Let OOD Feature Exploring Vast Predefined Classifiers
Authors: Kewen Xia
Xiaodong Yue
WeiZhipeng
Yaxin Peng
Zihao Li
Jianxiang Zhu
Jie Shi
PeilinXu
Venue: ICLR 2026 Poster (2026)
Keywords: Out of Distribution
Representation Learning
Neural Collapse
Evidential Deep Learning
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

Abstract: Real-world out-of-distribution (OOD) data exhibit broad, continually evolving distributions, rendering reliance solely on in-distribution (ID) data insufficient for robust detection. Consequently, methods leveraging auxiliary Outlier Exposure (OE) data have emerged, substantially enhancing generalization by jointly fine-tuning models on ID and large-scale OE data. However, many existing approaches primarily enforce orthogonality between ID and OE features while pushing OE predictions toward near-uniform, low-confidence scores, thus overlooking the controllability of representation geometry. We propose Vast Predefined Classifiers (VPC), which constructs a pre-specified Orthogonal Equiangular Feature Space (OEFS) to explicitly separate ID and OOD representations while capturing the rich variability of OOD features. We employ evidential priors to align ID features with their class-specific Equiangular Basic Vectors (EBVs), thereby preserving ID performance. In parallel, a new VEBV loss encourages OE features to explore the subspace spanned by Vast EBVs (VEBVs), enabling a rich characterization of diverse OOD patterns. This dual optimization, coupled with the prescribed geometric representation space, promotes optimal orthogonality between ID and OOD representations. Furthermore, we introduce the VPC Score, a discriminative metric based on the L2 activation intensity of features over the predefined classifiers. Extensive experiments across diverse OOD settings and training paradigms on benchmarks including CIFAR-10/100 and the ImageNet-1k, demonstrate strong and robust performance, validating VPC's effectiveness. Code is available at https://github.com/eightnight2049/VPC.

Published as a conference paper at ICLR 2026
LET OOD FEATURE EXPLORE VAST PREDEFINED CLASSIFIERS
Kewen Xia!, Xiaodong Yue Zhipeng Wei Yaxin Peng?, Zihao Lit, Jianxiang Zhu?, Jie Shi" & Peilin Xu!
"School of Computer Engineering and Science, Shanghai University, China
SSchool of Future Technology, Shanghai University, China
Institute of Artificial Intelligence, Shanghai University, China kewen_xi xi cout look com, yswant fly@shu. eeteshu.edu.cco cn
ABSTRACT
Real-world out-of-distribution (OOD) data exhibit broad, continually evolving distributions, rendering reliance solely on in-distribution (ID) data insufficient for robust detection. Consequently, methods leveraging auxiliary Outlier Exposure (OE) data have emerged, substantially enhancing generalization by jointly fine- tuning models on ID and large-scale OE data. However, many existing approaches primarily enforce orthogonality between ID and OE features while pushing OE predictions toward near-uniform, low- controllability of representation geometry. We propose Vast Predefined Classifiers (VPC), which constructs a pre-specified Orthogonal Equiangular Feature Space (OEFS) to explicitly separate ID and OOD representations while capturing the rich variability of OOD features. We employ evidential priors align ID features with their class-specific Equiangular Basic Vectors (sB thereby preserving ID performance. In parallel, a new VEBV loss encourages OE features explore the subspace spanned by Vast EBVs (VEBVs), enabling a rich characterization of diverse OOD) patterns. This dual optimization, coupled with the prescribed geometric representation space, promotes optimal orthogonality between ID) and OOD representations. Furthermore, we introduce the VPC Score, a discriminative metric based on the L2 activation intensity of features over the predefined classifiers. Extensive experiments across diverse OOD settings and training paradigms on benchmarks including CIFAR- 10/100 and the ImageNet- Ik, demonstrate strong and robust performance, validating VPC's effectiveness. Code is available at https: / http:/ttthuu..ooo ctwpuu..cccceeeee www.:iigggiigggtttgggggg
scores, thus overlooking the
INTRODUCTION
In open-world scenarios, deep neural networks (DNNs) must not only accurately recognize in- distribution (ID) samples encountered during training, but also robustly distinguish and detect unseen out-of-distributiin (OOD) data (Hendrycks & Gimpel) 2016 Liu et al. 2020 2021). Because real- world OOD data are often diverse and continuously evolving Ye al. 2025). relying solely on ID data during training is no longer sufficient to achieve desirable robustness. A recent and effective strategy is therefore leverage large-scale auxiliary outlier-exposure (OE) data and fine-tune the model jointly on ID and OE samples, which markedly improves generalization to OOD (Hendrycks)
al. 2018 Katz-Samuels al. 2022).
