we added a subsection titled \"Failure Modes and Marginal Gains\" in Appendix I (p. 36). It consolidates these insights, emphasizing that GCR’s benefits are limited under high label noise, severe class imbalance, or near-optimal baseline performance.\n\n### **2. Applicability to unsupervised and self-supervised learning**\n\nWe agree with the reviewer that our current GCR formulation relies on ground-truth labels for intra-class masking of the prediction graph. This approach was intentional to **first establish and validate GCR’s core effectiveness** in a fully supervised setting, where we demonstrated consistent, significant gains across diverse architectures and datasets.\n\nHowever, this reliance is not a fundamental limitation of GCR but **a characteristic** of the current implementation. The core mechanism, aligning feature graph geometry with prediction graph semantics, is flexible. The mask $\\mathbf{M}$ can be generated without ground-truth labels, enabling extensions to unsupervised or self-supervised learning.\n\nAs noted in our \"Limitations and Future Work\" (Appendix I, line 1348-1351), this is a promising research direction. Specifically, we propose two adaptations:\n\n- Pseudo-labeling: In semi/self-supervised settings, high-confidence model predictions can generate pseudo-labels to construct $\\mathbf{M}$, masking pairs sharing the same pseudo-label.\n\n- Unsupervised clustering: Feature representations can be clustered dynamically (e.g., via k-means), with $\\mathbf{M}$ derived from cluster assignments to enforce consistency among similar samples.\n\nWhile this work focuses on supervised learning to establish GCR’s principle, the framework is modular. Replacing the ground-truth mask with pseudo-labels or clustering-based masks readily extends GCR to unsupervised domains. We thank the reviewer for highlighting this important direction, which we have emphasized more clearly in the revised manuscript’s conclusion to highlight GCR’s broader potential and flexibility.\n\n### **3. Additional evaluations and comparisons**\n\nBelow, we provide comparisons between our GCR-augmented models and recent graph-based classification methods across CIFAR-10, CIFAR-100, and ImageNet-1K. The results show the consistent and complementary benefits of GCR when integrated into graph-based backbones.\n\n**(i) Results on CIFAR-10, CIFAR-100, and ImageNet-1K**\n\n||CIFAR-10|CIFAR-100|ImageNet-1k|\n|-|-|-|-|\n|ResNet18|||||\n|CNN2GNN[1]|95.51$\\pm$0.42|74.80$\\pm$0.81|60.12$\\pm$1.02|\n|CNN2GNN **+GCR**|95.87$\\pm$0.31|76.23$\\pm$0.38|62.47$\\pm$0.47|\n|CNN2Transformer[1]|95.79$\\pm$0.24|77.39$\\pm$0.20|71.12$\\pm$0.35|\n|CNN2Transformer **+GCR**|**95.96$\\pm$0.35**|**78.23$\\pm$0.30**|**72.33$\\pm$0.31**|\n|ResNet34|||||\n|CNN2GNN[1]|96.39$\\pm$0.41|77.87$\\pm$0.91|61.02$\\pm$0.77|\n|CNN2GNN **+GCR**|96.67$\\pm$0.36|78.14$\\pm$0.54|62.88$\\pm$0.46|\n|CNN2Transformer[1]|96.73$\\pm$0.37|80.10$\\pm$0.45|75.42$\\pm$0.15|\n|CNN2Transformer **+GCR**|**96.97$\\pm$0.36**|**81.27$\\pm$0.29**|**76.67$\\pm$0.26**|\n\n**(ii) Results on ImageNet-1K**\n\n|Model|ViG-Ti[2]|ViG-Ti **+GCR**|ViG-S[2]|ViG-S **+GCR**|ViG-B[2]|ViG-B **+GCR**|\n|-|-|-|-|-|-|-|\n|Accuracy|73.9|**74.9**|80.4|**81.7**|82.3|**84.0**|\n\n\nThese results highlight key insights:\n\n- **Consistent** across architectures: GCR improves performance across diverse backbones (CNN2GNN, CNN2Transformer, ViG), demonstrating broad applicability.