Controlling Neural Collapse Enhances Out-of-Distribution Detection and Transfer Learning
Authors: Md Yousuf Harun
Jhair Gallardo
Christopher Kanan
Venue: ICML 2025 poster (2025)
Keywords: OOD Generalization
OOD Detection
Neural Collapse
Simplex ETF
Representation Learning
Transfer Learning
Primary area: general_machine_learning->representation_learning
TL;DR: We developed a method for jointly optimizing the out-of-distribution detection and generalization based on the neural collapse phenomenon

Abstract: Out-of-distribution (OOD) detection and OOD generalization are widely studied in Deep Neural Networks (DNNs), yet their relationship remains poorly understood. We empirically show that the degree of Neural Collapse (NC) in a network layer is inversely related with these objectives: stronger NC improves OOD detection but degrades generalization, while weaker NC enhances generalization at the cost of detection. This trade-off suggests that a single feature space cannot simultaneously achieve both tasks. To address this, we develop a theoretical framework linking NC to OOD detection and generalization. We show that entropy regularization mitigates NC to improve generalization, while a fixed Simplex ETF projector enforces NC for better detection. Based on these insights, we propose a method to control NC at different DNN layers. In experiments, our method excels at both tasks across OOD datasets and DNN architectures.

Controlling Neural Collapse Enhances Out-of-Distribution Detection and Transfer Learning
Md Yousuf Harun
Jhair Gallardo
Christopher Kanan" 2
Abstract
OOD Detection
R = 0.77
Out-of-distribution (OOD) detection and OOD generalization are widely studied in Deep Neural Networks (DNNs), yet their relationship remains poorly understood. We empirically show that the degree of Neural Collapse (NC) in a network layer is inversely related with these objectives: stronger NC improves OOD detection but degrades gen- eralization, while weaker NC enhances general- ization at the cost of detection. This trade-off suggests that a single feature space cannot simul- taneously achieve both tasks. To address this, develop a theoretical framework linking NC to OOD detection and generalization. We show that entropy regularization mitigates NC to improve generalization, while a fixed Simplex Equiangular Tight Frame (ETF) projector enforces NC for bet- detection. Based on these insights, we propose method to control NC at different DNN layers. In experiments, method excels at both tasks across OOD datasets and DNN architectures.
OOD Transfer
R= - 0.60
Normalized OOD Detection Error
1.0
1.0
Normalized OOD Transfer Error
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.4
0.5
0.6
0.7
0.8
Stronger Neural Collapse (NC1)
Figure I: In this paper, we show that there is a close inverse relationship between OOD detection and generalization with respect the degree of representation collapse in DNN layers. This plot illustrates this relationship for VGG17 pre- trained on ImageNet- 100 using four OOD datasets, where we measure collapse and OOD performance for various lay- ers. For OOD detection, there is strong positive Pearson correlation (R 0.77) with the degree neural collapse (NC1) in DNN layer, whereas for OOD generalization, there is strong negative correlation (R -0.60). We rigorously examine this inverse relationship and propose method to control NC at different layers.
1. Introduction
(OOD) detection and OOD generaliza- tion are two fundamental challenges in deep learning. OOD detection enables deep neural networks (DNNs) to reject unfamiliar inputs, preventing overconfident mispredictions, while OOD generalization allows DNNs to transfer their knowledge new distributions. For applications like open- world learning, where a DNN continuously encounters new concepts, both capabilities are essential: OOD detection enables new concepts to be detected, while OOD generaliza- tion facilitates forward transfer to improve learning of these new concepts. Despite their importance, these tasks have primarily been studied in isolation. Here, empirically and theoretically demonstrate link between both tasks and
Neural Collapse (NC), as illustrated in Fig.
NC is phenomenon where DNNs develop compact and structured class representations (Papyan et al., 2020). While NC was first identified in the final hidden layer, later work has found that it occurs to varying degrees in the last K DNN layers (Rangamani et al., 2023; Harun et al., 2024; Sákeník et al., 2024). NC has a major impact on both OOD detection and generalization. Strong NC improves OOD detection by forming tightly clustered class features that enhance separation between in-distribution (ID) and OOD data (Haas et al., 2023; Wu et al., 2024b; Ming et al., 2022). Conversely, NC impairs OOD generalization by reducing feature diversity, making it harder to transfer knowledge to novel distributions (Kothapalli, 2023; Masarezyk et al., 2023; Harun et al., 2024). However, past work has consid-
'Rochester Institute of Technology University of Rochester. Correspondence to: Md Yousuf Harun Emh1023@rit.ed
Proceedings of the nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Controlling Neural Collapse Enhances
Detection and Transfer Learning
ered NC in the context of either OOD detection or OOD generalization individually, leaving open the question of how NC affects both tasks sinultaneously. To the best of our knowledge, no prior work has theoretically or empiri- cally examined this relationship.
and classify low-density test points as OOD (Lee al., 2018; Zisselman & Tamar, 2020; Choi et al., 2018; Jiang et al., 2023), More commonly, confidence-based approaches estimate OOD likelihood using model outputs (Hendrycks & Gimpel, 2016; Liang et al.. 2017; Liu et al., 2020), feature statistics (Sun et al., 2021; Zhu et al., 2022a; Sun et al., 2022), or gradient-based information (Huang et al., 2021; Wu et al., 2024a; Lee et al., 2023: Igoe et al., 2022).
Here, we establish that the NC exhibited by a DNN layer has an inverse relationship with OOD detection and OOD generalization: stronger NC improves OOD detection but degrades generalization, while weaker NC enhances gener- alization at the cost of detection performance. This trade-off suggests that single feature space cannot effectively opti- mize both tasks, motivating the need for a novel approach.
Since post-hoc methods depend on the representations learned during ID training, their effectiveness is fundamen- tally constrained by the quality of those features (Roady et al., 2020). Highly compact, well-separated ID representa- tions generally improve OOD detection by reducing feature overlap with OOD samples. For example, Haas et al. (2023) demonstrated that L2 normalization of penultimate-layer features induces NC, enhancing ID-OOD separability, Simi- larly, Wu et al. (2024b) introduced a regularization loss that enforces orthogonality between ID and OOD representa- tions, leveraging NC-like properties to improve detection. NECO (Ammar et al.. 2024), a post-hoc OOD detection method, leverages NC and the orthogonality between II) and OOD samples to achieve state-of-the-art performance. However, unlike our approach, NECO and other methods do not focus on OOD generalization or representation learning.
We propose a framework that strategically controls NC at different DNN layers to optimize both OOD detection and OOD generalization. We introduce entropy regularization to mitigate NC in the encoder, improving feature diversity and enhancing generalization. Simultaneously, we leverage a fixed Simplex Equiangular Tight Frame (ETF) projector to induce NC in the classification layer, improving feature compactness and enhancing detection. This design enables our DNNs to decouple representations for detection and generalization, optimizing both objectives simultaneously.
Our key contributions are follows:
1. We present the first unified study linking Neural Col- lapse to both OOD detection and OOD generalization, empirically demonstrating their inverse relationship and extending analyses of NC beyond the final hidden layer.
Another representation learning approach is to learn repre- sentations explicitly tailored for OOD detection by incorpo- rating OOD samples during training (Wu et al., 2024b; Bai et al., 2023; Katz-Samuels et al., 2022; Ming et al., 2022). These methods encourage models to assign lower confi- dence (Hendrycks et al., 2018) or higher energy (Liu et al., 2020) to OOD inputs. However, this approach presents sig- nificant challenges, as the space of possible OOD data is essentially infinite, making it impractical to represent all potential OOD variations. Moreover, strong OOD detec- tion performance often comes at the cost of degraded OOD generalization (Zhang et al., 2024), as representations op- timized for separability may lack the diversity needed for adaptation to novel distributions.
2. We develop a theoretical framework that explains how entropy regularization mitigates NC to improve OOD generalization. Additionally, we empirically demonstrate that a fixed Simplex ETF projector enforces NC, en- abling effective OOD detection.
3. In extensive experiments on diverse OOD datasets and DNN architectures, we demonstrate the efficacy of our method compared to baselines."
2. Background
2.1. OOD Detection
2.2. Transfer Learning and OOD Generalization
OOD detection methods aim to separate ID and OOD sam- ples by leveraging the differences between their feature representations. Most existing OOD detection methods are post-hoe, meaning they apply a scoring function to model trained exclusively on ID data, without modifying the train- ing process (Salehi et al., 2022). These methods inherently rely on the properties of the learned feature space to distin- guish ID from OOD samples.
Transfer learning and OOD generalization methods focus on learning features that remain effective across distribution shifts. Robust transfer is particularly important in open- world learning scenarios, where models must not only adapt to new distributions but also improve sample efficiency over time, a key requirement for continual learning. To facili- tate generalization, techniques such as feature alignment (Li et al., 2018b; Ahuja et al., 2021: Zhao et al., 2020; Ming et al., 2024), bosemble/mtta-learnig (Balaji et al., 2018; Li et al., 2018a; 2019; Bui al., 2021), robust optimiza- tion (Rame et al., 2022; Cha et al., 2021; Krueger al., 2021; Shi et al., 2021), data augmentation (Nam et al., 2021; Nuriel et al., 2021; Zhou et al., 2020), and feature disentan-
Post-hoc detection techniques can be broadly categorized based on the source of their confidence estimates. Density- based methods model the ID distribution probabilistically
Code:
gi thub, io/ncoodg
2

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Controlling Neural Collapse Enhances
Detection and Transfer Learning
glement (Zhang et al., 2022) have been proposed.
is trained on ID data drawn from Pxy. to produce logits, (a) which are used to predict labels. For robust operation in real-world scenarios, the model must classify samples from Pxy correctly and identify OOD samples from X'' which represents the distribution with no overlap with the ID label space, i.e., y n y" y==0. 0.
Key properties of learned features significantly impact gen- eralization to unseen distributions. Studies examining fac- tors that affect OOD generalization emphasize that feature diversity is essential for robustness (Masarczyk et al., 2023; Harun et al., 2024; Komblith et al., 2021; Fang et al., 2024; Ramanujan et al., 2024; Kolesnikov et al., 2020; Vishni- akov et al., 2024). Notably, recent work (Kothapalli, 2023; Masarczyk et al., 2023; Harun et al., 2024) suggests that progressive feature compression in deeper layers, linked to NC emergence, can hinder OOD generalization by reducing representation expressivity.
At test time, the objective of OOD detection is to determine whether given sample x originates from the ID or an OOD source. This can be achieved by using a threshold-basee decision rule through level set estimation, defined as:
S(x) 2 A OOOD S(x) <
ID,
Gx(x)
2.3. Neural Collapse
where S(-) is scoring function, and samples with S(x) > À are classified as ID, while those with S(x) < A are classi- fied as OOD. À denotes the threshold.
As noted earlier, NC arises when class features become tightly clustered, often converging toward Simplex ETF (Papyan et al., 2020; Kothapalli, 2023; Zhu et al., 2021; Han et al., 2022). Initially. NC was studied primarily in the final hidden layer, but later work demonstrated that NC manifests to varying degrees in earlier layers as well (Ranga- mani et al., 2023; Harun al., 2024). In image classification experiments, Harun et al. (2024) showed that the degree of intermediate NC is heavily influenced by the properties of the training data, including the number of ID classes, image resolution, and the use of augmentations.
On the OOD generalization part, the objective is to build a model fā : X' - KRK using the ID data such that it learns transferable representations and becomes adept both ID task and OOD downstream tasks. This is challenging problem since we do not have access to OOD data during training. In both OOD detection and OOD generalization, the label space is disjoint between ID and OOD sets.
Differences from prior works. Prior works (Zhang et al., 2024; Wang & Li, 2024; Bai et al., 2023) focusing on OOD detection and OOD generalization, define the problem dif- ferently than us. For OOD detection, they use "semantic OOD" data from X3' that has no semantic overlap with known label space from PID i.e., y' = 0. However, for OOD generalization, they use "covariate-shifled OOD" data from Pxyy that has the same label space as PID but with shifted marginal distributions pCov due to noise or corruption. Furthermore, they use additional semantic OOD training data, xx' during the training phase. Our problem definition is fundamentally more challenging and practical than the prior works because: (I) we aim to detect semantic OOD samples, PSEM without access to auxiliary OOD data during training, and (2) we aim to generalize to semantic OOD samples that belong to novel semantic categories.
NC can be characterized by four main properties:
1. Feature Collapse (NC1): Features within each class concentrate around a single mean, exhibiting minimal intra-class variability.
2. Simplex ETF Structure (NC2):: When centered at the global mean, class means lie on a hypersphere with max- imal pairwise distances, forming a Simplex ETF.
3. Self-Duality (NC3): The last-layer classifiers align tightly with their corresponding class means, creating a nearly self-dual configuration.
Nearest Class Mean Decision (NC4): Classification behaves like nearest-centroid scheme, assigning classes based on proximity to class means.
While NC's structured representations can aid OOD detec- tion by ensuring strong class separability (Haas et al., 2023; Wu et al., 2024b), the same compression may limit the feature diversity needed for generalization. One proposed explanation is the Tunnel Effect Hypothesis (Masarczyk et al., 2023), which suggests that as features become in- creasingly compressed in deeper layers, generalization to unseen distributions is impeded.
Evaluation Metrics. We define IDD generalization error (Ew), OOD generalization error (EGEN), and OOD detection error (ÉDET) as follows:
1. I. EiD == 1 - E(z. (( pm (I(((f(TT) = v}),
2. 1 EGEN :- 1 - B(z,g) -~ poco (I()(o(Z)) v}).
3. & Ê == Enon (1)
where II(-) denotes the indicator function, and the arrows indicate that lower is better. For OOD detection, ID) samples are considered positive. FPR95 (false positive rate at 95% true positive rate) is used as EoET. Details are in Appendix.
3. Problem Definition
We consider a supervised multi-class classification problem where the input space is X and the label space is y = (1, 2, 2..,.... KJ. A model parameterized by 0, so X + RK
OOD Detection. Following earlier work (Sun et al., 2021;
3