However, typical outlier-exposurg (OE)-based pipelines either encourage the model to produce uniform, low-confidence predictions on OE data (Hendrycks et al. 2018), or regularize post-hoc scores such maximum softmax probability (MSP) and energy Galues(Hendryce) & Gimpel 2016 Liu et al.] 2020). Beyond logit-based, gradient or neighborhood-
detectors provide
complementary signals (Huang et al. 2021 Sun et al.) 2022), but these methods fail to explicitly construct a controllable representation geometry during training (Liang et al. 2017 Sun et al.) 2021 Wang et al.) 2022a Djurisic et al.) (022). This leads two issues: (i) unstable and incomplete separation between in-distrbuttion (ID) features and OE features, which may bias predictions toward ID classes at test time (Wu et al. 2024); and (ii) limited representational capacity for the rich and

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Published as a conference paper at ICLR 2026
Festere A Feater
ie YERVs
Daai AST
25
OEUS
Estracter
Lgge
Irqat Data
Projection
Normalized
Predefuued Chosifiers
Loc
Training
VPC Scoret 1
Figure I: Overview of the Proposed VPC OOD Detector. During joint ID/OE fine-tuning or one-stage training, the normalized feature space is aligned to an Orthogonal Equiangular Feature Space (OEFS) with predefined EB V LENG aligns ID to EBVs, LVEBV attracts OOD into the VEBV subspace, Loc enforces separation, and OODness is scored by the L2-based VPC Score, defined as the absolute difference between the L2 activation magnitudes of the two subspaces.
evolving spectrum of OOD patterns, problem that becomes more prominent especially when the semantic space scales up (Huang & Li 2021).
To overcome these limitations, we propose Vast Predefined Classifiers (VPC). framework that explicitly represents OOD variability by allocating a large set of predefined optimal classifiers distinct from ID classifiers. Our design inspired by the phenomenon of Neural Collapse (NC)(Papyan et al.) 2020), where the class means of features and classifiers approach a Simplex Equiangular Tight Frame (ETF) (Han et al. 202T), as well as by the equiangular basis vectors (EBVs) based feature alignment optimization to induce this phenomenon (Shen et al. 20255 Markou et al. 2024). The central intuition that, to preserve the ID decision structure while accommodating the diversity real OOD under joint ID/OE fine-tuning, the representation space must embed a controllable geometric prior with scalable capacity. Naive uniformization of OE outputs may fail to ensure stable separability and lacks sufficient and directional coverage for complex OOD modes (Yang al. 2025 Tang et al. 2025).
To this end, we anchor OOD discrimination on predefined EBVs as class prototypes, defining an Orthogonal Equiangular Feature Space (OEFS) on high-dimensional unit hypersphere. In OEFS, EBVs enforce within-class collapse for ID features, while abundant VEBVs span a broad, activatable OOD subspace. Guided by evidential priors, align ID features toward EBVs; OOD- oriented attraction with orthogonality- based isolation encourages activation in the VEBV subspace, maintaining geometric separation from the ID subspace. This induces an interpretable pattern: ID) activates VB,s, whereas OOD activates VEBVs. Building on this, the VPC Score computes L22 activation intensity over the predefined classifier space as a class-agnostic OOD measure, decoupled from any ID classifier head. Our contributions are summarized as follows:
We propose Vast Predefined Classifiers (VPC) framework: using an Orthogonal Equiangular Feature Space (OEFS) to structurally decouple ID and OOD representations, and introducing a controllable, expandable geometric subspace for OOD without sacrificing ID) performance.
We propose an evidential prior guided Neural Collapse (ENC) loss that stabilizes ID rep- resentations, and OOD-oriented VEBV subspace attraction loss with an orthogonality constraint regularization loss to maintain consistent ID-OOD separation.
We introduce the VPC Score as class-agnostic metric, achieving robust discrimination across diverse OOD modes and outperforming MSP/uncertainty scores.
We obtain consistent performance gains on representative OOD detection settings, validating the synergistic benefits of geometric priors and evidential modeling.
RELATED WORK
Auxiliary OE Data based OOD Detection. With partial OOD data, Outlier Exposure (OE) improves
detection by co-training on ID and external samples. Classic OE (Hendrycks et al. 2018 drives OOD outputs toward uniformity; follow-ups either design loss, energy, or metric learning objectives (Liu eet al.] 2020 Bai et al. 2023 Ying et al.] (201)) or mine/synthesize harder outliers to sharpen the boundary (Ming et al. 2022a; Chen et al. [202 Wang et al.) 2024a 2023) Zheng et al.) 2023). A parallel line of research focuses post-hoc models further enhance separability, such methods
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based feature processing (Sun et al. 2021 Liang et al. 2017: Wang al. 2022a b). activation shaping (Djurisic et al.) 2022). density estimation (Peng et al. 2024). J. or energy- based scoring (Zhang et al. 2022). However, most OE methods emphasize output- level level uniformization and provide little control over the geometry of representations. Our VPC remedies this by imposing a geometric prior and allocating OOD features predefined, expandable subspace, trained jointly with OF.