\n\n- **Complementary** to existing methods: Its self-regularization mechanism enhances CNN2GNN and CNN2Transformer, indicating **orthogonality to existing graph-based pipelines**.\n\n- **Scalable** to large-scale tasks: **Notable gains** on ImageNet-1K (e.g., +1.7% with ViG-B) show GCR’s effectiveness in complex, real-world settings.\n\n- **Lightweight** and model-agnostic: GCR introduces no extra parameters and integrates easily into existing architectures.\n\nTogether, these findings reinforce GCR’s **novelty and practical value** as a flexible, self-prompted regularization framework applicable across scales and architectures.\n\n**(iii) Generalization to large-scale datasets**\n\nWe also conducted additional experiments on **ImageNet-1K** using transformer-based architectures (iFormer [3], ViT, and ViG [2]). Results, averaged over three runs, are now summarized in Appendix H.3.\n\n||iFormer-S|iFormer-B|ViT-B/16|ViG-B|\n|:-|:-:|:-:|:-:|:-:|\n|Baseline|83.4$\\pm$0.40|84.6$\\pm$0.45|74.3$\\pm$0.51|82.3$\\pm$0.42|\n|Early GCL|83.8$\\pm$0.31|85.0$\\pm$0.40|74.7$\\pm$0.44|82.8$\\pm$0.35|\n|Mid GCL|83.8$\\pm$0.39|85.5$\\pm$0.33|75.2$\\pm$0.36|83.0$\\pm$0.34|\n|Late GCL|84.5$\\pm$0.29|**86.1$\\pm$0.30**|**75.8$\\pm$0.33**|**84.0$\\pm$0.30**|\n|Early+Mid|84.3$\\pm$0.33|85.9$\\pm$0.38|75.6$\\pm$0.41|83.7$\\pm$0.33|\n|Mid+Late|**84.8$\\pm$0.28**|85.9$\\pm$0.37|75.6$\\pm$0.34|83.9$\\pm$0.30|\n|Early+Late|84.5$\\pm$0.30|85.2$\\pm$0.28|74.9$\\pm$0.33|83.5$\\pm$0.29|\n|Full GCL|84.3$\\pm$0.29|85.8$\\pm$0.26|75.5$\\pm$0.30|83.6$\\pm$0.27|\n\nKey results:\n\n- iFormer-S: GCR boosts performance from **83.4%** to **84.8%** (a **+1.4%** gain).\n\n- iFormer-B: GCR improves accuracy from **84.6%** to **86.1%** (a **+1.5%** gain).\n\n- Similar gains are observed with ViT-B/16 and ViG-B, confirming generality.\n\nThese improvements validate that GCR **scales effectively** to large, complex datasets and modern architectures.\n\nWe thank the reviewer again for their constructive feedback and positive assessment. \n\nWe believe these clarifications meaningfully strengthen the paper’s clarity, completeness, and overall impact.\n\n### References\n\n[1] Trivedy, V., & Latecki, L. J. (2023). Cnn2graph: Building graphs for image classification. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (pp. 1-11).\n\n[2] Han, K., Wang, Y., Guo, J., Tang, Y., & Wu, E. (2022). Vision gnn: An image is worth graph of nodes. Advances in neural information processing systems, 35, 8291-8303.\n\n[3] Si, C., Yu, W., Zhou, P., Zhou, Y., Wang, X., & Yan, S. (2022). Inception transformer. *Advances in Neural Information Processing Systems*, *35*, 23495-23509."