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Controlling Neural Collapse Enhances
Detection and Transfer Learning
Liu et al., 2020), we consider the energy-based scoring (Liu al., 2020) since it operates with logits and does require any fine-tuning or hyper-parametens, Scoring in energy- based models is defined as
among all layers. Whereas the encoder alone is utilized for OOD generalization, assuming encoder embedding f is most transferable among all layers. In the following subsec- tions, we will portray how we can build these collapsed and transferable representations.
k exp (x))
S(x) = - log L
4.1. Entropy Regularization Mitigates Neural Collapse
In this section, we provide a theoretical justification for using an entropy regularizer to prevent or mitigate interme- diate neural collapse (NC1) in deep networks. By "interme- diate" we mean that the collapse occurs in hidden layers.
where the k-th logit, (x:), denotes the model's 's confidence for assigning to class k. Note that Liu et al. (2020) uses the negative energy. meaning that OOD samples should obtain high energy, hence low S(x). See Fig. as an example.
OOD Generalization. For evaluating OOD generaliza- tion, we consider linear probing which is widely used to evaluate the transferability of learned embeddings to OOD datasets (Alain & Bengio, 2016; Masarczyk et al., 2023;
Setup and Notation. Let I. be the total number of layers
in our network, and é (1, , L the intermediate layer index. We denote the embedding (activation) in layer e for
the i-th sample X; as Z4,5 fe(x;), where Zí RRd Suppose we have K classes, labeled by 1..... K. We can view the random variable Ze (the layer-& embeddings) as distributed under the data distribution according to
Zhu et al., 2022b; Waldis et al., 2024; Grill et al., 2020;
He et al., 2020). For a given OOD dataset, we extract em- beddings from a pre-trained model. A linear probe (MIP classifier) is then attached to map these embeddings to OOD classes. The probe is trained and evaluated on OOD data.
pe( = Σ Th (z)
Mitigating NC Holps 000 Gonoralization
Increasing NC Helps Dotection
where Tu = Pr(y = k) is the class prior, and pe,*(z) is the class condiiooniiio distribution of Ze for label k.
Encoder
Projector
Classifier Head
Outputs
Inputs
Endeddinge fram
Endadding from Projezar
Intermediate Neural Collapse (NC1). Empirically, neu- ral collapse is observed when the within-class covariance of these embeddings shrinks as training proceeds. Formally, for each class k the distribution pt, concentrates around its class mean Me,e E resulting in:
Escoder
Implication of Neural Collapse. Mitigating NC in the encoder enhances OOD generalization whereas increasing NC in the projector improves OOD detection.
Figure 2:
Trace Eg,x) - 0, where Σe, = Cov 1 = k).
4. Controlling Neural Collapse
Typically, penultimate-laye embeddings from pre-trained DNN are used for downstream tasks. However, using the same embedding space for both OOD detection and OOD generalization is suboptimal due to their confficting objec- tives. We therefore propose separate embedding spaces at different onyers-one for OOD detection, another for OOD generalization. Specifically, we attach a projectos net- work s(-) to the DNN backbone f(() (the encoder) and add a classifien x, the encoder outputs f = f(x), e.g., a 512-dimensional vector for ResNet18. The projector then maps f to gog- = gf)), and finally the classifier produces logits h(g) E R
Although often highlighted in the penultimate layer, such collapse can appear across the final layers of a DNN (Ranga- mani et al., 2023; Harun et al., 2024).
Differential Entropy and Collapsing Distributions. For continuous random variable Z₃ E Re with density pr(z), the differential entropy is given by
H(pr) = - J pe(z) log dz Jd
head ii(-) on top. Given an input
It is well known that if Pe, collapses to a delta (or near-delta) around Pe then (PEA)) - -0o (Cover, 1999). Consequently, a mixture of such collapsing class- conditional distributions also attains arbitrarily negative en- tropy. The following proposition formalizes this point.
The encoder trained prevent NC and encourage trans- ferable representations for OOD generalization, while the projector is designed to induce NC, producing collapsed representations beneficial for OOD detection. A high-level illustration is provided in Fig. 2. For OOD detection and ID classification tasks, the entire network (h o g f) is utilized, assuming projector embedding g most discriminative
Proposition 4.1 (Entropy under Class-Condiitioall Col- lapse). Consider a mixture distribution pe(z) Σ TR (z) on Rd,, Suppose that, for each k. pe,k be- comes arbitrarily concentrated around a single point p
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Controlling Neural Collapse Enhances Out-of-Distribution Detection and Transfer Learning
In the limit where each pe,k approaches a Dirac delta, the differential entropy H (pe) diverges to -CC.
can be sensitive to outliers with large magnitudes. Therefore, in our method, the loss operates on the hyperspherical em- bedding space with the unit norm, i.e., 2 = z/ 2/||z||2. Note that unlike Lete acting on classifier head, Lreg is applied in encoder for mitigating NC.
Proof Sketch. If each & a family of densities approach- ing ((z-Ht,,): the individual entropies H (pt,s) go to -o.. The entropy of the mixture can be bounded above by the weighted sum of H (pt,)) plus a constant that depends the mixture overlap. Hence, the overall mixture entropy also diverges to -oc.
Although various loss functions including cross-entropy (CE) and mean-squared-enn (MSE) lead to NC, others pro- duce less transferable features than CE (Zhou al., 2022; Kornblith et al., 2021). We also find that CE outperforms MSE in both OOD detection and OOD generalization (see Table 9). Therefore, we consider CE loss for Lels in Equa- tion 1. It has been found that using label smoothing with CE loss intensifies NC properties when compared with the regular CE loss (Zhou et al., 2022; Kornblith et al., 2021). Therefore, we use label smoothing with CE loss to expedite NC properties in the projector and classifier head.
Appendix E contains the detailed proof.
Entropy Regularization to Mitigate Collapse. We see
from Proposition 4.1 that if all consicconiii distribu- tions collapse to near-deltas, the layer's overall density pu (x) has differential entropy H (pt) + -oo. Since standard clas- sification objectives can favor very tight class clusters (e.g..
In addition to Lreg mitigating NC, we consider alternatives to batch normalization (BN). In the context of learning trans- ferable representations in the encoder, batch dependency, es- pecially using BN, is sub-optimal as OOD data statistically differs from ID data. Therefore, for all layers in the encoder, we replace batch normalization with batch-independent alternative, particularly, a combination of group normaliza- tion (GN) (Wu & He, 2018) and weight standardization (WS) (Qiao et al., 2019) to enhance OOD generalization.
to sharpen decision boundaries), one can counteract this by maximizing H (pe).
Concretely, we augment the training loss (0) with negative-entropy penalty:
(O) (0) a all 0));
(L)
where > hyperparameter. As Στ, + 0 would force H (pe) 30 (cf. Proposition 4.1), the additional term -all (pt) becomes unboundedly large. Therefore, the model compelled to maintain nonzero within class variance for each class distribution, preventing complete layer collapse.
4.2. Simplex ETF Projector for Inducing Collapse
When a DNN enters into NC phase, the class-means con- verge to a simplex ETF (equinorm and maximal equiangu- larity) in collapsed layers (NC2 criterion). This implies that fixing the collapsed layers to be ETFs does not impair ID performance (Rangamani et al., 2023; Zhu et al., 2021). In this work, we induce NC in the projector to improve OOD detection performance. We do it by fixing the projector to be simplex ETF, acting as architectural inductive bias.
Since we do not have direct access to (z(z), we need to estimate H (pe) using data-driven density estimation ap- proach. In particular, prior work (Kozachenko & Leonenko, 1987; Beirlant et al., 1997) shows that the differential en- tropy can be estimated by nearest neighbor distances.
Given a batch of N random representations {z, \N the nearest neighbor entropy estimate is given by
Our projector comprises two MLP layers sandwiched be- tween encoder and classifier head. We set the projector weights to simplex ETFs and keep them frozen during train- ing. In particular, each MLP layer set to be a rank D 1 simplex ETF, where D denotes width or output feature di- mension. The rank D canonical simplex ETF is:
1 N
(pe)
log
N min
zill2 2+ xE[N]..stm
|Iz,,
where EC denotes the Euler constant. For purposes, we can simplify the entropy maximization objective by remov- ing affine terms, resulting in the following loss function:
D D-1 (ID D 1011)
Details on the projector are given in Appendix A.I. We further apply L2 normalization to the output of the projector since it constraints features to achieve equinormality and helps induce early neural collapse (Haas et al., 2023).
N
Crogg (0) = N
Σ log
min iE(N).
1.. 2il12
Total loss becomes: (0) = (0) + a (O) In- tuitively, Creg maximizes the distance between the nearest pairs in the batch, encouraging an even spread of represen- tations across the embedding space. The pairwise distances
While prior work has found that incorporating a projec- tor improves transfer in supervised learning (Wang et al., 2022a), the objective of our projector is to impede transfer.
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Controlling Neural Collapse Enhances Out-of-Distribation Detection and Transfer Learning
The difference is that projector is typically trained along with the backhone, whereas in our method the projector is configured as Simplex ETF and kept frozen during training.
Table 1: Main Results (Encover Vs. Projector). Vari- ous DNNs are trained on ImageNet- dataset (ID) and evaluated on eight OOD datasets. All models incorporate entropy regularization and the ETF projector to control NC. Reported &GEN (%) and TDA (%) are averaged over eight OOD datasets. A lower .NC indicates stronger neural collapse. + +DE-P and -Ag-p indicate % increase and % decrease respectively, when changing from the encoder (E) to projector (P).
5. Experimental Setup
Datasets. For ID dataset, we use ImageNet-100 (Tian et al., 2020)- subset (100 classes) of ImageNet- IK (Russakovsky et al., 2015). To assess OOD generalization and OOD detection, we study eight commonly used OOD datasets: NINCO (Bitterwolf et al., 2023), ImageNet-R (Hendrycks et al., 2021), CIFAR-100 (Krizhevsky & Hinton, 2014), Oxford 102 Flowers (Nilsback & Zisserman, 2008). CUB- 200 (Wah et al., 2011), Aircrafts (Maji et al., 2013), Oxford- IIT Pets (Parkhi et al., 2012), and 10 (Coates et al., 2011). Dataset details are given in Appendix B.
Model
Em
Neural Collapse
ECEN Avg.
EIT Avg.
NC1
NC2
.C33
NC44
VGG17
Projector
12.62
0.393
0.490
0.468
0.316
66.36
65.10
Encoder
15.52
2.175
0.603
0.616
5.364
41.85
87.62
Ag-P
-18669
-81.93
-18.74
-24.03
-94.11
+58.57
-25,70
ResNet18
Projector
16.14
0.341
0.456
0.306
0.540
63.08
69.70
Encader
20.14
1.762
0.552
0.555
10.695
47.72
86.
DNN Architectures. We train and evaluate representa- tive set of DNN architectures including VGG (Simonyan & Zisserman, 2015), ResNet (He et al., 2016), and ViT (Doso- vitskiy et al., 2020). In total, we experiment with five back- bones: VGG17, ResNet18, ResNet34, ViT-Tiny, and ViT- Small. Our projector composed of two MLP layers for all DNN architectures. Details are given in Appendix A.
Ag-i
19.86
-80.65
-17.39
-4.86
-94.95
+32.19
-19.11
VII-T
Projector
32.04
2.748
0.609
0.798
1.144
63.53
83.16
Encader
33.94
5.769
0.748
0.847
2.332
52.63
90.89
Ag SP
-5.60
-52.37
-18.58
-5.79
-50.94
+20.71
-8.50
setup will require major modifications, hence we cannot compare directly with them. We compare the proposed method with baselines that do not use any of our mech- anisms c.g., entropy regularization or fixed simplex ETF projector. Additionally, in Sec. 6.5, we include compar- ison with NECO (Ammar et al., 2024), a state-of-ther the- OOD detection method that leverages NC properties.
NC Metrics (NC1-NC4), We use four metrics, .NC1, NC2, .N3,, and NC4, as described in (Zhu et al., 2021; Zhou et al., 2022), to evaluate the NC properties of the DNN features and classifier. These metrics correspond to four NC properties outlined in Sec. 2.3. Note that NC1 is the most dominant indicator of neural collapse. We deseribe each NC metric in detail in the Appendix C.
6. Experimental Results
Training Details. In our main experiments, we train differ- ent DNN architectures e.g., VGG17, ResNet18, and ViT-T on ImageNet-100 for 100 epochs. The Entropy regular- ization loss Greg is modulated with a = 0.05. We use AdamW (Loshchilov, 2017) optimizer and cosine learning rate scheduler with a linear warmup of 5 epochs. For a batch size of 512, set the learning rate to 6 x 10-3 for VGG17, 0.01 for ResNer18, and 8 x 10-² for VIT-T. For all models, we set the weight decay to 0.05 and the label smoothing to 0.1. In all our experiments, we use 224 x 224 images. And, we use random resized crop and random hor- izontal flip augmentations. Linear probes are attached to the encoder and projector layers of pre-trained model and trained on extracted embeddings of OOD data using the AdamW optimizer and CE loss for 30 epochs. Additional implementation details are given in Appendix A.
Sec. 6.1 shows how controlling NC improves representa- tions for OOD detection and generalization. We compare with baselines in Sec. 6.2, and analyze the roles of entropy regularization and the ETF projector in Sec. 6.3 and Sec. 6.4, respectively. Sec. 6.5 presents comparison with NECO, and Sec. 6.6 summarizes additional results.
6.1. Impact of Controlling NC
We investigate whether controlling NC improves OOD de- tection and generalization by examining NC properties in the encoder and projector. Table I summarizes the results across eight OOD datasets. The projector, which exhibits lower NC values (i.e., stronger NC), achieves superior OOD detection (7. 17.73%-22.52%% margin) and lower ID error com- pared to the encoder. In contrast, the encoder's 's higher NC values (i.e., weaker NC) lead to better OOD generalization 110.9%%-44.51%% 51% margin) than the projector. Comprehen- sive results across OOD datasets are given in Appendix F.
Baselines. Recent work defines the problem differently where they focus on OOD detection and covariate OOD generalization (same labels but different input distribution). Our problem setup focuses on OOD detection and seman- tic OOD generalization (different labels and different in- put distribution). Adapting other methods to our problem
We also visualize the encoder and projector embeddings in Fig and 8 for deeper insights. Unlike encoder embeddings, projector embeddings cluster tightly around class means
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Controlling Neural Collapse Enhances Out-of-Distribation Detection and Transfer Learning
Encoder Embeddings
Projector Embeddiing
Table 3: Impact of Entropy Regularization. VGG17 mod- cls are trained on ImageNet-100 dataset (ID) and evaluated on eight OOD datasets. Both models share the same archi- tecture and use an ETF projector: the difference lies solely in the use of entropy regularization. NC metrics measured with encoder embeddings where entropy regularization is applied or omitted. Reported EGEN (%) and EDET (%) cor- respond to the encoder and projector, respectively and are averaged across eight OOD datasets.
Method
Eip 4.
Neural Collapse
t
EGEN
E
Figure 3: UMAP Visualization of Embedding. The projec- tor embeddings exhibit much greater NC (NC1- 0.393) than the encoder embeddings (NC1 = 2.175) as indicated by the formation of compact clusters around class means. For clarity, we highlight 10 ImageNet classes by distinct colors. For this, we use ImageNet- pre-trained VGG17.
NC1
NC2
NC3
NC4
Avg. 4
Avg.
No Reg.
13.46
1.31
0.72
0.62
5.18
44.56
67.46
Reg.
12.62
2.18
0.61
0.62
5,36
41.85
65.10
DNNs including VGG17, ResNet18, and ViT-T land on higher OOD detection error and OOD generalization error indicating that representations learned by these models can- not achieve both OOD detection and OOD generalization abilities. In contrast, our method shows significant improve- ments over these baselines. While being competitive in ID performance, our method controls NC unlike the baselines, and achieves better performance in OOD tasks. Particularly, OOD generalization is improved by 1.70% 7.69% (abso- lute) and OOD detection is improved by 7.01 % 29.82% (absolute). More comprehensive results across eight OOD datasets are given in Table 18 in the Appendix.
Table 2: Comparison with Baseline. Various DNNs e.g.. VGG17, R, and ViT-T are trained on ImageNet- 100 dataset (ID). Baseline models do not incorporate mecha- nisms like entropy regularization or the ETF projector to control NC. NC metrics are computed using the penultimate- layer embeddings. Reported EGEN (%) and DEE (%) are averaged over eight OOD) datasets.
Model
Em
Neural Collapse
EGEN
ÉpEr Avg. J
4
.C1
.C2
.C33
NC4
Avg.
VGG17
12.18
0.766
0.705
0.486
37.491
49.54
94.92
+Ours
12.62
0.393
0.490
0.468
0,316
41.85
65.10
ResNet18
15.38
1.1
0.658
0,590
31.446
49.42
97.40
6.3. Entropy Regularization Mitigates NC
+Ours
16.14
0.341
0.456
0.306
0.540
47.72
69.70
At first, we measure entropy and NC1 across all VGG17 lay- ers and observe that there lies strong correlation between entropy and NC1 (Pearson correlation 0.88). As illustrated in Fig. 4, stronger NC correlates with lower entropy whereas weaker NC correlates with higher entropy. This empirically demonstrates why using entropy regularization mitigates NC in the encoder. Next, we compare two identical VGG17 models, one uses entropy regularization and another omits it. The results are summarized in Table 3. Entropy regu- larization mitigates NC in the encoder, as evidenced by its higher NCC1, and achieves better performance in all criteria compared to the model without entropy regularization.
VIT-T
31.78
2.467
0.657
0.601
1.015
52.68
90.17
+Ours
32,04
2.748
0.609
0.798
1.144
52.63
83.16
(reflecting stronger NC1). Additionally, Fig. 5a in the Ap- pendix shows that the projector exhibits greater ID-OOD separation than the encoder. Finally, Fig. 5b and 6 in the Appendix show that the energy score distribution reveals the projector separates ID from OOD more effectively than the encoder across multiple datasets. These observations explain why the projector excels at OOD detection by ex- ploiting more collapsed features.
Finally, we analyze different layers of VGG17 and ResNet18 models and find that increasing NC strongly cor- relates with lower OOD detection error and reducing NC strongly correlates with lower OOD generalization error, as shown in Fig. and 7. Our experimental results validate that controlling NC effectively enhances OOD detection and OOD generalization abilities.
An implication of NC is that the collapsed layers exhibit lower rank in the weights and representations (Rangamani et al., 2023). In additional analyses provided in the Ap- pendix G, we observe that our entropy regularization im- plicitly encourages higher rank in the encoder embeddings and helps reduce dependence between dimensions (thereby promoting mutual independence).
6.2. Comparison with Baseline
6.4. Fixed Simplex ETF Projector Induces NC
We want to check how standard DNNs perform without any mechanisms to control NC. As depicted in Table 2, different
We train two identical models the same way (same hyperpa- rameters and training protocol), one of them uses a regular
7