Evidential Deep Learning. Evidential deep learning (EDL) characterizes classification uncertainty
via Dirichlet evidence formulation (Sensoy et al., 2018 drawing on Dempster Shafer theory
and subjective logic (Sentz & Ferson 2002 Josang 2016). It has been widely adopted in various
tasks (Gao al., 2024). ranging from open-set set recognition (Bao al. 2021; Wang al. 2024b
Yu et al.]|]].000 (2024) and continual learning (Aguilar et al. 2023) to multi-view classification (Liu et al.
6022||225ag bl while also being generalized to regression to model both aleatoric and epistemic uncertainty (Amini et al., 2020). Departing from the common view of EDL as an output-layer calibration tool, we embed evidential prior into feature- prature-rototype alignment: controlling the rate of evidence generation to ensure stable convergence of ID features to EBVs while mitigating overbias caused by optimization interference from OE samples and hard outliers. Coupled with our geometric constraints, VEBV subspace attraction and orthogonal separation from the ID subspace, evidential prior driven alignment produces a consistent train-test discrimination behavior.
Neural Collapse. Neural Collapse (NC) captures late-stage geometry within-clast collapse, simplex ETF means, and classifier means alignment(Papyan et al. 2020 Han et al. (202T) which has been ex-
ploited in incrementaal/continaal few-shot, and large-scale settings Yang et al. [0223] Seo et al. 2024
Shen et al. 2025). but is sensitive to imbalance and drift (Markou et al. 20244] Yan et al. 2024 Fang
al.) 202 For OOD, NC inspired two main approaches: one leverages the emergent phenomenon, such as in PFS (Wu et al., 2024), by applying constraints to the learned ID classifier weights. The second approach leverages the ideal geometry as prototypes (Shen et al., 2025). Recently, Zou et al.) (2025) utilized EBVs as a foundational framework for OOD detection and corrected the angular misalignment between EBV prototypes and data distributions. Furthermore, this framework relates to Prototype Learning (PI.), but with critical differences: most PL. methods (Peng et al. 2025 [uu et al.) 2024 rely on dynamic prototypes that are learned and updated, with OOD detection dependent on these dynamic ID representations. LPO (Zhou et al. 2021) introduces learnable OOD prototypes via manifold mixup. However, the limited diversity of internally synthesized features risks prototype collapse due to insufficient semantic separation from ID classes. While/Bojanower
& Joulin (2017) and Saadabadi et al. (2024) also exploit fixed target structures for unsupervised learning or dynamic assignment, our method adopts fixed prototype strategy which based on the ideal NC geometry and keep them fixed. This avoids the instability of dynamic prototypes while providing a more active geometric constraint than Wu et al. (2024). Crucially, we expand the ID ID-presetted EBVs into our OEFS to simultancously represent both ID and OOD, yielding interpretable separation and straightforward 1.2 activation activation-strength scoring rule.
METHOD
3.1 PRELIMINARY
Formalizing the OOD Detection Paradigm. Detecting inputs outside the training distribution is
critical for deploying reliable models in open environments (Hendrycks & Gimpel 2016 Liu et al.)
2020). We consider classification task with C' classes over input space X' and a corresponding ID
label space y (1, . C). The in-distribution (ID) data (zr, y) is drawn from a joint distribution
D (,,, )). Out- distribution (OOD) data Food is drawn from a marginal distribution D (X), which originates from different generative process Dout(& (X.Yoout) The core assumption of OOD
detection is that the OOD label space Vout is disjoint from the ID label space, i.c., y n Yout = 0. The objective to design a decision function GA X + (ID, OOD) based on scoring function S:
IDD
if S f) 2 X if Swx f) <
Gx(z)
OOD
where X is calibrated on ID data achieve target true positive rate. In practice, S can be instantiated by MSP (Hendrycks & Gimpel| 2016), energy (Liu et al. 2020), Mahalanobis distance (Lee et al. 2018), or k-NN distances (Sun et al. 2022).
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Revisiting the Outlier Exposure Paradigm. Recent advances demonstrate that exposing models to auxiliary OOD data during training significantly enhances OOD detection robustness (Hendrycks)
ct al. 2018 Liu et al.
2020 Wang et al. 20224). Let Daut denote an auxiliary OOD dataset satisfying supp Dot n supp Din) D and supp( ) o supp( Dont). The seminal Outlier Exposure (OF)
framework (Hendryeks al. 2018) regularizes the feature space by minimizing the KI. divergence
between OOD predictions and a uniform distribution:
(o = - Σ
1
log fz)) H(((x)),c))
j=1
where H. denotes cross-entropy, (ar is the j-th element of the model output, and Uc is the uniform distribution over C classes. The composite optimization objective becomes:
min (i Ccn(2, y) + ExE (O)
classlication
OOD regularization
This formulation establishes the foundational paradigm for OE-based OOD-aware training (Ndiour et al. 2020; Du et al., 2022: Wu et al. 2023). However, this paradigm suffers from fundamental limitations. First, it forces capacity- limited ID classifier (designed for K classes) into conflicting- objective dilemma: it must simultaneously perform fine-grat grained ID discrimination while rejecting massive, diverse OOD data via uniform, low-confidence predictions.