}, {"review_id": "EyRITG3sPf", "reviewer": "Reviewer_bJ5m", "summary": "This paper introduce a regularisation strategy (named GCR) which encourages the similarity between the features for elements in a batch at a certain layer to reflect the similarity between the output representations (i.e., softmax predictions). This is achieved by constructing a graph representing the similarity between the representations for the elements in a batch at a given layer (using the cosine similarity). The graph for the last layer considers only the relationships between elements with the same class and is used as a \"target\". The proposed regularisation loss is then computed by summing the losses between the similarity matrices at every layer and the target one.\nThe proposed regularisation only guides the training procedure and  does not affect the inference procedure.\n\nSome theoretical results are further provided to show how GCR acts as a \"good\" regularization by reducing the complexity of the model and aligning the Laplacians of the representations of the layers with the target one.\n\nThe authors test their method on classic image classification datasets (CIFAR, Tiny-Imagenet) on a large number of architectures including several vision transformers and convolutional network architectures. Both quantitative results (comparing the same architecture with an without GCR) and qualitative results (showing examples of relationships between representations extracted from models learned with and without GCR) are shown.\n\nThe idea behind the proposed method is based on the knowledge that earlier layers learn more basic features, while later layers learn more \"high-level\" and discriminative features. By then pushing the earlier representations to \"encode\" the same relationships extracted by the later layers, the proposed method should induce the learning of more discriminative patterns, which in turn should help generalization.", "questions": "- Could the authors please add information about the variance across runs of the results? I am happy to raise my score if this is added and the results remain positive\n- Cosine similarity is used to construct the similarity graphs. However, in high dimensions, cosine similarity may not be ideal as many vectors tend to become (nearly) orthogonal to each other. Furthermore it does not take magnitude into account, and it only measures linear relationships. Have the authors tried using any other way of measuring similarity? For example the use of kernels could allow for the introduction of non-linear relationships. I saw there is some limited discussion in the Appendix, but I think it's important to go more in depth on this\n- The abstract mentions that the proposed method reduces inter-class noise. The qualitative results do seem to hint at this, but are there any quantitative results to support this? For example it could interesting to show the confusion matrix of the models trained with and without GCR.\n- It is curious to me that in certain cases the best performance is achieved when GCR is applied to the earlier layers. I would expect the earlier layers to only have to focus on \"basic\" features, which shouldn't necessarily be aligned to the output features (as the distinction into different classes then happens in later layers which combine the basic features from earlier layers). Do you have any comment on why in some cases GCR on earlier layer works better than on later ones?", "limitations": "Yes limitations are discussed in the appendix. As