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Controlling Neural Collapse Enhances
Detection and Transfer Learning
1.0
Table 5: NECO Vs. Our Method. Various DNNs e.g., VGG17, ResNet18, and ViT-T are trained on ImageNet- dataset (ID). Reported OOD detection error, ÊDET (%) averaged across eight OOD datasets.
R
0.88
0.8
Entropy
0.66
Method
Avg. DDE (%) 4
0.4
VGGI7
R8
VIT-T
NECO
77.82
88.13
85.67
0.2
Ours
65.10 (-12.72)
69.70 (-18.43)
83.16 (-2.51)
0.4
0.5
0.6
0.7
0.8
0.9
<- Stronger Neural Collapse (NC1)
Table 6: Projector Configuration. VGG17 models with dif- ferent ETF projector configurations are trained on ImageNet- 100 (ID) dataset. D and W denote the depth and width of the projector, respectively. Reported EGEN (%) and EpET (%) are averaged over eight OOD datasets.
Figure 4: Neural collapse (NC1) correlates with entropy. The stronger the neural collapse, the lower the entropy and vice-versa. This suggests that increasing the entropy of the encoder's 's embeddings can help mitigate NC and enhance OOD generalization. Similar to Fig. 1, we analyze differ- ent layers of VGG17 networks that are pre-trained on the ImageNet- 100 (ID) dataset. R denotes the Pearson correla- tion coefficient.
Config.
Emn
Neural Collapse NC2
+
EGEN Avg. 4
EDET
+
.C11
NC3
NC4
Avg.
12,86
0.375
0.649
0,500
1.57
45.37
87.37
12.62
0,393
0.490
0.468
0.316
41.85
65.10
2
13.48
0.320
0.667
0.376
0.493
43.33
69.73
Table 4: Impact of Fixed ETF Projector and L2 Nor- malization on NC. The evaluation is based on projector embeddings of ImageNet- 100 pre-trained VGG17 networks. lower NC indicates higher neural collapse. Our model (highlighted) uses a fixed ETF projector with L2 normaliza- tion, whereas the baseline Plastic uses a trainable projector with L2 normalization, and the baseline No L2 norm uses fixed ETF projector but omits L2 normalization. Reported EDET (%) is averaged over eight OOD datasets.
6.5. State-of-the-art Comparison
To put our work in context with respect to existing methods, we compare our method with NECO (Ammar et al., 2024), a state-of-the-art OOD detection method based on NC. Since NECO does not address OOD generalization, we restrict this comparison OOD detection only. We train multi- ple DNN architectures on ImageNet-100 (ID) and evaluate their performance on eight OOD datasets. Remarkably, our method consistently outperforms NECO across all settings. As shown in Table 5, our approach reduces the average OOD detection error by an absolute margin 12.72% for VGG17, 18.43% for ResNet18, and 2.51% for ViT-T, highlighting its superior effectiveness. Comprehensive results across OOD datasets are presented in Table 22.
Projector
Em
Neural Collapse NC2
EDE Avg.
NC1
NC3
NC4
Plastic
15.10
0.498
0.515
0.428
1.422
74.00
Fixed ETF
12.62
0.393
0.490
0.468
0.316
65.10
No Ly norm
12.74
0.579
0.538
0.349
1.339
68.93
Lg norm
12.62
0.393
0.490
0.468
0.316
65.10
6.6. A Ablation Studies
trainable projector and the other one uses frozen simplex ETF projector. We summarize our findings in Table 4. Our results indicate that the fixed simplex ETF projector strength- ens NC more than regular plastic projector as evidenced by lower AC1. Consequently, the ETF projector outperforms plastic projector in OOD detection by an absolute 8.9%
Projector Design Criteria. Here ask: does a deeper wider projector achieve higher performance? Results are summarized in Table 6. We find that the projector with depth 2 performs better than shallower or wider projectors. Table 19 in the Appendix contains comprehensive results.
We also evaluate the impact of Le normalization on the projector embeddings. We train two models in an identical setting, the only variable we change is the L2 normaliza- tion. We observe that L2 normalization achieves a lower NC1 value (thereby strengthening NC) and 3.83% (abso- lute) lower OOD detection еттоr than its counterpart. These results demonstrate that using L2 normalization helps in- duce NC and thereby enhances OOD detection performance. Additional results are shown in Appendix H.
Group Normalization Enhances Transfer. While the im- pact of BN on NC has been studied in prior work (Pan & Cao, 2023; Ergen et al., 2022), we evaluate the effectiveness of both BN and GN within our framework. We compare BN with GN (GN is combined with WS) and show the results in Table 7. We find that GN helps mitigate NC in the encoder as indicated by a higher AC1 value than BN. This implies that, unlike GN, BN leads to stronger NC and impairs OOD transfer. This further confirmed by GN outperforming BN
8

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Controlling Neural Collapse Enhances Out-of-Distribuuie Detection and Transfer Learning
Table 7: Impact of Group Normalization. Reported EGEN (%) and EDET (%) are averaged across eight OOD datasets. The comparison is based on ImageNet-100 pre- trained VGG17 networks. Reported AC1 corresponds to the encoder.
7. Discussion
Our study highlights the impact of neural collapse on OOD detection and OOD generalization. Several promising direc- tions remain for future research. Extending our approach to open-world continual learning (Kim et al., 2025; Dong et al., 2024) presents an exciting challenge. While we fo- cused on architectural and regularization-basee techniques to control NC, another avenue is optimization-driven strate- gies. For instance, Markou et al. (2024) studies optimising towards the nearest simplex ETF to accelerate NC. Guiding NC enhance task-specific representations or disentangle conflicting tasks could improve robustness and generaliza- tion. Moreover, beyond standard loss functions, alternative formulations could be explored to regulate NC.
Method
AC1 1
En
EGEN Avg. +
EDET Avg. +
+
Batch Normalization
1.401
12.52
51.96
69.47
Group Normalization
2.175
12.62
41,85
65.10
Table 8: SGD Optimizer. VGG17 models are trained on ImageNet-100 100 (ID) dataset. Baseline VGG17 does not incor- porate mechanisms like entropy regularization or the ETF projector to control NC. NC metries are computed using the penultimate-layer embeddings. Reported EGEN (%) and DET (%) are averaged over eight OOD datasets.
Following prior work, our study primarily focused on vi- sion tasks and datasets. However, extending our method to other modalities such as audio and text represents a promis- ing direction for future research. While our experiments centered on classification tasks, the proposed method is inherently general and can be applied to other tasks, e.g., object detection or other regression tasks. The core of our approach-the the fixed ETF projector-is designed to enforce NC in the final layer, enhancing feature representations that benefit both classification and regression tasks. Furthermore, our entropy regularization is task-agnostic and seamlessly integrates with both classification and regression objectives.
Model
Em
Neural Collapse|
EGEN
ÈDET
NC1
.C22
NC3
.C4
Avg. 1
Avg. 1
VGG17
13.06
1.017
0.449
0.479
26.459
$7.17
89.69
+Ours
13.18
0.087
0.468
0.267
0.264
50.91
60.81
by 10.11% (absolute) in OOD generalization. Our results suggest that replacing BN is crucial for OOD generalization. Furthermore, using GN improves OOD detection by 4.37% (absolute). Table 17 includes comprchensive results.
Our study utilized nearest neighbor density estimation for entropy regularization. Exploring parametric and adaptive approaches could offer more robust regularization tech- niques for improving OOD generalization. We demon- strated that controlling NC improves OOD detection and generalization, but deeper theoretical understanding of this relationship is needed. Future work could establish theoreti- cal frameworks that unify OOD detection and generalization from an NC perspective, offering a more comprehensive view of representation learning under distribution shifts.
SGD Optimizer. While our main experiments employed the Adam W optimizer, we also evaluate the effectiveness of our method with the widely used SGD optimizer to ensure its robustness across optimization schemes. To this end, we train VGG17 models on ImageNet- dataset (ID) using SGD optimizer and assess their performance eight OOD datasets. As shown in Table 8, our method outperforms the baseline by 6.26% (absolute) in OOD generalization and by 88% (absolute) in OOD detection. Comprehensive results are provided in Appendix H.6.
MSE Vs. CE. Both MSE and CE are effective loss functions achieve NC proper- ties (Zhou et 2022). Unlike prior work, we evaluate their efficacy in both OOD detection and OOD generalization tasks. As shown in Table 9, CE outperforms MSE by 6.74% (absolute) in OOD detection and by 17.71% % (absolute) in OOD generalization. Our observations are consistent with prior work (Kornblith et al., 2021; Hui & Belkin, 2020).
Impact of Loss Functions:
8. Conclusion
In this work, established concrete relationship between neural collapse and OOD detection and generalization. Mo- tivated by this relationship, our method enhances OOD detection by strengthening NC while promoting OOD gen- eralization by mitigating NC. We also provided a theoretical framework to mitigate NC via entropy regularization. Our method demonstrated strong OOD detection and general- ization abilities compared to baselines that did not control NC. This work has implications for open-world problems where simultaneous OOD detection and generalization are critical. We hope our work inspires future efforts to develop more effective methods for building robust AI systems in open-world conditions.
Computational Efficiency. Our method is computation- ally efficient, introducing minimal overhead compared to standard DNNs. We assess efficiency by measuring training time and FLOPs relative to baseline models. As shown in Table 23, the additional cost remains below 0.3% across all cases- a negligible overhead given the substantial perfor- mance gains. Further details are provided in Appendix H.8.
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Controlling Neural Collapse Enhances Out-of-Distribation Detection and Transfer Learning
Acknowledgments. This work was partly supported by NSF awards #2326491, #2125362, and #2317706. The views and conclusions contained herein are those of the authors and should not be interpreted as representing any sponsor's official policies or endorsements. We thank Junyu Chen, Helia Dinh, and Shikhar Srivastava for their feedback and comments on the manuscript.
Bitterwolf, J., Müller, M., and Hein, M. In out? fixing imagenet out-of-distribution detection evaluation. ICML., 2023.
Bui, M.-H., Tran, T., Tran, A., and Phung. Exploiting Nomain-sppeifff features to enhance domain generaliza- tion. Advances in Neural Information Processing Systems, 34:21189-21221 2021.
Impact Statement
Cha, J., Chun, S., Lee, K., Cho, H.-C., Park, S., Lee, Y., and Park, S. Swad: Domain generalization by seeking flat minima. Advances in Neural Information Processing
This paper aims to contribute to the advancement of the field of Machine Learning. While our work has the potential to influence various societal domains, we do not identify any specifie societal impacts that require particular emphasis at this time.
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Controlling Neural Collapse Enhances Out-of-Distribation Detection and Transfer Learning
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Controlling Neural Collapse Enhances Out-of-Distribation Detection and Transfer Learning
Appendix
consisting of two MLP layers (512 - 2048 + 512) and finally added a classifier head. We use ReLU activation between projector layers. We replace BN with GN+WS in all layers. For GN, we use 32 groups in all layers.
We organize the Appendix as follows:
Appendix A describes the implementation details. It describes the DNN architectures (VGG, ResNet, and ViT), feature extraction for linear probing, training, and evaluation details of both pre-training and linear probing in various experiments.
We used the entire ResNet-18 18 or ResNet-34 as the encoder and attached projector (512 + 2048 + 512) similar to the VGG networks mentioned above. We replace BN with GN+WS in all layers. For GN, we use 32 groups in all layers.
ResNet.
Appendix B provides details on the datasets used in this paper. In total, we use datasets.
ViT. We consider ViT-Tiny/Small (5.73M/21 1.73M/21185M
1.85M param- eters) as the encoder for our experiments. The projector comprising two MLP layers configured as fixed ETF Sim- plex and added after the encoder. Following (Beyer et al., 2022), we omit the learnable position embeddings and in- stead use the fixed 2D sin-cos position embeddings. Other details adhere to the original ViT paper (Dosovitskiy et al., 2020).
Appendix C describes four neural collapse metries (NC1 -C44 used in this paper.
Appendix D presents comprehensive comparison be- tween MSE and CE.
Appendix E contains proof on the implication of NC entropy.
ViT-Tiny Configuration: patch size=16, embedding dimension=192, # heads=3, depth=12. Projector has output dimension=192 and hidden dimension=768, (192 + 768 + 192). We use ReLU activation be- tween projector layers. The number of parameters in ViT-Tiny + projector is 6.02M.
Appendix F provides a comprehensive comparison be- tween the encoder and projector across different archi- tectures.
Appendix G provides detailed analyses on entropy reg- ularization and neural collapse.
ViT-Small Configuration:
patch size=16, embedding dimension=384, # heads=6, depth= 12. Projector has output dimension=384 and hidden dimension=1536. (384 + 1536 - 384). We use ReLU activation be- tween projector layers. The number of parameters in ViT-Small + projector is 23,03M.
Additional experiments and analyses are summarized in Appendix H. The mechanisms of controlling NC have been examined.
Appendix I includes the list of 100 classes in the ImageNet-100 dataset.
A.2. Feature Extraction For Linear Probing
A. Implementation Details
In experiments with CNNs, at each layer I, for each sample, extract features of dimension H1 x Wi x Ct, where Hi. Wi, and Cj denote the height, width and channel dimensions respectively. Next, following (Sarfi et al., 2023), we apply 2 x 2 adaptive average pooling on each spatial tensor (Hi X Wi). After average pooling, features dimension 2 2 x C₁ are flattened and converted into a vector of dimension 4C₁, Finally, linear probe is trained on the flattened vectors. In experiments with ViTs, following (Raghu al., 2021), apply global vverage-pooling (GAP) to aggregate image tokens excluding the class token and train linear probe on top of GAP tokens. We report the best error (%) on the test dataset for linear probing at each layer.
In this paper, we use several acronyms such as NC: Neu- ral Collapse, ETF- Equiangular Tight Frame, ID In- LR Learning Rate, WD: Weight Decay, GAP: Global Average Pooling, :N: Group Normalization, BN: Batch Normalization, WS : Weight Standardization, CE: Cross Entropy, MSE Mean Squared Error, FPR False Positive Rate.
Distribution, OOD
We use the terms "OOD generalization" and "OOD transfer" interchangeably.
A.I. Architectures
VGG. We modified the VGG-19 architecture create our VGG-17 encoder. Additionally, we removed two fully connected (FC) layers before the final classifier head. And, we added an adaptive average pooling layer Aw..ddaaiieAvgPol22l which allows the network to ac- cept any input size while keeping the output dimensions the same. After VGG-17 encoder, we attached a projector
A.3. VGG Experiments
VGG ID Training: For training VGG on ImageNet-100, we employ the AdamW optimizer with a LR of x 10-3 and WD of 5 x 10-2 for batch size 512. The model is trained for 100 epochs using the Cosine Annealing LR scheduler with