More critically, the persistent OOD regularization severely impedes the alignment of ID features. While ID) features should ideally converge Neural Collapse (NC) state for maximal separability, the OE objective interferes with this convergence by forcing the classifier to also manage OOD inputs. The resulting feature space is thus a compromise, sacrificing the compactness and stability of ID representations. This analysis reveals a critical need for solution beyond mere output regularization that can structurally decouple ID and OOD representations. This would protect the ID feature geometry's ss convergence while providing dedicated OOD space. Overcoming this limitation is the primary motivation for our VPC framework.
3.2 ORTHIOGONAL EQUIANGULAR FEATURE SPACE
The core intuition of VPC is to explicitly engineer the feature space geometry, creating structural separation between ID and OOD representations. We move beyond OOD detection paradigms that rely solely on ID class weights/elassiffers. Instead, we propose the Orthogonal Equiangular Feature Space (OEFS). a pre-specified representation space that is conceptually partitioned prior.
This space is mathematically constructed from single set of K + V total prototype vectors, W K+V {w;} ) i=1 generated on the unit hypersphere. To ensure maximal and uniform separation between all prototypes, these vectors are structured as a simplex Equiangular Tight Frame (ETF) (Papyan al. 2020 Yang et al. 2023 Markou et al. (2024, Shen et al. 2025). This construction provides the geometric foundation for the OEFS and is defined by the relation:
K + V K + V
T Wk, Wk,
8k..ks K + - 1ki,k f1. K+V),
(3)
where 6x, A2 = 1 if 11 = 12 and 0 otherwise. This guarantees all vectors share the same l norm and any two distinct vectors have an inner product of -VKKKV
The key to our method is the functional partitioning of this unified set of prototypes. The first K' vectors are designated Equiangular Basic Vectors (EBVs); they form the ID subspace and serve as the stable, fixed prototypes for the K in-distribution classes. The remaining V vectors are designated as Vast EBVs (VEBVs), establishing& a dedicated OOD subspace. Crucially, this subspace is geometrically orthogonal to the ID boundaries to ensure non-interference with ID tasks, while simultaneously offering a vast array of distinct geometric anchors to resolve the finennrained semantic variations inherent in OOD data.
Building on this preset geometry, we adopt a dual feature-alignment strategy (Sec. 3.3 3.4) to induce distinct activation patterns EBVs and VEBVs for ID and OOD samples, respectively. This drives ID features toward Neural Collapse geometry while granting OOD features sufficient directional resolution to differentiate complex patterns: theoretical discussion of Neural Collapse is provided in the appendix I.I. This geometric separation is measured by our VPC Score 3.3] a discriminative metric designed to quantify the activation of feature within these distinct subspaces.
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3.3 EVIDENTIAL PRIOR GUIDED FEATURE ALIGNMENT
In guiding ID features toward the target subspace, a central challenge is that OE features with rich variability can interfere with the model at any time and destabilize the representation space. To counter this, we propose an Evidential Neural Collapse framework that reformulates feature alignment as geometry-driven evidence accumulation process.
Unlike standard evidential learning (Sensoy et al. 2018; Jøsang 2016 Sentz & Ferson 2002) where evidence stems from learnable logits g( 0). we instantiate a geometry-drive evidence metric derived directly from the cosine proximity between features and the predefined Vs. This couples the accumulation of evidence with the rigor of geometric aligmment:
Gij = exp (
gj g3(xx1,0)
Predetined VPC
id T m
uight
exp
Leurmblle Classifier Logits
Goometric Alignment Logits
Here, the evidence magnitude ei reflects the angular alignment intensity of the normalized feature riddd towards the j-th prototype uigby, scaled by temperature
Crucially, we fuse this geometric evidence with a uniform prior to form the Dirichlet parameters of E + 1. This unit prior acts
angular regulator: it injects uniform geometric buffer that prevents the model from collapsing onto specific directions too abruptly due to optimization interference from OE samples. By maximizing the following ENC likelihood, we achieve calibrated convergence where evidence growth is strictly governed by reliable geometric support:
K
(EN
= => yis (log S7 - log añ,)):
(5)
1=1
where S = =E
=== a, is the total evidential strength and Vi.j is the one-hot label. This objective effectively mitigates over-confidence and ensures stable ID-EBVs alignment, which we further
theoretically justify via optimality and stability analyses in Appendix A.2.1 A.2.2
3.4 DUAL-SUBSPACE FEATURE ATTRACTION AND ORTHOGONALITY CONSTRAINTS
To fully exploit the benefits of abundant auxiliary OE samples for OOD detection, we move beyond the traditional OE loss that only enforces uniformization over ID classifier weights, and establish dual mechanism of subspace attraction and orthogonality constraints. On the one hand, OE features to enrich OOD representations;
guided to enter the VEBV subspace they are enforced to be orthogonal to ID EBVs to preserve ID classification performance. For the tiispace-attractionn objective, design the VEBV loss by quantifying the Fuclidean distance between the normalized OE feature mi and the subspace spanned by the VEBVs:
the other hand,
LVEBV (z
(moeT ayybv)),
(6)
where vbyebr denotes the j-th VEBV in OEFS. The summand is the squared cosine similarity between the normalized OE feature and each VEBV: The square root (an &z norm) measures the activation strength within the VEBV subspace. Minimizing the negative of this norm maximizes the activation of outlier features in the VEBV subspace, forcing them to distribute activation over multiple VEBVs rather than collapsing onto single direction, thereby capturing the diversity of OOD modes.