mentioned in my question I think aspect related to the choice of metric should be expanded in my opinion", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- The paper is well written and easy to understand\n- The qualitative results nicely present the possible benefits of the method\n- The theoretical results are a nice addition to provide some possible motivations for GCR, although they do not bring any practical insight\n\nWeaknesses:\n- The quantitative results, contrary to the qualitative ones, show very minor performance improvements. The largest ones are only a few percentage points, while most of them are even smaller.\n- The authors mention that results are averaged over 3 runs, but the results do not show the variance over the multiple runs. Given the small differences in the results (as expressed in the previous point), showing the variance (and possibly increasing the number of runs to 10) is fundamental to be able to judge the effectiveness of the proposed method\n- The abstract mentions that the proposed method reduces inter-class noise, but no quantitative results are shown (see question below for suggestions)\n- Only cosine similarity is used to construct the similarity metric (see question below for more details and suggestion)", "was_revised": true, "final_justification": "The authors have provided significant new results addressing all my concerns. I thus have raised my score to Accept", "rebuttal": "We thank the reviewer for the feedback and appreciation of the paper's clarity and contributions. \n\nBelow, we address all raised points.\n\n### **1. Variance across 10 runs**\n\nAs requested, we report variance across 10 runs. Due to rebuttal space limits, we present Tiny ImageNet results here; **all tables in the revised paper include means and standard deviations over 10 runs**.\n\nTiny ImageNet:\n\n||ViT/32|ViT/16|CeiT|MViT_XXS|MViT_XS|MViT|Swin|MNet|R18SD|SER18|R34|Mean|\n|-|-|-|-|-|-|-|-|-|-|-|-|-|\n|Baseline|37.79$\\pm$0.35|40.05$\\pm$0.33|49.95$\\pm$0.29|49.28$\\pm$0.29|51.58$\\pm$0.27|52.68$\\pm$0.27|54.27$\\pm$0.25|57.81$\\pm$0.25|63.49$\\pm$0.26|65.65$\\pm$0.24|67.51$\\pm$0.25|53.64$\\pm$9.62|\n|Early GCL|39.02$\\pm$0.29|40.98$\\pm$0.19|51.22$\\pm$0.20|50.11$\\pm$0.28|51.33$\\pm$0.26|53.91$\\pm$0.22|54.88$\\pm$0.25|57.93$\\pm$0.21|63.81$\\pm$0.19|66.52$\\pm$0.22|67.79$\\pm$0.19|54.32$\\pm$9.39|\n|Mid GCL|38.61$\\pm$0.23|40.95$\\pm$0.19|50.30$\\pm$0.19|49.92$\\pm$0.26|51.43$\\pm$0.22|53.88$\\pm$0.20|55.23$\\pm$0.24|57.63$\\pm$0.20|64.03$\\pm$0.22|65.66$\\pm$0.23|67.62$\\pm$0.20|54.11$\\pm$9.38|\n|Late GCL|37.98$\\pm$0.28|40.35$\\pm$0.25|50.82$\\pm$0.20|49.77$\\pm$0.21|51.99$\\pm$0.23|54.10$\\pm$0.19|55.47$\\pm$0.21|57.87$\\pm$0.23|63.79$\\pm$0.19|65.85$\\pm$0.25|67.61$\\pm$0.19|54.18$\\pm$9.56|\n|Early+Mid|39.08$\\pm$0.25|41.26$\\pm$0.18|50.25$\\pm$0.25|49.73$\\pm$0.22|51.57$\\pm$0.19|53.91$\\pm$0.23|54.95$\\pm$0.19|57.49$\\pm$0.19|64.18$\\pm$0.20|65.86$\\pm$0.24|67.74$\\pm$0.23|54.18$\\pm$9.32|\n|Mid+Late|38.44$\\pm$0.18|40.52$\\pm$0.28|50.09$\\pm$0.25|50.55$\\pm$0.18|51.48$\\pm$0.21|53.90$\\pm$0.20|55.62$\\pm$0.23|57.65$\\pm$0.21|64.29$\\pm$0.17|65.95$\\pm$0.19|67.58$\\pm$0.21|54.19$\\pm$9.52|\n|Early+Late|38.34$\\pm$0.23|40.71$\\pm$0.21|50.70$\\pm$0.25|50.23$\\pm$0.20|51.36$\\pm$0.18|53.57$\\pm$0.21|54.89$\\pm$0.21|57.93$\\pm$0.19|63.88$\\pm$0.19|65.83$\\pm$0.17|67.75$\\pm$0.25|54.11$\\pm$9.47|\n|Full