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Controlling Neural Collapse Enhances Out-of Diitri--Dittiitttion Detection and Transfer Learning
linear warmup of 5 epochs. In all experiments, use CE and entropy regularization (a = 0.05) losses. However, in some particular experiments comparing CE and MSE, we use MSE loss (x=15, M-60) and entropy regularization loss (a = 0.05). In the experiments with SGD optimizer and CE loss, we set LR to 0.2 and WD to 10-4 for batch size 512.
detection performance by measuring the fraction of OOD samples misclassified as II) where threshold, À is chosen when the true positive rate is 95%. Both OOD detection and OOD generalization tasks are evaluated on the same OOD test set.
Percentage Change. To capture percentage increase or de- crease when switching from the encoder (E) to the projector (P), we use
VGG Linear Probing: We use the AdamW optimizer with flat LR of 1 x 10-3 and WD of 0 for batch size 128. The linear probes are trained for 30 epochs. We use label smoothing of 0.1 with the cross-entropy loss.
AP-P- (P E) F.
X 100.
A.4. ResNet Experiments
Normalization for different OOD datasets. In our correla- tion analysis between NC and OOD detectionngennrralzzaionn (Fig. 1 and 7), we use min-max normalization for layer- wise OOD detection errors and OOD generalization errors which enables comparison using different OOD datasets. For a given OOD dataset and a DNN consisting of total L layers, let the OOD detection/ generalization error for a layer / be E,. For L layers we have error vector E [E,, Eg, EE]
ResNet ID Training: For training ResNet-18/34 we em- play the AdamW optimizer with an LR of 0.01 and a WD of 0.05 for batch size 512. The model is trained for 100 epochs using the Cosine Annealing LR scheduler with a linear warmup of 5 epochs. We use CE and entropy regular- ization (a- 0.05) losses.
In the linear probing experiment, use the AdamW optimizer with LR of 1 x 10-3 and WD of 0 for batch size 128. The linear probes are trained for 30 epochs. We use label smoothing of 0.1 with cross- entropy loss.
ResNet Linear Probing:
which is then normalized by
E min E
E,
max E)- min E)
Effective Rank. We use RankMe (Garrido et al., 2023) to measure the effective rank of the embeddings.
A.5. VIT Experiments
VIT ID Training: For training ViT-Tiny, we employ the Adam optimizer with LR of 8 x 10-4 and WD of 5 x 10-2 for batch size 256. The LR is scaled for n GPUs according batehsize
B. Datasets
ImageNet-100, ImageNet-100 (Tian et al., 2020) is a sub- set of ImageNet-IK, (Deng et al., 2009) and contains 100 ImageNet classes. It consists of 126689 training images (224 x 224) and 5000 test images. The object categories present in ImageNet- are listed in Appendix I.
to: LR X n 512 We use an LR of x 10- for ViT-Small when the batch size is 256. We use the Cosine Annealing LR scheduler with warm-up (5 epochs). We train the ViT-Tiny/Small for 100 epochs using CE and entropy regularization (a = 0.05) losses. Following (Raghu et al., 2021; Beyer et al., 2022), we omit class token and instead use GAP token by global average- sorrage-pooling image tokens and feed GAP embeddings to the projector.
CIFAR-100. CIFAR- (Krizhevsky & Hinton, 2014) is a dataset widely used in computer vision. It contains 60, 000 RGB images and 100 classes, each containing 600 images. The dataset is split into 50, 000 training samples and 10, 000 test samples. The images in CIFAR- 100 have a resolution of 32 x 32 pixels. Unlike CIFAR-10. 10, CIFAR-100 has a higher level of granularity, with more fine-grained classes such as flowers, insects, household items, and variety of animals and vehicles. For linear probing, all samples from both the training and validation datasets were used.
VIT Linear Probing: We use the AdamW optimizer with LR of 0.01 and WD of 1 x 10-4 for batch size 512. The lin- ear probes are trained for 30 epochs. We use label smoothing of 0.1 with cross-entropy loss.
Augmentation. We use random resized crop and random flip augmentations and 224 x 224 images as inputs to the DNNs.
NINCO (No ImageNet Class Objects). NINCO (Bitter- wolf et al., 2023) is a dataset with 64 classes. The dataset is curated to eliminate semantic overlap with ImageNet- IK dataset and is used to evaluate the OOD performance of the models pre-trained on imagenet- 1 K. The NINCO dataset has 5878 samples, and we split it into 4702 samples for training and 1176 samples for evaluation. We do not have a fixed number of samples per class for training and evaluation datasets.
In experiments with CE loss, use label smoothing of 0.1.
A.6. Evaluation Criteria
FPR95. The OOD detection performance is evaluated by the FPR (False Positive Rate) metric. In particular, we use FPR95 (FPR at 95% True Positive Rate) that evaluates OOD
2

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Controlling Neural Collapse Enhances Out-of-Distributie Detection and Transfer Learning
ImaneitttRendittoon (ImageNet-R), ImageNet-R incor- porates distribution shifts using different artistic ren- ditions of object classes from the original ImageNet dataset (Hendrycks et al., 2021). We use a variant of ImageNet-R dataset from (Wang et al., 2022b). ImageNet-R a challenging benchmark for continual learning, transfer learning, and OOD detection. It consists of classes with different styles and intra-class diversity and thereby poses significant distribution shifts for ImageNet- IK pre-trained models (Wang et al., 2022b). It contains 200 classes, 24000 training images, and 6000 test images.
a self-dual configuration.
Nearest Class Mean Decision (NC4); The classifier operates similarly to the nearest class-center (NCC) decision rule, assigning classes based on proximity to the class means.
Here, we describe each NC metric used in our results. Let Ng denote the global mean and l4, the e-th class mean of the features, {Ze,a) at layer 1, defined as follows:
1 MG nC
n
1I1 T 12
Ze.i.
He :==
Zes (LS0SC).
CUB-200. CUB-200 composed of 200 different bird species (Wah et al., 1). The CUB-200 dataset comprises a total of 11,788 images, with 5,994 images allocated for training and 5,794 images for testing.
We drop the layer index from notation for simplicity. Also bias is excluded for notation simplicity. Feature dimension is d instead of d + 1.
Aircrafts-100. Aircrafts or FGVCAircrafts dataset (Maji et al., 2013) consists of 100 different aircraft categories and 10000 high-resolution images with 100 images per category. The training and test sets contain 6667 and 3333 images respectively.
Within-Class Variability Collapse (NC1); It measures the relative size of the within-class covariance Ww with respect to the between-class covariance ΣB of the DNN features:
1 nC
Ew
(Ze 16)) (Ze, -mo²² Rdddd
Oxford Pets-37. The Oxford Pets dataset includes total of 37 various pet categories, with an approximately equal number of images for dogs and cats, totaling around 200 images for each category (Parkhi et al., 2012).
f=1
C
Σn = C
Σ
(m. - -MG) E BRddd
cml
Flowers- 102. The Flowers- 102 dataset contains 102 flower categories that can be easily found in the UK. Each cate- gory of the dataset contains 40 to 258 images. (Nilsback & Zisserman, 2008)
The NC1 metric is defined as:
NC1 C trace Eww Σ33)
STL-10. STL-10 has 10 classes with 500 training images and 800 test images per class (Coates et al., 2011). I).
where Σ/ is the pseudo-inverse of Σg. Note that NC1 is the most dominant indicator of neural collapse.
For all datasets, images are resized to 224 x 224 to train and evaluate DNNs.
Convergence to Simplex ETF (NC2): quantifies the fg distance between the normalized simplex ETF and the normalized WWT. T as follows:
C. Neural Collapse Metrics
WW
Io (
1
Neural Collapse (NC) describes a structured organization of representations in DNNs (Papyan et al., 2020; Kotha- palli, 2023; Zhu et al., 2021; Rangamani et al., 2023). The following four criteria characterize Neural Collapse:
NC2 = |W |e
VC-I
where W E Cxd denotes the weight matrix of the learned
classifier.
1. Feature Collapse (NC1):: Features within each class concentrate around a single mean, with almost no vari- ability within classes.
Convergence to Self-Duality (NC3): It measures the &z distance between the normalized simplex ETF and the nor- malized WZ:
WZ
Ic do-2)
2. Simplex ETF Structure (NC2): Class means, when centered at the global mean, are linearly separable, maximally distant, and form a symmetrical structure on a hypersphere known as a Simplex Equiangular Tight Frame (Simplex ETF).
NC3 == WZ VC-
where Z [Z] Pg Zc Hcl E RdxC is the cen-
tered mlass-mean matrix.
Simplification to NCC (NC4); measures the collapse of bias b:
3. Self-Duality (NC3):: The last-layer classifiers align closely with their corresponding class means, forming
NCC = |b+ Wpall
3

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Controlling Neural Collapse Enhances Out-of-Distriaatis Detection and Transfer Learning
Table 9: Comparison between MSE and CE. VGG17 net- works are trained on ImageNet-100 dataset (ID). For OOD generalization we report GEN (%) whereas for OOD detec- tion we report DET (%). both are averaged over eight OOD datasets. All models incorporate entropy regularization and the ETF projector control NC. A lower NCC indicates stronger neural collapse. + A and - - indicate % increase and % decrease respectively, when changing from the encoder (E) to projector (P).
Then, the differential entropy of the mixture
K
pe(z; c) =
Th A
diverges -00 in the limit that is,
lim Hpr((z;)) = -CC
c-0
Method
Ei
Neural Collapse
NC1
NC2
.C3
.C1
Avg.
Avg.
Detailed Proof. We begin by considering the mixture distri-
CE Loss
bution
Projector
0.393
0.468
0.316
K
Encoder
15.52
2.175
0.603
0.616
5.364
41.85
87.62
T PL, c); Pe(z; Σ
Ag-p
18.69
81.93
-8.74
-2.00
99.11
+58.57
-25.70
R=1
MSE Loss
Prujector
14.04
0.469
0.743
0.2799
0.382
70.87
71.84
For each ks assume that the density pe, (r, e) satisfies
Fnender
14.74
2.267
0.843
0.673
10,773
59,56
88.88
Ag-PP
4.75
-79.31
-11.86
-58.54
-96.45
+18.99
-19.17
lim PL,(Z;)) = 8(z - Mr,x), 6,,,),,,),
c-U
and, importantly, that its differential entropy diverges
D. Mean Squared Error vs. Cross-Entropy
lim H (pe,e(Z; = -00.
c-0
Prior work (Kornblith et al., 2021) finds that MSE rivals CE in ID classification task but underperforms CE in OOD transfer. However, the comparison between CE and MSE in OOD detection task remains unexplored. In this work, we find that CE significantly outperforms MSE in both OOD transfer and OOD) detection tasks. As shown in Table 9, MSE underperforms CE by 6.74% (absolute) in OOD detec- tion and by 17.71% % (absolute) in OOD generalization. Our OOD generalization results et al. (2021). CE also obtains lower ID) error than MSE, thereby showing good overall performance.
A concrete example is when pe, (2; E) is a Gaussian with covariance eI. that case.
H ((pex(z;))) dt
log((Tre))
which clearly tends to 0o as E +
consistent with Kornblith
Step 1: Introduce Latent Class Variable. Define discrete random variable K taking values in {1...., K} with Pr(K = k) = Ty. Then, the joint distribution of (Z, K) is given by
In terms of inducing neural collapse, both MSE and CE are effective and achieve lower NC values (i.e., stronger NC). However, our results suggest that CE does a better job than MSE in enhancing NC without sacrificing OOD transfer. We find MSE to be sensitive to the hyperparameters. The comparison on all OOD datasets is shown in Table 10.
plz, c) T (e:e)
Step 2: Apply the Chain Rule for Differential Entropy. Using the chain rule for differential entropy, we have
E. Formal Proposition: Collapsing Implies Entropy -00
H K) H(K) + H(Z K).
Here, the entropy the discrete variable K is
Proposition E.1 (Entropy under Class Conditional Col- lapse). Consider mixture of K class-conditional densities (p; K in Te with mixture weights k Sup- pose that for each k. the density pe,x (z; e) is member of a family indexed by > 0 that converges in the weak sense to Dirac delta, Le..
H(K)
Tk log Tk.
which is finite since there are only finitely many classes. The conditional entropy given by
K
H K) Σ
Th H (pe,(z;)))
lim (Z;e) = -(zz- (Mex)) c0
km

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Controlling Neural Collapse Enhances Out-of-Distribation Detection and Transfer Learning
Step 3: Relate the Entropy of the Mixture to the Condi- tional Entropies. By standard property of conditional entropy, we have
sets. This demonstrates that, unlike the encoder, the pro- jector can intensify NC and is adept at OOD detection. We show the energy distribution of ID and OOD sets in Fig. 5b and 6. In all comparisons, we observe that the projector outperforms the encoder in separating ID and OOD sets based on energy scores.
(Z) H HK] 2) S H(Z, K).
Thus, the entropy of the mixture satisfies
H (pr(z;e)) - (Z) S H(K)
Σ Th (pea(zzze).
F.2. ResNet Experiments
The detailed ResNet18/34 results are given in Table 11. Our findings validate that NC can be controlled in vari- ous ResNet architectures for improving OOD detection and OOD generalization performance. Additionally, NC shows strong correlation with OOD detection and OOD general- ization as illustrated in Fig. 7.
Step 4: Conclude that the Mixture Entropy Diverges to Since H (K) is finite constant and for each -C.
lim H (pe,x c)) = -00 c+0
it follows that
We also visualize embeddings extracted from the encoder and projector of the ResNet18 8 model. As depicted in Fig. 8, projector embeddings exhibit much greater neural collapse than encoder embeddings.
K
lim
Th H (DE,((2;)) = -00.
Therefore,
lim H (pe(z;e)) -00.
F.3. VIT Experiments
-
As shown in Table 12, the projector outperforms the en- coder in OOD detection by absolute 7.73% (ViT-Tiny) and 9.23% (ViT-Small). Whereas the encoder outperforms the projector in OOD transfer by absolute 10.90% (ViT-Tiny) and 11.56% (ViT-Small). This demonstrates that controlling NC improves OOD detection and generalization in ViTs.
Discussion. The essential idea is that even though the mixture might appear to smooth the singular behavior of in- dividual condiisonti distributions, the overall entropy is still governed by the weighted sum of the entropies of its components. Because each component entropy diverges to -oo, the entire mixture's entropy must also diverge to -00, up to the finite additive constant H (K).
G. Analysis on Entropy Regularization
This completes the proof.
Table 13 presents the detailed comparison between a model using the entropy regularization vs another model omitting it. We observe that using entropy penalty enhances OOD transfer by 2.71% % (absolute), OOD detection by 2.36% (ab- solute), and ID performance by 0.84% (absolute).
F. Comprehensive Results (Encoder Vs. Projector)
F.1. VGG Experiments
Additionally, we analyze the impact of the entropy regular- ization loss coefficient on the ID and OOD transfer. Table 14 shows that increasing coefficient increases OOD transfer and rank of embeddings. This suggests that entropy regu- larization helps encode diverse features and reduce redun- dant features, encouraging utilization of all dimensions. Al- though entropy regularization is not sensitive to coefficient, over-regularivation may hurt ID) performance. Thereby, any non- non-aggressive coefficient can maintain good performance in both and OOD tasks.
The detailed VGG17 results are given in Table 10. VGG re- sults demonstrate that the encoder effectively mitigates NC for OOD generalization and the projector builds collapsed features and excels at the OOD detection task. The results also confirm that NC properties can be built using both CE and MSE loss functions.
Qualitative Comparison. We compare and visualize en- coder embeddings and projector embeddings using UMAP. We also visualize the energy score distribution of ID and OOD data. The analysis is based on the VGG17 model pre-trained the ImageNet-100 100 (ID) dataset and evaluated on OOD datasets: NINCO-64, flowers-102, and STL-10. We observe the following:
We also analyze the impact of entropy regularization on encoder embeddings during the training phase. During each training epoch, we measure the NC1 criterion, entropy, and effective rank of encoder embeddings. These experiments are computationally intensive for large-scale datasets. There- fore, we perform small-scale experiments where we train VGG17 models on the ImageNet- 10 (10 ImageNet classes) classess
In Fig. 5a, the UMAP shows that projector embeddings nicely separate ID and OOD sets whereas encoder embed-
dings exhibit substantial overlap between ID and OOD
5