In addition, we introduce an orthogonality constraint (OC) loss to strictly regulate OE feature alignment within the ID subspace. Formally, it minimizes the KI. divergence between the predicted distribution over ID EBVs and a uniform distribution:
K
Loc(s,)) = K
Σ
log(p, oe-
Here, pee- is computed by applying Softmax to the scaled cosine similarities s, (m mehh) )/T. By virtue of the Simplex ETF structure, this uniformity objective effectively translates into geometric orthogonality constraint (proof provided in Appendix A.3). This ensures that OE features reside the null space of ID prototypes, safeguarding IDD convergence and creating rigorous decision boundary.
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3.5 DISCRIMINATIVE SCORING VIA ACTIVATION INTENSITY
The geometric partitioning induced by the OEFS enables a principled and discriminative scoring function. This function quantifies the distinct activation patterns of ID and OOD features across the predefined classifier subspaces. We define two metrics based on the L.2 norm of feature's projectionn onto the ID and OOD subspaces, respectively:
e (z,)
T [m,, wi m,
(8)
eyeby (z.) [m, w1 m, up 1
Here, e5T (x:z) and egeb measure the activation intensity of feature mi within the ID (EBV) and OOD (VEBV) subspaces, respectively.
Following optimization with our proposed loss, IID features align with their EBVs, yielding high geby and negligible Eyeo activations. OOD features are conversely guided into the VEBV subspace, producing the opposite pattern. This resulting dichotomy motivates the VPC Score:
(v)c( -a
geh (I:) + Eyeby (x:);
Consequently, the VPC score serves as a continuous measure of OOD likelihood, allowing for the separation of ID and OOD samples via a simple threshold. In contrast to methods that rely on the statistical properties of logits implicitly learned, the VPC Score is derived directly from the prior geometric structure of the feature space. By measuring the relative activation intensity across orthogonal subspaces, it provides a robust and interpretable measure of distributional shift, grounded in the model's explicit geometric constraints.
4 EXPERIMENTS
In this section, we first evaluate our method on the large-scale ImageNet-1K 1k benchmark and the widely-used CIFAR-IO/100 benchmarks (Krizhevsky al. (009) to assess its performance (Sec. 4.1 We then examine variety of model architectures to further verify its effectiveness (Secs. 4.2 4.3 and
4.4. Additional ablation studies are reported in Sec. 4.5 More theoretical analysis, results appear in appendix A
B
We begin by detailing the experimental setup.
OOD Datasets. For CIFAR benchmarks, we use 300K auxiliary samples from 80 Million Tiny Images (Toralba et al. 2008) and evaluate on five standard test datasets with disjoint categories:
SVHN (Ntzzer et al. 2011), LSUN (Yu et al.) 2015), iSUN (Xu et al. 2015), Texture (Cimpoi et al.,
2014). and Places365 (Zhou et al.) 2017). For ImageNet, we utilize the ImageNet-21k-p validation
subset as auxiliary data and test on four wedell-reeoognized datasets: iNaturalist (Van Horn et al.
2018 SUN (Xiao et al. 2010), Places (Zhou et al. 2017). and Textures (Cimpoi al.) 2014).
Pre-training Sctups. We employ Wide ResNet-40-2. ResNet-18, and DenseNet- 121 Zagoruyko & Komodakis 2016; He et al. 2016: Huang et al. (017)) as backbones for the CIFAR benchmarks, training for 200 epochs with a batch size of 128, initial learning rate 0.1, momentum 0.9, weight decay 0.0005, and cosine learning-rate schedule. For the ImageNet-Ik 1k benchmark, in the comparative experimental methods, we use standard pre-trained ResNet-50 (He et al.) (016) and ViT-B-16 models as backbones. In the VPC experiments, we follow the standard practice for E-V-based training (Shen) et al. 2025). Unlike the conventional cross-entropy + learnable classifier head, our method directly maximizes the similarity between features and predefined classifiers. Accordingly, we remove the final classification layer and add an extra projection layer to match the dimensional requirements of OEFS; see the Appendix A.4 B.3 for details.