GCL|38.38$\\pm$0.22|40.80$\\pm$0.18|49.92$\\pm$0.20|50.16$\\pm$0.17|51.87$\\pm$0.19|54.01$\\pm$0.19|54.87$\\pm$0.19|57.64$\\pm$0.20|64.10$\\pm$0.19|66.01$\\pm$0.15|67.66$\\pm$0.18|54.13$\\pm$9.49|\n\nResults confirm **consistent gains** with **low variance** and **statistical stability**. \n\n### **2. Measuring similarity**\n\nWe chose cosine similarity to emphasize directional alignment, which is more semantically meaningful and robust to nuisance factors (e.g., brightness) than raw magnitude. This aligns with common practice in representation learning, where angular relationships often capture class structure more effectively.\n\nWhile kernel methods (e.g., RBF, polynomial) offer expressive similarity functions, our GCLs operate on features already shaped by deep non-linear transformations. Thus, we prioritize simplicity and generality: cosine is efficient, **hyperparameter-free**, and preserves our goal of making GCLs a lightweight, plug-and-play regularizer.\n\nWe tested multiple kernels on MobileNet with CIFAR-100 and found cosine consistently outperforms others, further supporting our design choice. \n\n||Cosine|RBF|Polynomial|Sigmoid|Laplacian|\n|-|-|-|-|-|-|\n|Baseline|65.95$\\pm$0.25|65.95$\\pm$0.25|65.95$\\pm$0.25|65.95$\\pm$0.25|65.95$\\pm$0.25|\n|Early GCL|**67.53$\\pm$0.21**|66.66$\\pm$0.28|66.59$\\pm$0.29|66.63$\\pm$0.30|66.42$\\pm$0.28|\n|Mid GCL|**67.91$\\pm$0.19**|67.04$\\pm$0.24|66.97$\\pm$0.24|67.01$\\pm$0.36|66.80$\\pm$0.29|\n|Late GCL|**68.32$\\pm$0.20**|67.45$\\pm$0.23|67.38$\\pm$0.29|67.42$\\pm$0.29|67.21$\\pm$0.31|\n|Early+Mid|**67.62$\\pm$0.23**|66.75$\\pm$0.24|66.68$\\pm$0.21|66.72$\\pm$0.27|66.51$\\pm$0.29|\n|Mid+Late|**68.26$\\pm$0.18**|67.39$\\pm$0.23|67.32$\\pm$0.23|67.36$\\pm$0.31|67.15$\\pm$0.27|\n|Early+Late|**67.21$\\pm$0.24**|66.34$\\pm$0.21|66.27$\\pm$0.28|66.31$\\pm$0.28|66.10$\\pm$0.26|\n|Full GCL|**68.25$\\pm$0.21**|67.38$\\pm$0.22|67.31$\\pm$0.25|67.35$\\pm$0.27|67.14$\\pm$0.25|\n\nWe’ve expanded Appendix H.4 to reflect this rationale.\n\n### **3. GCR reduces inter-class noise**\n\nWe now provide quantitative evidence supporting our claim that GCR reduces inter-class noise, beyond prior visualizations.\n\nSilhouette score: Higher values indicate tighter intra-class clustering and clearer separation from other classes. Separability ratio: Measures inter-class vs. intra-class distance; higher is better.\n\n- Clustering metrics on CIFAR-10 show GCR improves feature separability and cohesion, with higher silhouette and separability ratios across 10 models (e.g., ResNet34: Silhouette($\\uparrow$) from 0.60 $\\rightarrow$ 0.73, SepRatio ($\\uparrow$) from 3.10 $\\rightarrow$ 4.41), confirming clearer class boundaries.