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Controlling Neural Collapse Enhances Out-of oiitrf-Disttrbation Detection and Transfer Learning
Table 10: Comprehensive VGG Results. VGG17 models are trained on ImageNet-100 dataset (ID) and evaluated on eight OOD datasets. All models incorporate entropy regularization and the ETF projector to control NC. For OOD transfer we report GEN (%) whereas for OOD detection we report DET (%). lower NCC indicates stronger neural collapse. The same color highlights the rows to compare.
Method
ED IN
Neural Collapse NC2
OOD Datasets
NC1
NC3
NC4
IN-R 200
CIFAR
Flowers 102
NINCO
CUB
Airerafts
Pets
STL
Avg.
100
100
200
100
CE Loss
Transfer Error
Projector
12.62
0.393
0.490
0.468
0.316
91.38
65.72
64.51
64.97
82.22
97.42
43.17
21.51
66.36
Encoder
15.52
2.5
0.603
0.616
5.364
71.52
47.24
25.10
24.32
63.67
67.81
21.56
13.55
41.85
Defection Error L
Projector
12.62
0.393
0.490
0.468
0.316
60.85
48.23
42.35
67.69
56.51
99.04
76.32
69.84
65.10
Encoder
15.52
2.175
0.603
0.616
5.364
67.17
98.14
81.76
84,95
84,57
99.70
97.36
87.34
87.62
Loss
Transfer Error
Projector
14.04
0.469
0.743
0.279
0.382
87.18
70.33
82.16
55.95
90.35
97.09
55.93
28.01
70.87
Encoder
14.74
2.267
0.843
0.673
10.773
83,22
60.55
66.27
40.73
78.89
88.27
36.17
22.41
59.56
Detection Error
Projector
14.04
0.469
0.743
0.279
0.382
63.75
48.02
61.18
69.50
74.58
99.10
84.57
74.06
71.84
Encuder
14.74
2.267
0.843
0.673
10.773
93.32
62.42
77.55
92.14
95.77
99.19
99.13
91.50
88.88
Encoder Embeddings
Projector Embeddings
Encoder (FPR9S: 84.95%)
Projector (FPR95: 67.69%)
1.0
ID Data
ID Data
ID Data
DO0 Data
0OD Data
00D Data
0.8
Density
0.8
0.4
0.2
OOD
6
7
B
Energy Score
Energy Score
UMAP of Emheddings
(h) Energy Score Distribution
Figure 5: ID & OOD Data Visualization. In (a), The projector exhibits greater separation between ID) and OOD embeddings than the encoder. For clarity, we show 10 ImageNet classes ID data and 64 classes from the NINCO dataset as OOD data. In (b). The projector achieves higher energy scores (and lower FPR95) for ID data. For ID and OOD datasets, we show ImageNet- 100 and NINCO-64 respectively.
subset for 100 epochs. We evaluate two cases: one with entropy regularization and another without entropy regular- ization.
Without the entropy regularization, the entropy of encoder embeddings does not improve. Also, the effective rank ends up at low value (as low as the number of ID classes). The low rank is sign of strong neural collapse and suggests that the encoder uses few feature dimensions to encode in- formation with huge redundancy in other dimensions. This degeneracy of embeddings impairs OOD transfer. Entropy regularization counteracts this and improves OOD transfer.
The results are illustrated in Fig. 9, We find that entropy regularization achieves higher NC1 values during training compared to the model without any regularization. Thus, it helps mitigate NC during training, thereby contributing to OOD generalization. These findings align with our theoreti- cal analysis showing entropy as an effective mechanism to prevent NC in the encoder.
H. Additional Experimental Results
Entropy regularization also increases the entropy and effec- tive rank of the encoder embeddings. This demonstrates that entropy regularization helps encode diverse features, ensuring the features remain sufficiently "spread out."
H.1. Fixed ETF Projector Vs. Learnable Projector
In Table 15, we observe that the fixed ETF projector shows higher transfer error (2.47% absolute) than the plastic projector but outperforms the plastic projector in ID error
6

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Controlling Neural Collapse Enhances Out-of-Distribation Detection and Transfer Learning
Encoder (FPR95: 81.76%)
Projector (FPR95: 42.35%)
Encoder (FPR95: 87.34%)
Projector (FPR95: 69.84%)
1.2-
ID Data
Data
ID Data
ID Data
0OD Data
OOD Data
OOD Data
OOD Data
1.0
Density
0.8-
Density
0.6
0.4
0.4-
0.2
0.2
0.0
10
15
0.0
5
10
15
8
Energy Score
Energy Score
Energy Score
Energy Score
(a) OOD Dataset: Flowers- 102
(h) OOD Dataset: STI.-10
Figure 6: Energy Score Distribution. The projector creates a greater separation between ID and OOD data and achieves a lower FPR95 than the encoder. For better OOD detection, ID data should obtain higher energy scores than OOD data. For ID and OOD datasets, we show ImageNet-100 and Flowers- 102/ STL- 10 respectively. The energy scores are calculated based on logits from the VGG17 model pre-trained on ImageNet-100.
Table 11: Comprehensive ResNet Results. ResNet models trained on ImageNee-100 dataset (ID) and evaluated on eight OOD datasets. All models incorporate entropy regularization and the ETF projector to control NC. For OOD transfer we report GEN (%) whereas for OOD detection we report & DET (%). A lower NC indicates stronger neural collapse.
Model
ED L
Neural Collapse NC2
OOD Datasets
IN
NC1
NC3
NC4
IN-R 200
CIFAR
Flowers 102
NINCO
CUB
Aircrafts
Pets
STL
Avg.
100
100
64
200
100
37
10
ResNet18
Transfer Error U
Projector
16.14
0.341
0.456
0.306
0.540
86.65
60.33
63.92
$0.09
81.79
94.36
43.15
24.32
63.08
Encoder
20.14
1.762
0.552
0.555
10.695
74.17
53.33
31,37
28.15
68.85
81.61
27.72
16.56
47.72
Delection Error
Projector
16.14
0.341
0.456
0.306
0.540
67.92
61.21
71.18
71.09
23.20
99.28
81.41
82.29
69.70
Encoder
20.14
1.762
0.552
0.558
10.695
71.50
96.44
86.27
84.78
65.48
99.43
95.86
89.63
86.17
ResNet34
Transfer Error
Projector 14.54
0.252
0.672
0,294
0.324
83.93
58.65
64.41
44.05
81.65
93.58
43.64
22.87
61.60
Encoder
17.20
0.737
0.634
0.871
22.587
76.97
54.45
41.47
33.33
71.25
82.00
16.45
50.65
Delection Error
Projector
14.54
0.252
0.672
0.294
61.72
60.05
47.94
66.24
67.59
98.35
83.78
78.49
70.52
Encoder
17.20
0.737
0.634
0.871
22.587
69.67
93.07
70.59
76.87
83.02
99.34
97.17
90.75
85.06
(2.48% absolute) and OOD detection error (8.9% absolute). fixed ETF projector should intensify NC and hinder OOD transfer and our fixed ETF projector fulfills this goal.
training. We perform small-scale experiments since large- scale experiments are compute-intensive. We train two VGG17 models the ImageNet- 10 (10 ImageNet classes) subset where model uses T.2 normalization and another does not. During training, measure the NC1 metric for the encoder embeddings. The impact of L2 normalization on NC1 is exhibited in Fig. 9d. We find that L2 normaliza- tion helps intensify NC during training. Consequently, promotes better OOD detection.
H.2. Impact of L2 Normalization on NC
We verify whether L2 normalization effectively induces more neural collapse and improves OOD detection. We analyze two VGG17 models pre-trained on ImageNet-100 dataset where other omits it. The results are summarized in Table 16. We find that Le normalization induces more NC as evidenced by the lower NC1 value than its counterpart. Consequently, L2 normalization improves OOD detection by 3.83% (absolute). Also, it achieves lower ID error than the compared model without La normalization.
model uses L2 normalization and the
H.3. Batch Normalization Vs. Group Normalization
In this experiment, we analyze how batch normalization and group normalization perform within our framework. We find that group normalization (combined with weight stan- dardization) outperforms batch normalization by 10.1 10.11% (absolute) in OOD generalization and by 4.37% (absolute) in
Next, we analyze how L2 normalization impacts NC during

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Controlling Neural Collapse Enhances
Detection and Transfer Learning
Table 12: Comprehensive VIT Results. ViT-Tiny (6.02M) and ViT-Small (21.62M) are trained on ImageNet-100 dataset (ID) and evaluated on eight OOD datasets. All models incorporate entropy regularization and the ETF projector to control NC. For OOD transfer we report EGN (%) whereas for OOD detection we report DET (%). A lower NCC indicates stronger neural collapse.
Model
EI IN 100
Neural Collapse
OOD
Datasets
NC1
NC2
NCC
.N44
IN-R 200
CIFAR
Flowers 102
NINCO
CUB
Airerafts 100
Pets
STL
Avg.
100
64
200
ViI-Tiny
Transfer Error 1
Projector Encoder
32.04
2.748
0.609
0.798
1.144
87.37
60.71
64.61
39.71
80.00
92.00
54.27
29.55
63.53
33.94
5.769
0.748
0.847
2.332
82.28
52.00
42.94
30.36
63.15
84.31
44.86
21.13
52.63
Detection Error
Projector
32.04
2.748
0.609
0.798
1.144
81.12
60.81
77.55
82.40
79.05
99.10
95.15
90.06
83,16
Encoder
33.94
5.769
0.748
0.847
2.332
83.80
96.76
87.65
93.11
82.14
99.10
95.75
88.79
90.89
ViT-Small
Transfer Error I
Projector
31.28
0.822
0.522
0.712
0.962
86.57
58.46
64.51
39.20
78.25
90.70
53.86
29.30
62.61
Encoder
33.40
1.610
0.601
0.740
2.814
80.53
49.68
40.49
29.93
61.08
81.28
44.45
20.98
51.05
Detection Error
Projector
31,28
0.822
0.522
0.712
0.962
76.03
58.79
75.20
81.97
82.46
98.50
95.42
88.74
82.14
Encoder
33.40
1.610
0.601
0.740
2.814
82.47
96.84
90.39
92.60
86.00
99.25
94.36
89.04
91.37
OOD Detection 0.56
Encodier Embeddiigg
Projector Embeddings
OOD Transfer
R -..55
Normalized OOD Detection Error
1.0
0.7
0.6
0.8
NNNN Comman OOD Transfer
0.5
0.6
0.4
0.4
0.3
0.2
0.1
Figure 8: Visualization of Embedding (ResNet18). In this UMAP, projector embeddings exhibit greater neural collapse (NC1 = 0.341) than the encoder embeddings (NC1 1.762) as indicated by the formation of tight clusters around class-means. For clarity, highlight 10 ImageNet classes by distinct colors. The embeddings are extracted from ImageNet-100 pre-trained ResNet18.
0.0
0.3
0.4
0.8 Stronger Neural Collapse (NC1)
0.5
0.6
0.7
0.9
Figure 7: Lower NC1 values (indicating stronger neu- ral collapse) correlate with lower OOD detection error but higher OOD transfer error, and vice versa. This sug- gests that stronger neural collapse improves OOD detection, while weaker neural collapse enhances OOD generaliza- tion. We analyze various layers of ResNet18, pre-trained ImageNet-100 100 dataset (ID), and evaluate them on four OOD datasets. R denotes the Pearson correlation coefficient.
performance and plays a crucial role in OOD generalization.
H.4. Comparison with Baseline
Our experimental results show that our method significantly improves OOD detection and OOD transfer performance across all DNN architectures. We summarize the results in Table 18. We evaluate VGG17, ResNet18, and VIT-T baselines 8 OOD datasets and compare them with models. The absolute improvements over VGG17 baseline are 7.69% (OOD generalization) and 29.82% (OOD detec- tion). Similarly, our method outperforms other DNNs in all criteria. Our results corroborate our argument that control- ling NC enables good OOD detection and OOD general-
OOD detection (see Table 17). This demonstrates that batch normalization leads to less transferable representations.
Moreover, group normalization achieves higher NC1 value (i.e., lower neural collapse) than batch normaliza- tion, thereby mitigating NC effect and enhancing OOD generalization. Group normalization also achieves ID per- formance similar to that of batch normalization. Our results demonstrate that group normalization achieves competitive
8

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Controlling Neural Collapse Enhances Out-of-Distribation Detection and Transfer Learning
Table 13: Entropy Regularization Vs. No Entropy Regularization. VGG17 models are pre-trained on ImageNet-100 dataset (ID) and evaluated on eight OOD datasets. All models incorporate an ETF projector. Entropy regularization loss with coefficient, a is applied in the last encoder layer. The same color highlights the rows to compare. All metrics except NC are reported in %. The lower the NC value, the stronger the neural collapse. For OOD transfer we report & EGEN (%) whereas for OOD detection we report DET (%)
Method
Ei IN
Neural Collapse
OOD
Datasets &
NC1
NC2
NC3
.O4
IN-R 200
CIFAR
Flowers
NINCO
CUB
Airerafts
Pets
STL
Avg.
100
100
102
64
200
100
37
10
Transfer Error |
No Reg. (a 0)
Projector Encoder
13:46
0.260
0.636
0.369
0.883
84.30
60.73
65.69
45.15
82.90
93.73
40.56
22.85
61.99
15.24
1.308
0.719
0.619
5.184
73.52
49.26
37.06
25.51
64.31
69.58
22.24
15.00
44,56
Reg, (a 0.06)
Prujectur
12.62
0.393
0.490
0.468
0.316
91.38
65.72
64.51
64.97
82.22
97.42
43,17
21.51
66.36
Encoder
15.52
2.175
0.603
0.616
5.364
71.52
47.24
25.10
24.32
63.67
67.81
21.56
13,55
41.85
Reg. (n 0.1)
Projector
13.04
0.428
0.671
0.340
0.320
93.62
66.00
55,29
79.25
81.84
97.09
46,96
23:00
67.88
Encoder
16.12
2.861
0.538
0.636
6.677
73.05
48.61
27.84
22.62
61.91
70.21
22.87
13.83
42.62
Detection Error
No Reg, (a 0)
Projector
13.46
0.260
0.636
0.369
0.883
65.22
54.32
45.20
67.18
52.37
98.41
84.38
72.58
67.46
Encoder
15.24
1.308
0.719
0.619
5.184
74.22
99.75
85.10
88.52
92.99
98.59
95.34
92.14
90.83
Reg. (a 0.05)
Projector
12.62
0.393
0.490
0.468
0.316
60.85
48.23
42.35
67.69
56.51
99.04
76.32
69.84
65.10
Encoder
15.52
2,175
0.603
0.616
5.364
67.17
98,14
81.76
84.95
84.57
99.70
97.36
87.34
87.62
Reg. (a = 0.1)
Projector
13.04
0.428
0.671
0,340
0.320
61.13
54.69
43.14
64.63
50.73
98.74
82,42
71.51
65.87
Encoder
16.12
2.861
0.538
0.636
6.677
68.72
94.67
85.78
87.76
85,49
98.92
95.15
86.28
87.85
ization performance. It is also evident that a single feature space cannot simultaneously achieve both OOD detection and OOD generalization abilities.
absolute 28.88% in OOD detection. Also, we observe that our encoder reduces NC and enhances OOD generalization by an absolute 13.86% compared to the projector. Whereas the projector intensifies NC and improves OOD detection by an absolute 25.34% compared to the encoder. While SGD intensifies NC more than AdamW, AdamW achieves better overall performance (Table 18 in Appendix).
H.S. Projector Design Criteria
Here we study the design choices of the projector net- work. We want to know how depth and width impact the performance. For this, we examine projectors consisting of a single layer (depth=1, 512d), two layers (depth=2, 512d + 2048d + 5124),, three layers (depth=3, 512d -> 2048d + 2048d + 512d), and wider variant (width=2, 512d + 4096d + 512d). All of these variants are trained in identical settings and only the projector is changed. We train VGG17 networks on ImageNet- 100 dataset (ID) and on 8 OOD datasets. The results are shown in Table 19. The projector with depth 2 outperforms other variants across all evaluations.
H.7. Fixed ETF Classifier Vs. Plastic Classifier
We investigate how using fixed ETF classifier head im- pacts NC and OOD deteetioongenneralization performance. We train two identical models consisting of our proposed mechanisms to control NC, the only thing we vary is the classifier head. One model consists of a plastic (learnable) classifier head which is our proposed model and the other consists of an ETF classifier head. The ETF classifier head is configured with Simplex ETF and frozen during train- ing. We train VGG17 networks on ImageNet-100 (ID) and evaluate them on 8 OOD datasets.
evaluate OOD detection/generalizzan
H.6. SGD Optimizer
In our experiments, we mainly used AdamW optimizer and thereby we want to verify if our method works well with other commonly used optimizers e.g., SGD. For this, we train VGG17 models on ImageNet-100 dataset (ID) with the SGD optimizer and evaluate them eight OOD datasets. As shown in Table 20, our method outperforms the baseline by an absolute 6.26% in OOD generalization and by an
Table 21 shows results across all OOD datasets, where the plastic classifier outperforms the fixed ETF classifier by 4.39% (absolute) in OOD detection and by 15.6% in OOD generalization. The plastic classifier also outperforms ETF classifier in the ID task. Our results suggest that imposing NC in the classifier head is sub-optimal for enhancing OOD detection and generalization.
9