Two-stage Training Setting, Fine-tuning setup. For the CIFAR- 10 and 100 benchmarks, we initialize the network with the hest-performing checkpoint from the pretraining stage and fine-tune for 50 epochs with auxiliary OE data (Hendryeks et al. 2018)) (ID batch size 128; OOD batch size 256; initial learning rate 0.07; momentum 0.9; weight decay 0.0005; cosine annealing schedule). For the ImageNet benchmark, we use pre-trained models from Pytorch and pre-trained models following the (Shen et al.) 2025) training setup as the initial networks, respectively, and then fine-tune for epochs with ID/OOD batch size 64, initial learning rate le-4, momentum 0.9, weight decay 0.0005, and a cosine schedule. For our VPC method, the temperature т used in all loss functions is set to O.I. All other settings follow the original paper's setup.

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Table I: Results on ImageNet- Ik benchmark with auxiliary OOD data. The best result is in bold.
Far-ODD Datasets
Near-OOD Dataseds
Model
Method
INaturslist
Testures
SUN
Planes
Average
ID Aee
FPRMS,
AUBOCT
FFRS5)
AUROC1
FPR95.
AUBOCT
FFR55
ALRRCC
AUROCI
OF (Hendryckee al
2015
68.60
65.
58.85
82:60
61.75
62,90
20.70
30.55
59.98
$3.60
76,00
Fre RuGoue
7020
49.40
68.40
59.50
82:35
62:40
62.20
71.
30.25
60.68
83.40
75. 75
RrsNe110
DAI Wannerr
48.00
80.05
58.00
82:95
61:30
63:15
67.82
30.75
58.78
83.95
75.00
PTS Wire 2024
45.30
$9.20
56.50
83,10
61.00
83.25
67.50
30.95
57.85
94.13
76,02
Ours: YPC
43.90
91.29
$6,80
85.00
60.20
83.10
66.30
$1.50
$6,50
84,70
76.11
OF (Hadlyykkstal 2016
42.15
90.38
$2.45
85.85
65.80
82.20
70.35
80.85
$7.69
84.82
80.02
42:75
90.03
53.15
85.50
65.25
81.95
70.95
80.50
58.28
84.50
79.85
VIT-B-16
DAL Manderus uetun prome
40.65
90.86
51.15
86.10
65.07
82.30
70.25
80.90
56.78
85.04
50.09
PES Wireeal 0i s004
40.5
50.80
51.30
86.05
65.35
K2.25
70.20
30.95
56.93
85.01
50.13
Ours: VTC
48.00
91.10
31.30
86.00
65.20
82.30
68.30
$1.40
56.20
85.20
80.33
Table 2: Results CIFAR-I0 and CIFAR-100 with WideResNee-40-2. The best result is in bold.
Far-OOD Datusets
Method
SVIN
ISUIN
Tranunos
Phacs
Averagoo AUROCT
AUROCT
NPROS]
AUROCT
With vanilla trvining
MSP Sadreekee Grupe
10
44.22
93.61
27.56
96.12
69:62
$5.29
60.02
83.53
65.68
86.25
53.42
89.96
Each
31.81
94.65
4,50
95.05
$0.06
89.75
49.68
90.00
42.38
90.82
35.69
92.85
Mahr 12 1oc
0013
42.67
90371
18.95
95 05
25.58 56
93.76
25.22
92.81
56.78
69. 14
40.20
85.58
KNN N1l 31
44.76
92.55
27.38
95.35
41 S4
91.24
37.54
92.82
49.33
87.89
40.57
91.97
With contractive INITLnng
ct 20:07
17.39
97.00
6.75
98.05
12.55
94.20
40.00
92.05
20.47
96,17
6.76
7.45
20-003
22.85
95.75
43700
91.91
21.36
96. 16
KNN
3.25
17.85
13.87
97.92
92.98
O6 Hefferciieee
2015
1.05
99.23
97.76
99.05
0.95
3.35
98.72
Want
99.36
0.95
1.35
99.02
97.39
PPS
Wirers
1.10
98.74
0.35
99.61
1.35
20
2.85
94.55
7.75
97.17
2.68
95.56
Ourk:
0.85
99.62
0.45
99.50
1.10
99.50
2.25
99.38
97.91
2.27
99.18
CIFAR:100
With vatillu training
MSP K Gampe
2016
74.79
79.64
56.72
$0.45
93.35
36.92
85.75
68.48
$3.24
71.95
79.07
72.69
75.77
Maln
77.73
98
54.11
73.37
75.39
90.05
71.33
75.13
2009
64.50
25.88
70.62
86.74
77.11
85.05
CIDE
16.87
96.23
45.45
60.01
49.29
57.48
68.39
K3.19
KNN-
32.50
47.41
99.82
45.05
79.25
87.55
WIB auniliary OOD Cata
06 Hendryysse
2016
28.95
95.00
10.95
97.95
49.55
$9.29
41.50
91.57
49.75
89.37
36.14
92.76
2nrmcca ( Le 2007
25.00
96. 18
11.50
41.4
91.67
48.10
83.00
36.30
87.66
40.34
91.09
DAL Wenctti aieim FURRTE
19.30
93.75
16.20
96.71
30.70
93.85
43.15
91.36
55.10
88.39
32.89
93.21
PF3 ora
Ourk: 2014
24.70
95.81
12.65
97.78
38.35
91.44
46.20
91.32
$1.85
90.33
34.35
93.33
9.95
97.08
45.05
80.67
03.65
One-stage Training Setting. Beyond the stepwise pretraining fine-tuning paradigm for OOD models, we additionally investigate a one-stage training scheme to explore stable separation between ID and OOD samples under more entangled optimization setting. On the CIFAR-10 benchmark, we jointly train on ID and OE data for 150 epochs; on CIFAR-100, we train for 200 epochs. All other settings follow the two-stage experiments.