\n\n|Model|Baseline Silhouette|+GCR Silhouette|Baseline SepRatio|+GCR SepRatio|Baseline Confidence|+GCR Confidence|\n|:-|-:|-:|-:|-:|-:|-:|\n|densenet121|0.4724|0.5001|2.2278|2.3325|0.9746|0.9805|\n|shufflenet|0.2806|0.4083|1.7692|2.0472|0.9568|0.9619|\n|squeezenet|-0.1245|-0.0825|1.0008|1.0494|0.9603|0.9660|\n|resnet34|0.6032|0.7314|3.1015|4.4144|0.9801|0.9870|\n|resnet50|0.5314|0.6186|2.5480|3.2294|0.9789|0.9835|\n|resnet101 |0.5641|0.6069|2.7705|3.0793|0.9803|0.9859|\n|resnext50 |0.5298|0.5604|2.6323|2.7941|0.9788|0.9814|\n|resnext101|0.5668|0.6951|2.8387|3.8703|0.9811|0.9880|\n|googlenet |-0.0255|-0.0055|1.1982|1.2065|0.9720|0.9749|\n|Avg|0.3776|**0.4481**|2.2319|**2.6692**|0.9737|**0.9788**|\n\n- Confusion matrices show reduced inter-class confusion. For instance, 'cat–dog' confusion drops from 0.09 to 0.07, and diagonal accuracies improve across several classes (e.g., 'auto': 0.96 $\\rightarrow$ 0.97).\n\n**Baseline**\n||plane|auto|bird|cat|deer|dog|frog|horse|ship|truck|\n|:-|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:--:|:-:|:-:|\n|**plane**|**0.93**|0.01|0.02|0.01|||||0.03|0.01|\n|**auto**|0.01|**0.96**||||||||0.03|\n|**bird**|0.02||**0.88**|0.03|0.02|0.01|0.02|0.01|||\n|**cat**|0.01||0.02|**0.83**|0.02|0.08|||0.01|0.01|\n|**deer**|0.01||0.01|0.02|**0.93**|0.01|0.01|0.01|||\n|**dog**|||0.01|0.09|0.02|**0.86**||0.01|||\n|**frog**|0.01||0.02|0.01|0.01||**0.95**||||\n|**horse**|||0.01|0.02|0.01|0.02||**0.94**|||\n|**ship**|0.02|0.01|||||||**0.95**|0.01|\n|**truck**|0.01|0.04|||||||0.01|**0.94**|\n\n**+GCR**\n||plane|auto|bird|cat|deer|dog|frog|horse|ship|truck|\n|:-|:-:|:-:|:-:|:-:|:-:|:-:|:-:|:--:|:-:|:-:|\n|**plane**|**0.94**||0.02|0.01|||||0.01|0.01|\n|**auto**||**0.97**||||||||0.02|\n|**bird**|0.01||**0.89**|0.03|0.02|0.02|0.01|0.01|||\n|**cat**|0.01||0.01|**0.85**|0.01|0.08||0.01|0.01|0.01|\n|**deer**|0.01||0.01|0.02|**0.94**|0.01|0.01|0.01|||\n|**dog**|||0.01|0.07|0.01|**0.88**||0.01|||\n|**frog**|0.01||0.02|0.01|0.01||**0.95**||||\n|**horse**|||0.01|0.02|0.01|0.01||**0.94**|||\n|**ship**|0.02||||||||**0.96**|0.01|\n|**truck**|0.01|0.02||0.01|||||0.01|**0.95**|\n\nWe have included these analyses in Appendix H.5.\n\n### **4. GCR on earlier layers**\n\n**Part 1**: While later layers are more semantic, we find GCR sometimes works best early, especially on Tiny ImageNet and low-capacity models, due to several factors:\n\n- Early features **show higher noise and misalignment**, which GCR’s adaptive weighting naturally targets.\n\n- Early semantic regularization **helps prune spurious low-level features**, setting the network on a better optimization path.\n\n- Prediction-driven self-prompting lets final-layer structure refine earlier layers via **backpropagated relational signals**.\n\nShallow models benefit more from early guidance, as they downsample aggressively and lack strong inductive biases.\n\nThese points are now clarified in Sec 3.2 and Appendix H.6.\n\n**Part 2**: We present additional experiments below.\n\n- [Exp 1] GCR **corrects** the largest points of misalignment\n\nTo test whether GCR’s impact correlates with a layer’s semantic misalignment, we trained a CeiT model on Tiny ImageNet without GCR and measured the baseline discrepancy $\\delta(l) = ||F(l) - P||_F^2$ for each block $l$. We then applied GCR to individual blocks and recorded the top-1 accuracy gain $\\Delta\\text{Acc}(l)$.