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Controlling Neural Collapse Enhances Out-of-Distribation Detection and Transfer Learning
Table 14: Entropy Regularization Loss Coefficient. We examine the impact of entropy regularization on the OOD generalization of a regular VGG17 without any projector. VGG17 models are pre- pre-trained on the ImageNet- 10 (10 ImageNet classes) IID dataset and evaluated on eight OOI) datasets. denotes the entropy regularization loss coefficient. Effective rank corresponds to the penultimate embeddings where entropy regularization is applied or omitted. For OOD generalization, we report EGEN (%).
Reg. Coeff.
EDD
Rank
GEN +
a
IN
IN
IN-R
CIFAR
Flowers
NINCO
CUB
Aircrafts
Pets
STL
Avg.
10
10
200
100
102
64
200
100
37
10
0
9.20
2211.99
94.62
83.77
72,45
65.56
86.76
89.32
80.32
49.44
77.78
0.1
9.80
2964.39
90.72
75.58
57.94
50.09
79.96
84.13
72.72
39.79
68.87
0.2
10.20
3170.92
90.25
72.77
57.84
50.85
79.48
84.16
70.43
37.71
67.94
0.6
12.00
3761.33
88.33
68.73
50.29
47.45
77.10
82.57
67.73
39.00
65.15
1.0
12.80
4815.32
88.38
67.81
50.29
47.11
77.48
81.64
68.27
38.74
64.96
Table 15: ETF Fixed Projector Vs. Plastic Projector. VGG17 models trained on ImageNNet-100 dataset (ID) and evaluated eight OOD datasets. All models incorporate entropy regularization and a projector (plastic/fixed ETF). The same color highlights the rows to compare. For OOD transfer we report &GEN (%) whereas for OOD detection we report DET (%).
Projector
Em 4
Neural Collapse
OOD Datasets
IN
NC1
NC2
NC3
.C4
IN-R 200
CIFAR 100
Flowers 102
NINCO
CUB
Airerafts
Pets
STL
Avg.
100
200
100
Transter Error +
Plastic
Projector
15.10
0.498
0.515
0.428
1.422
87.52
64.83
79.71
53.32
87.00
93.46
48.76
28.04
67.83
Encoder
23.64
13.953
0.526
0.833
6.697
69.43
45.12
20.00
23.55
57.90
60.10
25.40
13.52
39.38
Fixed ETF (Ours)
Projector
12.62
0.393
0.490
0.468
0.316
91.38
65.72
64.51
64.97
82.22
97.42
43.17
21.51
66.36
Encoder
15.52
2.175
0.603
0.616
5.364
71.52
47.24
25.10
24.32
63.67
67.81
21.56
13.55
41.85
Detection Error
Plastic
Projector
15.10
0.498
0.515
0.428
1.422
63.05
47.87
62.45
70.07
80,88
98.95
89.37
79.25
74,00
Enender
23.64
13.953
0.526
0.833
6.697
81.27
98.82
93.33
86.48
79.98
99.40
91.25
93.88
90.55
Fixed ETF (Ours)
Projector
12.62
0.393
0.490
0.468
0.316
60.85
48.23
42.35
67.69
56.51
99.04
76.32
69.84
65.10
Encoder
15.52
2.175
0.603
0.616
5.364
67.17
98.14
81.76
84.95
84.57
99.70
97.36
87.34
87.62
H.8. Computational Efficiency & Scalability
same training time (see Table 23).
Our proposed method is computationally efficient and does require higher computational costs than standard DNNs. It introduces two additional components compared to stan- dard DNN architecture and training protocol:
FLOPs. In terms of FLOPs floating-poin operations per second), both our method and the baseline require almost the same amount of computation. For FLOPs analysis, we use DeepSpeed 2 with the same GPU (single NVIDIA RTX A5000) across compared models. As shown in Table 23, the overhead introduced by our method remains below 0.3% in all cases, which we believe is trivial and well-justified given the observed performance gains.
1. Entropy regularization applied at the encoder's output.
2. A frozen ETF projector (two MLP layers) following the DNN backbone.
For entropy regularization, we employ an efficient batch- level nearest neighbor distance computation, which incurs negligible computational overhead during training. Regard- ing the ETF projector, since it remains frozen and does not undergo gradient updates, it does not introduce any notice- able training costs beyond those of the baseline DNN.
Scalability, Here we ask: does the proposed method scale to deeper architectures? Our method is inherently compatible with deeper architectures since the ETF projector (two MLP layers) can be seamlessly integrated into encoders of any depth. Additionally, while deeper DNNs typically exhibit stronger NC in their top layers, our entropy regularizer effectively mitigates NC in encoders of any depth. As shown
Training Time. When training DNNs on ImageNet-106 (ID dataset) for 100 epochs using four NVIDIA RTX A5000 GPUs, both our method and the baseline require almost the
Pnttps: / gi thub. com/
10

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Controlling Neural Collapse Enhances Out-of-Distrlbation Detection and Transfer Learning
Stronger Neural Collapse (NC1)
With Entropy Reg.
-1.0
0.6
Without Entropy Reg.
1.5
0.5
-2.0
Entropy
-2.
0.4
-3.0
0.3
-3.5
With Entropy Reg.
-4.0
Without Entropy Reg.
0.2
20
40
60
80
100
20
60
100
Epoch
Epoch
Impact of Entropy Regularization NC1
(b) Entropy Dynamics
Stronger Neural Collapse (NC1)
30
With Entropy Reg.
0.9
Without L2 Norm
Without Entropy Reg.
With L2 Norm
25
Effective Rank
20
15
10
0.5
0.4
0
20
40
60
80
100
20
40
60
80
100
Epoch
Epoch
(c) Effective Rank Dynamics
(d) Impact of L2 Normalization on NC1
Figure 9: Analyzing entropy regularization L2 normalization. (a) Entropy regularization reduces neural collapse (indicated by higher NC1 values) the encoder. (b) Entropy regularization increases the entropy of encoder embeddings otherwise entropy remains unchanged. (c) Entropy regularization increases the effective rank of encoder embeddings otherwise effective rank remains low the number of classes (i.e., 10 ImageNet classes). (d) T-2 normalization increases neural collapse (indicated by lower NCI values) in the projector. For this analysis, we train VGG17 networks on the ImageNet- 10 subset (10 ImageNet classes) for 100 epochs.
in Table 11, our method performs effectively with both R and ResNet34, highlighting its scalability. Similar trend is observed for VIT' (Table 12).
spider, theater curtain, honeycomb, garter snake, wild boar, pedestal, bassinet, pickup, American lobster, sarong, mouse- trap, coyole, hard disc, chocolate sauce, slide rule, wing, cauliflower, American Staffordshire terrier, meerkat, Chi- huahua, lorikeet, bannister, tripod, head cabbage, stinkhorn, rock crab, papillon, park bench, reel, toy terrier, obelisk, walking stick, cocktail shaker, standard poodle, cinema, car- bonara, red fox, little blue heron, gyromitra, Dutch oven, hare, dung beetle, iron, bottlecap. lampshade, mortarboard, purse, boathouse, ambulance, milk can, Mexican hairless, goose, boxer, gibbon, football helmet, car wheel, Shih-Tzu, Saluki, window screen, English foxhound, American coot, Walker hound, modem, vizsla, green mamba, pineapple, safety pin, borzoi, tabby, fiddler crab, leafhopper, Chesa- peake Bay retriever, and ski mask.
Classes of ImageNet-100 ID Dataset
We list the 100 classes in the ID) dataset, ImageNet- 100 (Tian et al., 2020). This list can also be found at: https: /ttps://github.. httttbbtuuuutggg///////tt.... com/ httlli..ccooobbbbbb Http:HobbbiLoogg/////o htce/blobbmaatter//mageeeett
Rocking chair, pirate, computer keyboard, Rotweiler, Great Dane, tile roof, harmonica, langur, Gila monster, hognose snake, vacuum, Doberman, laptop, gasmask, mixing bowl, robin, throne, chime, bonnet, komondor, jean, moped, tub, rotisserie, African hunling dog, kuvasz, stretcher, garden

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Controlling Neural Collapse Enhances cuttof-Diisrrbbation Detection and Transfer Learning
Table 16: L2 Normalization. VGG17 models are trained on ImageNet-100 dataset (ID) and evaluated on eight OOD datasets. All models incorporate entropy regularization and the ETF projector to control NC. The same color highlights the rows to compare. For OOD detection, we report DET (%).
Method
Ei 4
Neural Collapse I NC2
DET
IN
NC1
NC3
NC4
IN-R 200
CIFAR
Flowers
NINCO
CUR
Aircrafts
Pets
STT.
Avg.
100
100
102
64
200
100
37
10
No L2 Norm
Projector
12.74
0.579
0.538
0,349
1.339
57.43
49.41
62.35
69.81
58.04
99.58
85.28
69.53
68.93
Encoder
14.70
1.788
0.633
0.823
10.643
77.08
96.77
91.18
92.35
89.47
99.64
89.51
85.31
90.16
Lg Norm
Projector 12.62
0.393
0.490
0.468
0.316
60.85
48.23
42.35
67.69
56.51
99.04
76.32
69.84
65.10
Encoder
15.52
2.175
0.603
0.616
5.364
67.17
98.14
81.76
84.95
84,57
99.70
97.36
87.34
87.62
Table 17: Batch Norm Vs. Group Norm. VGG17 models are trained on ImageNet- dataset (ID) and evaluated on eight OOD datasets. All models incorporate entropy regularization and the ETF projector to control NC. The same color highlights the rows to compare. Group norm integrated with weight standardization, All metries except NC are reported in percentage. For OOD transfer we report GEN (%) whereas for OOD detection we report DET (%).
Method
Ew IN 100
Neural Collapse NC2
OOD Datasets
NC1
NC3
NC4
IN-R
CIFAR
Flowers
NINCO
CUB
Airerafts
Pets
STI.
Avg.
200
100
102
64
200
100
37
10
Transfer Error
Batch Norm
Projector
12.52
0.372
0.669
0,263
0.536
89.43
66.00
63.14
64.46
83.00
94.57
38.65
21.30
65.07
Encoder
14.54
1.401
0.605
0,590
25.611
78.02
53.34
49.51
33.25
74.08
85.27
25.46
16.75
51.96
Group Norm
Projector
12.62
0.393
0.490
0.468
0.316
91.38
65.72
64.51
64.97
82.22
97.42
43.17
21.51
66.36
Encoder
15.52
2.175
0.603
0.616
5.364
71.52
47.24
25.10
24.32
63.67
67.81
21.56
13.55
41.85
Detection Error +.
Batch Norm
Projector
12.52
0.372
0.669
0.263
0.536
$7.30
74.62
44.12
66.33
65.14
99.19
75.93
73.13
69.47
Encoder
14.54
1.401
0.605
0.590
25.611
92.17
99.77
91.08
91.41
99.48
98.62
85,39
93.26
93.90
Group Norm
Projector
12.62
0,393
0.490
0.468
0,316
60.85
48.23
42.35
67.69
56,51
99.04
76.32
69.84
65,10
Encoder
15.52
2.175
0.603
0.616
5.364
67.17
98.14
81.76
84,95
84,57
99.70
97.36
87.34
87.62
Table 18: Comprehensive Comparison with Baseline. Various DNNs are trained on ImageNet- 100 dataset (ID) and
evaluated on eight OOD datasets. Baseline models do not incorporate mechanisms like entropy regularization or the ETF projector to control NC. NC metrics are computed using the penultimate- embeddings. For OOD transfer we report EGEN (%) whereas for OOD detection we report DET (%).
Model
Neural Collapse AC2
OOD Datasets
IN
AC1
AC3
AC4
IN-R 200
CIFAR
Flowers 102
NINCO
CUB
Aircrafts 100
Pets
STL
Avg.
100
100
64
200
Transfer Error
VGG17
12.18
0.766
0.705
0.486
37.491
75.60
50.11
42.75
29.17
71.35
84.13
27.58
15.65
49.54
VGG17+0urs
12.62
0.393
0.490
0.468
0.316
71.52
47.24
25.10
24.32
63.67
67.81
21.56
13.55
41.85
Detection Error I.
VGG17
12.18
0.766
0.705
0.486
37.491
96.02
97.16
97.94
93.11
95.19
98.59
87.33
94.05
94.92
VGGI7+0urs 12.62
0.393
0.490
0.468
0.316
60.85
48.23
42.35
67.69
56.51
99.04
76.32
69.84
65.10
Transfer Error
ResNet18 15.38
1.11
0.658
0.590
31.446
75.75
49.48
41.37
30.02
69.80
82.75
29.63
16.53
49.42
ResNet18+Ours 16.14
0.341
0,456
0.306
0.540
74.17
53.33
31.37
28.15
68.85
81.61
27.72
16.56
47.72
Detection Error
ResNet18
15.38
1.11
0.658
0,590
31.446
98.40
98.85
98.33
96.68
96.60
99.67
92.40
98.25
97.40
ResNet18+Ours 16.14
0.341
0.456
0.306
0.540
67.92
61.21
71.18
71.09
23.20
99.28
81.41
82.29
69.70
Transfer Error
VI--T
31.78
2.467
0.657
0.601
1.015
82.18
52.64
41.67
32.74
63.48
81.61
45.11
22.00
$2.68
ViT-T+Ours
32.04
2.748
0.609
0.798
1.144
82.28
$2.00
42.94
30.36
63.15
84,31
44.86
21.13
52.63
Detection Error 1
VIT-T
31.78
2.467
0.657
0.601
1.015
85.18
91.70
87.06
89.54
87.78
98.35
91.77
89.99
90.17
Vir-T+Ours
32.04
2.748
0.609
0.798
1.144
81.12
60.81
77.55
82.40
79.05
99.10
95.15
90.06
83.16
12

---

Controlling Neural Collapse Enhances Out-of- Detection and Transfer Learning
Table 19: Projector Design Criteria. VGG17 models are trained on ImageNet- 100 dataset (ID) and evaluated eight OOD datasets. All compared projectors are configured as fixed simplex ETFss And, entropy regularization is used in all cases. The same color highlights the rows to compare. Our final model has depth 2 and performs better than other variants. For OOD transfer we report &GEN (%) whereas for OOD detection we report & DET (%)
Criteria
EiD 1
Neural Collapse NC2
OOD Datasets
IN
NC1
NC3
NC4
IN-R
CIFAR
Flowers 102
NINCO 64
CUB 200
Aircrafts 100
Pets
STL
Avg.
100
200
100
37
Transfer Error +
Depth=1
Prujector
12.86
0.375
0.649
0,500
1.157
90.27
64.61
60.88
55.02
81.12
96.34
44.34
23.04
64.45
Encoder
16.34
1.673
0.667
0.589
7.936
74.08
50.61
30,00
28.06
66.74
71.95
25.73
15.75
45.37
Depth=2 (Ours)
Projector
12.62
0.393
0.490
0.468
0.316
91.38
65.72
64.51
64.97
82.22
97.42
43.17
21.51
66.36
Encoder
15.52
2.175
0.603
0.616
5.364
71.52
47.24
25.10
24.32
63.67
67.81
21.56
13.55
41.85
Width=2
Prujector
13.48
0.320
0.667
0.376
0.493
89.88
66.46
64.51
53.40
82,50
95.77
41.76
23.90
64.77
Encoder
16.46
2.341
0.607
0.646
5.899
73.05
50.61
27.25
25.60
64.84
67.87
22.35
15.10
43.33
Detection Error
Depth=1
Prujector
12.86
0.375
0.649
0.500
1.157
80.15
95.98
81.68
84.18
92.75
98.38
73.62
92.24
87.37
Encoder
16.34
1.673
0.667
0.589
7.936
62.72
95.04
84.65
84.95
92.22
99.43
89.75
83.66
86.55
Depth=2 (Ours)
Projector
12.62
0.393
0.490
0.468
0.316
60,85
48.23
42.35
67.69
56,51
99.04
76.32
69.84
65,10
Encoder
15.52
2.175
0.603
0.616
5.364
67.17
98. 14
81.76
84.95
84.57
99.70
97.36
87.34
87.62
Width=2
Projector
13.48
0.320
0.667
0.376
60.83
$1.96
67.77
$7.70
99.52
79.29
75.33
69.73
Encoder
16.46
2.341
0.607
0.646
80
97.64
89.61
83,42
88.89
98.89
98.58
94.39
89.78
Table 20: Comprehensive Results with SGD Optimizer. VGG17 models are trained on ImageNet-100 dataset (ID) using the SGD optimizer. Baseline models do not incorporate mechanisms like entropy regularization or the ETF projector to control NC. lower NC value indicates stronger neural collapse. For OOD transfer we report &GNN (%) whereas for OOD detection we report DET (%). OOD performance averaged across eight OOD datasets. The same color highlights the compare.
Model
ED L
Neural Collapse NC2
OOD Datasets
IN
.C11
NC3
NC4
IN-R 200
CIFAR 100
Flowers 102
NINCO
CUB
Aircrafts
Pets
STL
Avg.
100
64
200
100
37
10
Transfer Error
VGG17
Encoder
13.06
1.017
0.449
0.479
26.459
82.10
56.82
59.71
39.20
77.70
88.21
35.00
18.58
57.17
VGG17+0urs
Projector
13.18
0.087
0.468
0.267
0.264
84.48
72.35
47.28
85.88
94.66
42.85
25.93
64.77
Encoder
15.36
0.459
0.804
0.972
3.898
78.20
57.27
45.59
32.06
74.27
74.49
27.58
17.80
50.91
Detection Error
VGG17
Encoder
13.06
1.017
0.449
0.479
26.459
81.63
93.34
87.25
86.31
94.89
97.93
84.76
91.40
89.69
VGG17+Ours
Projector
13.18
0.087
0.468
0.267
0.264
52.28
35.80
60.59
67.60
67.77
67.35
67.52
67.60
60.81
Encoder
15.36
0.459
0.804
0.972
3.898
93.58
95.11
68.43
84.01
87.42
87.16
87.00
86.48
86.15
13