Scoring functions. We compare the classic MSP (Maximum Softmax (obubbility)Hennryyee &
Gimpel, 2016), the EDL (Evidential Deep Learning) Uncertainty Score (Sensoy al. 2018). and our proposed VPC Score. In the default configuration, VPC Score uses = and 3 100. We
also evaluate single-subss subspace variants: VPC Score" (a (a== 1, 0, ID-EBVs subspace) and VPC Scoret (a = 0, 3 = 100, DOD-VEBY Vs subspace).
Compared Methods. We compare our method with post-hoc approaches, contrastive learning based methods, and auxiliary OOD data based methods. The post-hoc methods include MSP (Hendrycks) Gimpel, 2016)), Energy (Liu al.] 2020), Maha (Teee al. (018). and KNN (Sun al. 2022). The contrastive learning based methods include CSI (Tack al. 2020). CIDER (Ming al. 2022b
and KNN+ (Sun et al. 2022). The auxiliary OOD data based methods include OE (Hendrycks et al.
2018), Energy-OE (Liu et al. 2020), DAL (Wang al.) 2024a and PFS (Wu et al. 2024). For other methods, we adopt their suggested setups for faimesss.
Evaluation Metrics. We report three classic metrics commonly used in OOD detection (Hendrycks & Gimpel 2016 2016|Liana [iang et al., 2017): (i) FPR95 (False Positive Rate at 95% True Positive Rate): the FPR for OOD samples when the TPR for ID samples reaches 95%. (ii) AUROC (Area Under the Receiver
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Operating Characteristic Curve): the area under the TPR-FPR curve over all thresholds. (iii) AUPR (Area Under the Precision Recill Curve): the area under the precision- curve, emphasizing the balance between OOD detection accuracy and coverage.
4.1
MAIN RESULTS
Tables D and present the primary results on the large-scale ImageNet-IK Ik benchmark and the widely- used CIFAR- CITAR-IO/100 10/100 benchmarks, respectively. Compared with conventional supervised or contrastive learning methods that rely solely on IID data, incorporating auxiliary outlier exposure (OE) data during training significantly reduces FPR95 and enhances AUROC, clearly highlighting the value of OE-based training OOD detection research. We compare our approach against several state-of-the-
art OE-based methods, including the original OE method (Hendrycks et al. 2018), the Energy-based OE method (Energy-OE (Liu et al. 2020),, the Distributional Adversarial Learning (DAL (Wang et al., (0244a), and the PFS method leveraging Neural Collapse (NC) (Papyan et al. 2020) properties. Our proposed VPC achieves superior performance across all benchmarks. On ImageNet-Ik Ik (Table I VPC demonstrates robust advantages. With the ResNet50 backbone, VPC achieves an average FPR95 of 56.50% significantly outperforming the next-best baseline (PFS) by 1.35% while also attaining the highest AUROC. This advantage is maintained on the ViT-B- 16 backbone, where VPC (56.20%) surpasses the strongest competitors (DAL at 56.78% and PFS at 56%93% while also achieving the highest AUROC (85. .20%) and maintaining competitive ID accuracy. This strong performance is mirrored on the CIFAR benchmarks (Table 2). On CIFAR-10, VPC reduces the average FPR95 to 2.27%, clearly surpassing the best-performing baseline (PFS at 2.68%). On CIFAR- 1000 VPC achieves an average FPR95 of 32.04%. outperforming both DAL (32.89%) and PFS (34.35%), and again reaches the highest AUROC score of 93.65% among the compared methods.
4.2 TWO-STAGE TRAINING
Under the two-stage pretraining fine-tuning setup (Table 3}. our method achieves overall superiority on CIFAR-10 with NedeReesNet-4-22 and DenseNet-121 122, and attains the best FPR95 on ResNet- 18 while its AUROC/AUPR are slightly below DAL. On CIFAR-100 it shows robust advantages on WideRes.Net44-22 (best FPR95 and AUROC, with AUPR tied with PFS) and on 1enseNet-121 (best on all three metrics), whereas on ResNet-18 it minimizes FPR95 but trails PFS in overall ranking and PR area. Collectively, these results indicate that our approach effectively suppresses OOD false acceptance while improving overall separability.