\n\nResults show that early layers, bridging low-level features to class concepts, exhibit the highest misalignment and largest gains:  \n\n- Block 1: $\\delta_1 = 0.45$, gain +1.2% \n- Block 2: $\\delta_2 = 0.30$, gain +0.9% \n\nA **strong Pearson correlation** of **0.62** between $\\{\\delta(l)\\}$ and $\\{\\Delta\\text{Acc}(l)\\}$ quantitatively confirms that GCR is more effective where feature-prediction misalignment is greater.\n\n- [Exp 2] early GCR **creates more robust foundational features**\n\nNext, we tested whether early GCR creates more robust features that benefit later layers, a \"feature cleaning\" effect. We trained two ShuffleNet models (5 blocks each) and then froze the regularized blocks to evaluate their standalone quality:\n\n- Model A (Early-GCR): GCLs on Blocks 1-2, frozen after 100 epochs, then fine-tuned remaining blocks.\n\n- Model B (Late-GCR): GCLs on Blocks 4-5, frozen and fine-tuned similarly.\n\n|| Pre-freeze Top-1 | Post-freeze Top-1 | Performance Drop |\n|:-|:-:|:-:|:-:|\n|Model A|66.8%|66.1%|**0.7%**|\n|Model B|66.4%|65.1%|**1.2%**|\n\nModel A’s smaller drop shows early GCR features are more robust and semantically coherent, reducing reliance on later regularization. Model B’s larger drop suggests its performance depends more on continued late-stage regularization, with earlier features remaining entangled.\n\n**Part 3**: We provide evaluations on ImageNet-1K.\n\n||iFormer-S|iFormer-B|ViT-B/16|ViG-B|\n|:-|:-:|:-:|:-:|:-:|\n|Baseline|83.4$\\pm$0.40|84.6$\\pm$0.45|74.3$\\pm$0.51|82.3$\\pm$0.42|\n|Early GCL|83.8$\\pm$0.31|85.0$\\pm$0.40|74.7$\\pm$0.44|82.8$\\pm$0.35|\n|Mid GCL|83.8$\\pm$0.39|85.5$\\pm$0.33|75.2$\\pm$0.36|83.0$\\pm$0.34|\n|Late GCL|84.5$\\pm$0.29|**86.1$\\pm$0.30**|**75.8$\\pm$0.33**|**84.0$\\pm$0.30**|\n|Early+Mid|84.3$\\pm$0.33|85.9$\\pm$0.38|75.6$\\pm$0.41|83.7$\\pm$0.33|\n|Mid+Late|**84.8$\\pm$0.28**|85.9$\\pm$0.37|75.6$\\pm$0.34|83.9$\\pm$0.30|\n|Early+Late|84.5$\\pm$0.30|85.2$\\pm$0.28|74.9$\\pm$0.33|83.5$\\pm$0.29|\n|Full GCL|84.3$\\pm$0.29|85.8$\\pm$0.26|75.5$\\pm$0.30|83.6$\\pm$0.27|\n\nThese results show that GCR scales well and confirm our core insight: aligning feature geometry with prediction semantics strengthens generalization.\n\nAll insights and findings **have been incorporated** into the revised manuscript."}]

Author rebuttal:

Meta-review: This paper proposes Graph Consistency Regularization (GCR), a lightweight and parameter-free framework that aligns intermediate feature similarity graphs with prediction-based graphs to enhance semantic structure and generalization across architectures. Initially, reviewers appreciated the clarity, theoretical grounding, and versatility of the method, but raised concerns about novelty, reliance on its core assumption, limited large-scale evaluation, and small observed improvements. The authors provided extensive rebuttals including new ImageNet-1K experiments, theoretical complexity analysis, variance across multiple runs, quantitative evidence of reduced inter-class noise, and proofs with anti-collapse mechanisms, which satisfactorily addressed most concerns. Overall, I recommend acceptance as the paper is technically solid and broadly applicable, though future work could more explicitly examine its limitations in noisy or imbalanced settings and explore extensions to unsupervised learning.
Decision: Accept (poster)