---

Controlling Neural Collapse Enhances Out-of-Distribation Detection and Transfer Learning
Table 21: Fixed ETF Classifier Vs. Plastic Classifier, VGG17 models trained on ImageNet-100 dataset (ID) and evaluated on eight OOD datasets. All models incorporate entropy regularization and the ETF projector to control NC; only the classifier (final layer) differs, being either trainable or fixed ETF. The same color highlights the rows to compare. All metrics except NC are reported in percentage. For OOD transfer we report GEN (%) whereas for OOD detection we report DET (%).
Classifier
EID
Neural Collapse
OOD Datasets
(Last layer)
IN
NC1
NC2
NCC
.VC4
IN-R 200
CIFAR 100
Flowers 102
NINCO
CUB
Aircrafts 100
Pets
STL
Avg.
100
64
200
Transfer Error 1
Fixed ETF
Prujector
13.56
0.088
0.702
0.374
0.379
98. 18
84.28
92.25
96.94
96.86
97.60
72.23
36.59
84.37
Encoder
16.40
3.794
0.773
0.786
54.24
82.47
63.19
55.98
36.31
81.00
88.36
31.18
20.88
57.42
Plastic (Ours)
Projector
12.62
0.393
0.490
0.468
0.316
91.38
65.72
64.51
64.97
82.22
97.42 42
43.17
21.51
66.36
Encoder
15.52
2.175
0.603
0.616
5.364
71.52
47.24
25.10
24.32
63.67
67.81
21.56
13.55
41.85
Detection Error
Fixed ETF
Projector
13.56
0.088
0.702
0.374
0.379
73.80
26.45
73.04
68.20
55.80
98.98
96.05
63.56
69.49
Encoder
16.40
3.794
0.773
0.786
54.24
81.03
98.98
81.57
87.25
97.29
86.48
93.11
90.59
Plastic (Ours)
Projector
12.62
0.393
0.490
0.468
0.316
60.85
48.23
42.35
67.69
56.51
99.04
76.32
69.84
65.10
Encoder
15.52
2.175
0.603
0.616
5,364
67.17
98.14
81.76
84.95
84.57
99.70
97.36
87.34
87.62
Table 22: Comprehensive Comparison with NECO. We compare our method against SOTA OOD detection method NECO (Ammar et al., 2024). Since NECO does not address OOD generalization, we do not compare OOD generalization performance. Here, various DNNs are trained on ImageNet-100 100 dataset (ID) and evaluated on eight OOD datasets. NC metrics are computed using the penultimate-layer embeddings. lower NC value indicates stronger neural collapse.
Model
ED +
Neural Collapse
OOD Detection Error pET (%)
IN
NC1
NC2
AC3
AC4
IN-R
CIFAR
Flowers
NINCO
CUB
Airerafts
Pets
STL
Avg.
100
200
100
102
64
200
100
37
10
VGG17
NECO
12.18
0.766
0.705
0.486
37.491
83.20
26.41
87.94
78.74
83.45
83.53
96.76
82.49
77.82
Ours
12.62
0.393
0.490
0.468
0.316
60.85
48.23
42.35
67.69
56.51
99.04
76.32
69.84
65.10
ResNet18
NECO
15.38
1.11
0.658
0.590
31.446
93.88
66.81
94.12
90.05
90.05
90.05
90.05
90.05
88.13
Ours
16.14
0.341
0.456
0.306
0.540
67.92
61.21
71.18
23.20
99.28
81.41
82.29
69.70
VIT-T
NECO
31.78
2.467
0.657
0.601
1.015
79.90
74.76
82.84
84.44
85. 82.12
98.50
92.67
87.10
85.67
Ours
32.04
2.748
0.609
0.798
1.144
81.12
60.81
77.55
82.40
79.05
99.10
95.15
90.06
83.16
Table 23: Compute Overhead. We compare our method with baseline DNNs in terms of FLOPs and training time. Training time (wall-clock) is reported in minutes.
Model
FLOPs Comparison
Time Comparison
FLOPs | % Increase
Time (Mins) 1
% Increase
VGG17
4,955.622,740,132.. 132,886
307.80
VGG17 + Ours
4956,972.705.684
+0.0272%
307.98
+0.0585 %
ResNet18
161,500,,10,8255472
136.81
ResNet18 + Ours
462,031,,42,,44.0000
+0.1152%
137.04
+0.1681 %
ResNet34
931,123,885,195228
140.89
ResNet34 Ours
931,655.516,833.799
+0.0571%
141.26
+0.2626%
ViT-T
271.301,725,913.088
63.03
VIT-T + Ours
271.376.414,5399776
+0.0275%
63.18
+0.2380
VIT-S
1,068,921 ,092,308,992
0068,921,0922308.999
102.02
VIT-S Ours
1,069,219.622.567.0044
+0.0279%
102.24
+0.2156%
14


Reviews:
[{"review_id": "6vqcAM1WOW", "reviewer": "Reviewer_NGXx", "summary": "The paper proposes a novel claim that: stronger NC(neural collapse) improves OOD detection but degrades generalization, while weaker NC enhances generalization at the cost of detection. The explanation of above claim is sufficient and the experiment proves the statement.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful reviews and valuable feedback. Please let us know if you have any questions or suggestions. We would be happy to address them.", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "Yes", "theoretical_claims": "I do not check proofs for theoretical claims.", "experimental_designs_or_analyses": "The experiments are conducted on ImageNet-100 with different architectures. The balance of generalization and detection performance is clear and verify their statement about the generalization-detection trade-off.", "relation_to_broader_scientific_literature": "None", "essential_references_not_discussed": "The related work is sufficiently listed and discussed.", "other_strengths_and_weaknesses": "The paper is well organized and expressed, and easy to understand.", "other_comments_or_suggestions": "None", "questions_for_authors": "None", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FyZLeILWYd", "reviewer": "Reviewer_BRBv", "summary": "This paper studies the role of neural collapse on the out-of-distribution (OOD) detection and generalization tasks. The authors also propose an entropy regularization technique to control the degree of collapse in intermediate layers and show that stronger collapse in the final layers can aid in OOD detection but can hurt OOD generalization.\n\n## update after rebuttal\n\nThe authors have addressed my concerns and have also assured that the code will be open-sourced. Overall I appreciate the extra experiments that they did during the rebuttal to further justify their approach (when compared to previous works). I also increased my score from 2 -> 3 after the rebuttal.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful reviews and valuable feedback. We have carefully considered your concerns and tried to address them. Below, we have provided detailed responses to each review separately.\n\n# Clarifications on Tables 2 and 3\n\n - In Table 3, we isolate the effect of entropy regularization while keeping all other components identical. Specifically, we compare a baseline VGG17 with our method, where both models include the ETF projector, and only entropy regularization is either applied or omitted. In contrast, in Table 2, the baseline VGG17 does not include the ETF projector, which explains the differences in ID/ OOD performance between the first rows of the two tables.\n\n - The second rows of Tables 2 and 3 correspond to the same model (VGG17 + ETF projector + entropy regularizer). The NC values differ because, in Table 2, we measure NC after the projector, whereas in Table 3, we report NC at the encoder. Since entropy regularization is applied to the encoder, Table 3 explicitly compares its effect on encoder representations.\n Table 1 reports NC values for both the encoder and the projector (first two rows), which remain consistent with Tables 2 and 3.\n\n We will add more details to the table captions to enhance clarity. Please let us know if further clarification is needed.\n\n# Comparison with NECO\n\nWe have included NECO [1] in our related work and discussed its relevance. NECO is a *post-hoc OOD detection method* based on Neural Collapse, requiring a pre-trained DNN. It does not focus on OOD generalization and representation learning. In contrast, our method is designed to learn good representations during training to improve both OOD detection and generalization.\n\nReference: [1] Ammar et al., NECO: Neural collapse based out-of-distribution detection, ICLR, 2024\n\n**Our Method Vs. NECO:** To compare against NECO, we trained various DNNs on ImageNet-100 (ID) and evaluated them on eight OOD datasets. Our method outperforms NECO in average OOD detection error by an absolute 12.72\\% for VGG17, 18.43\\% for ResNet18, and 2.51\\% for ViT-T.\n\n**Below, we report the avg. OOD detection error (\\%):**\n| Method | VGG17 | ResNet18 | ViT-T |\n|----------|----------|----------|----------|\n| NECO | $77.82$ | $88.13$ | $85.67$ |\n| Ours | $\\mathbf{65.10}$ | $\\mathbf{69.70}$ | $\\mathbf{83.16}$ |\n\nWe appreciate the reviewer’s suggestion and have incorporated the results in our paper.\n\n# Additional Suggestions\n\n- We have revised our statement about the theoretical framework explaining how a fixed simplex ETF projector enforces NC for OOD detection. We have stated that this analysis is empirical. We thank the reviewer for the valuable feedback.\n\n- We have revised this statement ``How BN or GN impacts neural collapse is unexplored in previous work.\" since BN has been studied in prior work [2]. We thank the reviewer for pointing this out.\n\n- Reference: [2] Ergen, T., et al. ``Demystifying Batch Normalization in ReLU Networks: Equivalent Convex Optimization Models and Implicit Regularization.\" ICLR, 2022.\n\n# Additional Weaknesses\n\n**Why is comparison with prior works difficult?**\n\n- Prior works typically focus on either OOD detection or OOD generalization, whereas we aim to improve both objectives. Some recent works address both objectives, but they differ significantly in problem formulation and methodology. As detailed in Sec. 3 and 5, we cannot directly compare against such methods because:\n\n1. They rely on additional OOD training data, whereas our method works without such data. Adapting these methods for comparison would require substantial modifications, including redefined training objectives and architectures, which are beyond the scope of our work. Therefore, a direct *apple-to-apple* comparison is difficult.\n\n2. For OOD generalization, prior work *solely* focuses on *co-variate shifted OOD* data that has similar label space as ID data (e.g., a car in *sunny weather* as ID vs. a car in *rainy weather* as OOD). In contrast, we focus on *semantic shifted OOD* data that does not overlap with ID labels. Thus, unlike our method, previous methods are not designed to handle semantic OOD data.\n\nWe would appreciate the reviewer’s feedback on whether this justification sufficiently addresses the concern.\n\n# Questions\n\n- **Clarifications on Table 12:** Table 12 examines the impact of entropy regularization on the OOD generalization of a regular VGG17 without any projector. The regularization is applied to the last encoder layer (penultimate layer) before the final classifier layer. All models are compared identically. We are not sure if we understand the last part of the question relating to the VGG+ETF projector because we are not comparing it in Table 12.\n\n- Additionally, in Table 11, we compare how entropy regularization impacts OOD generalization when VGG17 models include ETF projectors.\n\nWe appreciate the reviewer's insightful feedback and suggestions. Please let us know if further clarifications are needed.", "claims_and_evidence": "Experiments and analysis with various model architectures and datasets justify the empirical claims (some concerns are mentioned in the following sections). However, the authors do not justify their claims about a \"theoretical framework explaining how a fixed Simplex ETF projector enforces NC for OOD detection\". This claim must be revised to state that such an analysis is only experimental.", "methods_and_evaluation_criteria": "The NC and evaluation metrics, datasets, and models are relevant to the analysis.", "theoretical_claims": "The authors present the proof of their Proposition 4.1 in Appendix E. (on the entropy regularization tending to -inf with stronger collapse in the features).", "experimental_designs_or_analyses": "The design of the experiments is valid and sound. However, I have some concerns with the reported numbers and would like the authors to clarify the following:\n\n1. Why are the baseline VGG17 results for $\\mathcal{E}_{ID}$ in Table 2 (first row) different from the non entropy regularization case in Table 3 (first row)? The captions do not indicate any major difference.\n\n2. Next, the second row in Table 2 (i.e, the proposed method with VGG17 backbone + fixed ETFs and entropy regularization) has the same ID and OOD performance as a VGG17 model with entropy regularization in the second row of Table 3. Does the latter also employ the fixed ETFs as projectors? If yes, what is the difference between these two models? How can the ID and OOD performance be the same even though the NC values are different?", "relation_to_broader_scientific_literature": "This paper analyzes the role of neural collapse (NC) on OOD generalization and detection tasks in NNs. In the NC literature there has been extensive work on analyzing the optimality of collapse in intermediate and final layers and the implications on transfer learning and in-domain generalization. However, the study on OOD detection+generalization and NC is still in its nascent stage. This paper is well positioned to present an initial unifying perspective (at least empirically) and encourage future efforts in this direction.", "essential_references_not_discussed": "1. A relevant previous effort on NC and OOD-Generalization [1] is not discussed in the paper. Can the authors provide some comparisons with this paper ?\n\n2. I have noticed that in Section 6.5 (in Appendix), the authors state that \"How BN or GN impacts neural collapse is unexplored in previous work\". This is not valid as previous works such as [2] have indeed studied this aspect. It would be great if the authors can rectify this statement.\n\n[1] Ammar, Mouin Ben, et al. \"NECO: NEURAL COLLAPSE BASED OUT-OF-DISTRIBUTION DETECTION.\" The Twelfth International Conference on Learning Representations. 2024.\n\n[2] Ergen, T., et al. \"Demystifying Batch Normalization in ReLU Networks: Equivalent Convex Optimization Models and Implicit Regularization.\" International Conference on Learning Representations. 2022.", "other_strengths_and_weaknesses": "The paper is generally well-written and easy to follow. However, a lack of comparison with previous work is a bit concerning. In particular, for a given OOD detection/generalization task, I am not convinced that a comparison cannot be done with any of the previous work in the literature. I would like to know the authors' justification on this aspect.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "1. The entropy regularization applied to a standard VGG17 without any fixed ETF layer weights is shown in Table 12. Is the regularization applied to the last layer in this experiment? If yes, then it is not a fair comparison with VGG17 + 2 layer ETF architecture since technically the regularization should be applied to the last but second/third layer (depending on the usage of classifier head). Can the authors clarify this?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "9B3ftalHhd", "reviewer": "Reviewer_WkZc", "summary": "This paper empirically shows a trade-off between Out-of-distribution (OOD) detection and OOD generalization on multi-class classification task in the deep neural network (DNN) with in-distribution (ID) training data: a network can either impose a stronger neural collapse (NC) and improves OOD detection or weakens NC to enhances OOD generalization. \n\nTo enhance both aspects, this paper proposes a pipeline to train the backbone for OOD generalization and add a projector and linear probing after it for OOD detection, hence the backbone which encodes the embeddings only needs to be trained once. (See Figure 2 in the paper)\n\nTo modify the architecture to the transfer learning setting, this paper uses a novel *entropy regularization*, which mitigates NC, \nand replaces batch-normalization (BN) by group normalization (GN) plus weight standardization (WS). These changes further improves OOD detection and generalization performance.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful reviews and constructive feedback. We have carefully considered your concerns and tried to address them. Below, we have provided detailed responses to each review separately.\n\n# Clarifications\n\n**Non-vision datasets or tasks**\n\nMost prior work on OOD detection and/or OOD generalization has focused on vision datasets, and we align with this precedent. However, we recognize the importance of evaluating non-vision tasks for broader applicability. Due to time and computational constraints, we were unable to extend our work to non-vision domains within the rebuttal period. We consider this an important direction for future work and will discuss potential extensions to other modalities, such as audio and text.\n\n**Code and reproducibility**\n\nAt the time of submission, our code was not fully polished, which is why we did not include it. However, we plan to release a cleaned and well-documented version upon acceptance.\n\n**Performance with optimizers other than AdamW**\n\nWe have addressed this concern in Q2.\n\n# Additional Suggestions\n\n - We have revised the statement ``How BN or GN impacts Neural Collapse is unexplored in prior work.\" since BN has been studied in previous work [1]. We thank the reviewer for pointing this out.\n\n - **Essential References Not Discussed:** We have cited the relevant papers [1, 2] suggested by the reviewer. We have included NECO [2] in our related work and discussed its relevance. We thank the reviewer for the valuable feedback.\n\n  - **References:** \n 1. Pan, Leyan, and Xinyuan Cao. \"Towards understanding neural collapse: The effects of batch normalization and weight decay.\" arXiv preprint arXiv:2309.04644 (2023).\n 2.  Ammar, Mouïn Ben, et al. \"Neco: Neural collapse based out-of-distribution detection.\" ICLR, 2024.\n\n - **Comparison with NECO:**  NECO [2] is a SOTA OOD detection method based on Neural Collapse. To compare against NECO, we trained various DNNs on ImageNet-100 (ID) and evaluated them on eight OOD datasets. Our method outperforms NECO in average OOD detection error by an absolute 12.72\\% for VGG17, 18.43\\% for ResNet18, and 2.51\\% for ViT-T. More details are given in responses to reviewer BRBv. We have included these results in our paper. We thank the reviewer for the valuable suggestions.\n\n - We have improved the clarity of Figure 4 by flipping the axes, as per the reviewer’s suggestion.\n\n# Questions\n\n**Q1. Could authors include non-vision experiments?**\n\nWe have addressed this concern in the clarifications section above. We welcome any further thoughts on this.\n\n**Q2. Could authors perform experiments with SGD or Adam Optimizer?**\n\nRunning all experiments with SGD or Adam would be computationally expensive due to our extensive evaluation across five DNN architectures and eight OOD datasets. However, we conducted ablation studies by training VGG17 models with an SGD optimizer and evaluating them on eight OOD datasets. \nOur SGD results confirm that our method consistently improves OOD detection and generalization performance.\n- In particular, our method outperforms the baseline by an absolute 6.26\\%\nin OOD generalization and by an absolute 28.88\\% in OOD detection.\n- Also, we observe that our encoder reduces Neural Collapse (NC) and enhances OOD generalization by an absolute 13.86\\% compared to the projector. Whereas the projector intensifies NC and improves OOD detection by an absolute 25.34\\% compared to the encoder. \n- While SGD intensifies NC more than AdamW, AdamW achieves better overall performance (Table 16 in Appendix).\n\n**SGD Optimizer Results:**\n\n| Model | $\\mathcal{E}_{\\text{ID}}$ | NC1 | NC2 | NC3 | NC4 | $\\mathcal{E}_{\\text{GEN}}$ | $\\mathcal{E}_{\\text{DET}}$ |\n|----------|----------|----------|----------|----------|----------|----------|----------|\n| VGG17 | $13.06$ | $1.02$ | $0.45$ | $0.48$ | $26.46$ | $57.17$ | $89.69$ |\n| +Ours   | $13.18$ | $0.09$ | $0.47$ | $0.27$ | $0.26$  | $\\mathbf{50.91}$ | $\\mathbf{60.81}$ |\n\nWe have included these results in our paper. We will try to conduct more experiments with SGD/ Adam and include the results in the final paper. We would appreciate the reviewer’s feedback on whether this analysis sufficiently addresses the concern.\n\n**Q3. Why is there a gap in correlation figures?**\n\nWe observe this trend for VGG17 but not for ResNet18 (Fig. 7 in Appendix). We have double-checked our data and code, they seem fine. The potential reason could be the limited data points, typically more data points would show better trends. It might be better if we combine Fig. 1 and Fig. 7.\n\n**Q4. Could the paper include a formal proof of Proposition 4.1?**\n\nYes, we have included a detailed formal proof of Proposition 4.1 in Appendix E. We thank the reviewer for the valuable feedback and insightful questions.\n\nWe appreciate the reviewer's insightful feedback and suggestions. Please let us know if further clarifications are needed.", "claims_and_evidence": "The claims are supported clearly and evidently by various experiments and ablation tests.", "methods_and_evaluation_criteria": "The paper uses 8 commonly used OOD datasets, 3 DNN architectures trained on AdamW with cosine learning rate scheduler and linear warmup. (See Section 5)\n\nI think it is rather standard for vision OOD detection/generalization task.", "theoretical_claims": "There is only one theoretical claim, namely Proposition 4.1, with proof sketch in the main text which promises a detailed proof in Appendix E. While in Appendix E, the proof is still rather sketchy. I still believe the proposition is true, but I hope the proof can be written in a more formal way in the revised edition.", "experimental_designs_or_analyses": "While I generally acknowledge the validity of the experiments, I would like to point out that:\n\n1. The datasets used are all vision tasks. It would be more complete if the paper could draw similar conclusions on OOD detections/generalizations for other tasks with other input forms like audio or text. \n\n2. This paper only used AdamW as the optimisers, while previous work, to name a few, [1,2,3], mainly used SGD or Adam, to showcase neural collapse. I personally also experiment on neural collapse (with the same architectures but datasets smaller than those in this paper), and find out that AdamW does not show neural collapse. For example, in Table 4, the NC metrics NC2, NC3 are about 0.5, which are significantly away from zero, since the metric is normalised between 0 and 2. If SGD or Adam is used as optimisers, the metrics could drop to the magnitude by 1 or 2. I still believe the novel findings in this paper, like entropy regularization and encoder-projector-classifier pipeline, are significant works on OOD context, but since the title includes the term neural collapse, we should be careful on the experiments and NC metrics.\n\n\n---\n[1] Papyan, V., Han, X., and Donoho, D. L. Prevalence of neural collapse during the terminal phase of deep learning training. Proceedings of the National Academy of Sciences, 117(40):24652–24663, 2020.\n\n[2] Zhu, Z., Ding, T., Zhou, J., Li, X., You, C., Sulam, J., and Qu, Q. A geometric analysis of neural collapse with unconstrained features. Advances in Neural Information Processing Systems, 34:29820–29834, 2021.\n\n[3] Masarczyk, W., Ostaszewski, M., Imani, E., Pascanu, R., Miło´s, P., and Trzcinski, T. The tunnel effect: Building data representations in deep neural networks. Advances in Neural Information Processing Systems, 36, 2023.", "relation_to_broader_scientific_literature": "This paper clearly relates to the neural collapse phenomenon introduced by [1]. I am not an expert in OOD and transfer settings so I cannot draw more conclusions on this.\n\n---\n[1] Papyan, V., Han, X., and Donoho, D. L. Prevalence of neural collapse during the terminal phase of deep learning training. Proceedings of the National Academy of Sciences, 117(40):24652–24663, 2020.", "essential_references_not_discussed": "I think most of the essential related works are cited. A few more minor but related work which might be worth to cite can be:\n\n[4]: also about OOD and NC. Would be interesting to see which method could yield better performance.\n\n[5]: in contrary to line 421-422, \"How BN or GN impacts neural collapse is unexplored in previous work.\", BN has been explored in [5].\n\n----------\n\n[4] Ammar, Mouïn Ben, et al. \"Neco: Neural collapse based out-of-distribution detection.\" arXiv preprint arXiv:2310.06823 (2023).\n\n[5] Pan, Leyan, and Xinyuan Cao. \"Towards understanding neural collapse: The effects of batch normalization and weight decay.\" arXiv preprint arXiv:2309.04644 (2023).", "other_strengths_and_weaknesses": "The contributions of this paper seem original and it is delivered in a clear way. The experiments' description is very detailed.", "other_comments_or_suggestions": "Just a minor stuff, I would suggest to flip the axes of Figure 4: the dependent variable is NC1, which is usually shown on the y-axis.", "questions_for_authors": "Regarding my concerns on the experimental designs, I would like to ask:\n\n1. Could the paper include non-vision datasets in the revised edition to show that the contribution is universal among different data type?\n\n2. Could the experiment be repeated with different optimisers like SGD or Adam to see if the the effect is intensified, neutralised or unchanged on OOD performance?\n\n---\n\nregarding Figure 1 and 4, I would also like to know:\n\n3. why is there a gap between NC1 in the middle of the plots where no instance can achieve NC1 around 0.5? Is there an explanation for this or could it be a bug in the code?\n\n---\n\nregarding my concern in theoretical claims,\n\n4. could the paper include a formal proof of Prop. 4.1?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sAJspk3CKw", "reviewer": "Reviewer_Cfr3", "summary": "This paper explores the relationship between Neural Collapse, Out-of-Distribution detection, and OOD generalization. It provides that strong NC enhances OOD detection but damages generalization, while weaker NC has the opposite effect. In order to balance these objectives, the authors use a method that controls NC at different neural network layers.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thoughtful review and valuable feedback. We have carefully considered your concerns and tried to address them. Below, we have provided detailed responses to each review separately.\n\n# Weaknesses\n\n**W1. Is the proposed method computationally efficient?**\n\nYes, the proposed method is computationally efficient. It introduces two additional components compared to standard DNN architecture and training protocol:\n\n1. Entropy regularization applied at the encoder’s output.\n2. A frozen ETF projector (two MLP layers) following the DNN backbone.\n\n- For *entropy regularization*, we employ an efficient batch-level nearest neighbor distance computation, which incurs negligible computational overhead during training. \n- Regarding the *ETF projector*, since it remains frozen and does not undergo gradient updates, it does not introduce any noticeable training costs beyond those of the baseline DNN.\n\nTo quantify this, when training DNNs (VGG17, ResNet18/34, ViT-T/S) on ImageNet-100 (ID dataset) for 100 epochs using four NVIDIA RTX A5000 GPUs, both our method and the baseline require the same training time. Moreover, in terms of FLOPs (floating-point operations per second), both our method and the baseline require the same amount of computation. This indicates that our method does not require higher computational costs than standard DNNs. Below, we report the FLOPs and training time:\n\n| Model | Compute (FLOPs $\\times 10^{15}$) | Time (Hours) |\n|----------|----------|----------|\n| VGG17   | 4.96   | 5.13   |\n| VGG17+Ours   | 4.96   | 5.13   |\n| ResNet18   | 0.46   | 2.28   |\n| ResNet18+Ours   | 0.46   | 2.28   |\n| ResNet34   | 0.93   | 2.35   |\n| ResNet34+Ours   | 0.93   | 2.35   |\n| ViT-T   | 0.27   | 1.05   |\n| ViT-T+Ours   | 0.27   | 1.05   |\n| ViT-S   | 1.07   | 1.70   |\n| ViT-S+Ours   | 1.07   | 1.70   |\n\nWe will include this comparison in the final paper and appreciate the reviewer’s suggestion to provide this computational analysis.\n\n\n**W2. Does the proposed method scale to deeper architectures?**\n\n- Our method is inherently compatible with deeper architectures since the ETF projector (two MLP layers) can be seamlessly integrated into encoders of any depth. Additionally, while deeper DNNs typically exhibit stronger Neural Collapse (NC) in their top layers, our entropy regularizer effectively mitigates NC in encoders of any depth. \n- As demonstrated in Table 9 (Appendix), our method performs effectively with both ResNet-18 (13M) and ResNet-34 (23M), highlighting its scalability. Also, as shown in Table 10 (Appendix), our method improves OOD detection and generalization performance for both ViT-Tiny (6M) and ViT-Small (23M), demonstrating its scalability.\n\n\n# Questions\n\n**Q1. What is the computational overhead of the proposed method compared to standard classifiers?**\n\nAs detailed in W1, our method does not increase computational overhead compared to standard DNNs.\n\n\n**Q2. Can this approach be extended to other tasks such as detection, i.e., is the proposed approach agnostic to the task at hand?**\n\nYes, the proposed method is agnostic to the task at hand. Although we primarily focus on classification tasks, our method is applicable to other tasks, e.g., objection detection or other regression tasks. Our method employs an ETF projector after the DNN backbone (encoder). This ETF projector is designed to increase Neural Collapse in the last layer, thereby improving feature representations for downstream tasks, including classification and regression tasks. Our entropy regularization is also task-agnostic and compatible with both classification and regression objectives. We will discuss this in the main text. We thank the reviewer for the insightful questions. \n\nWe appreciate the reviewer's insightful feedback and suggestions. Please let us know if further clarifications are needed.", "claims_and_evidence": "The paper claims that NC has an inverse relationship with OOD detection and generalization and that controlling NC at different layers improves both objectives. These claims are supported by empirical evidence demonstrating:\n\n- strong correlation between NC and OOD detection -- generalization performance\n- improved results across multiple datasets when NC is manipulated using entropy regularization and a simplex ETF projector\n- theoretical justification linking entropy regularization to preventing NC + demonstrating the benefits of an ETF projector for OOD detection", "methods_and_evaluation_criteria": "The proposed method appears to be well-motivated and aligns with the problem at hand. Comparisons with baseline methods are performed on standard OOD benchmarks (e.g., NINCO, CIFAR-100, ImageNet-R). Metrics such as OOD detection error, OOD generalization error, and NC properties (NC1-NC4) are used. A range of architectures (VGG17, ResNet18, ViT) to ensure robustness of findings across varying architectures of different nature.", "theoretical_claims": "The theoretical component of the paper appears to be solid, with well-executed derivations explaining how entropy regularization mitigates NC and how a Simplex ETF projector enforces NC. The proofs seem to be accurate -- though further clarity in some steps of the entropy collapse argument could improve readability.", "experimental_designs_or_analyses": "The experiments appears to be well-structured and executed, with comprehensive comparisons to baselines and ablation studies. The encoder with entropy regularization maintains feature diversity for better OOD generalization. The projector with a fixed ETF structure enhances OOD detection by inducing NC and the projector, _o_ in _f-o-g_, structure appears to be well evaluated. he authors also examine the hyperparameter and architectural sensitivity.", "relation_to_broader_scientific_literature": "The paper builds on literature in NC, OOD detection, and transfer learning and cites key works and recent advances in ETF-based learning. It extends prior studies by linking NC explicitly to both detection and generalization tasks and proposing a practical mecanism to control NC at different network layers.", "essential_references_not_discussed": "The paper appears to be covering most relevant prior work", "other_strengths_and_weaknesses": "- Providing a detailed analysis of computational efficiency and trade-offs could benefit the evaluation of the proposed framework\n- Discussing potential limitations in scaling the method to deeper architectures wrt the NC could enhance clarity", "other_comments_or_suggestions": "No comments", "questions_for_authors": "- What is the computational overhead of the proposed method compared to standard classifiers?\n- Can this approach be extended to other tasks such as detection, i.e., is the proposed approach agnostic to the task at hand?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper studies the relationship between Neural Collapse (NC) and out-of-distribution (OOD) detection and generalization. It demonstrates that strong NC improves OOD detection but harms generalization, while weaker NC has the opposite effect. To balance these objectives, the authors then propose a method to control NC across different neural network layers.

The idea of fixing classifiers to induce NC and the impact of NC on OOD generalization have been explored before, but this paper makes a valuable contribution by leveraging NC to improve both OOD detection and generalization, and by demonstrating the effectiveness of controlling NC in achieving these goals.
Decision: Accept (poster)