Table 3: Two-stage Training Results CIFAR- 10 and CIFAR-100 with WideResNee-40-2, ResNet- 18, DenseNet-121 121. Metrics are reported as /TROC//TURO////UPRR
Method
CIFAR-1O ResNet-18
DenseNet-22 121
WideRosNat-10-2
CIFAR-100
ResNet-18
DenseNet121
OE Homdrycks et
2018
3,64/99.0/999
3.16/98, 36/9967
2.81/98.86/9
36. 06.14992.7699
48.75/89.366 36/97 54
36,46/93 228.2399888 53
Emerg-
3,75
3,8W/8 26/99
345/98.71/
71
40.3499
46.34/80,
44.57/91
21
DAI
Waitr crai
3,17A88 8439
98.96/99.77
2.5899.77 72/99.77 71
80.9999 21699
44,57/90 87/7
36. 75/9) 66/97 68
PFS Wil oral M1.4
2.68/9.66-999
3,08/98 72/99.72
2 2.779844770 66
34.3599.33999 33/9.53
40 10.15/ 92.64/98.41
43.80/90.960 96/97.99999 .99
Olis
2.27/99. 18/999118/99.1
334498.22/99.8
211999.8/99.77
32.049966999853
38.96/9..0/98..22
31.17/94.0.99871
4.3 ONE-STAGE TRAINING
As reported in Table we further investigate a single-stage training paradigm, which differs from the prior two-stage setup. The goal is to train OOD detectors directly with fewer steps and hyperpa- rameters, this direct setup has been underexplored because exacerbates the difficulty of controlling representation geometry. When jointly optimizing ID and OE data without stage separation, our method attains highly competitive results across the board. This corroborates the effectiveness of imposing the OEFS geometric prior on the representation space and, via outimiittmizttion strategy, strengthens VPC's ability to suppress representation drift under noisy gradients while maintaining stable ID/OOD decision boundaries.
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Table One-stage Training Results on CIFAR-I0 and CIFAR-100 with WideResNeet44-22, ResNet- 18, DenseNet-12) 121. Metrics are reported as FPRO5//UROCC/AURR;
Mothod
WideRosNett40-3
CIPAR-10
ResNet-18
DensNee-121
WileRexNet-00-2
CIFAR-100
RosNet-18
DonseNetl
Hendrycks
3.12/9661/992
35,43822 67498
8058
37.77/92. 68/96.37
Fnoeeco norer ai
2,29008, 79/99 72
3.6608 21/9
3,098.7660977
080228 76/M
177991996000 18
12/22//33 30/98.44
DAL Wank et at 2020
2,95 88/99889977 75
/4/9999
2.73/97.7/99. S3
32,31192 28/98 24
43. 11.99176/99..16 16
36, 36.55/92 63/98.31
PHS Wo
2,44996 (8798999999 68
3.649% 72/99 69
2-42/98955999
32, 84834 22/98
41. 11.14491.79996 14
39.91/91 76/96 15
2.01.99.19/9988
3.45%98.3/99.73
2.29/9.7409977
32.15993.6/9085
38.41/92.69998422
32.31/93.41/99854
4.4 DIFFERENT SCORE FUNCTIONS
This section provides detailed comparison between the proposed VPC Score and range of scoring functions across multiple network architectures (see Table (). Built upon the twin-subspace representation of OEFS, VPC employs a class-agnostic L2 activation magnitude as the separation signal to enhance OOD discriminability. On CIFAR-100, VPC Score achieves the best AUROC and AUPR on all three backbones and reaches, or closely approaches, the lowest FPR95. By contrast, MSP suffers from bias because OOD features cannot remain orthogonal to the ID classifier weights; EDL Prob introduces evidence priors that partially alleviate this bias yet remains suboptimal; and Uncertainty aggregates evidence strength without imposing explicit geometric constraints on the representation space. Leveraging the orthogonal subspace-induced class-agnostic L2 activation, VPC Score delivers consistent gains across nearly all architectures while retaining interpretability and strong generalization. Additional discussion of VPC Score is provided in Appendix A.3
Table 5: Different score functions on CIFAR-I0 and CIFAR- 100 with WideResNet-40-2, ResNet-18, DenseNet-1211 Metrics are reported as FPROS//AUROC//AURRT;
Score Function
CIFAR-I0
CIFAR-I00
Wid:ResNet-10-2
ResNet-18
DenseNet-121
WideReaNet-10-2
ResNet-18
DenseNet-121
MSP
9.74/99.8/99..8
8119998.1/999.5
2,46/88 99/99.79
33.95/92.3698811
44.75/89.79/97 793389979//744 44
01.65/92.79/98.0
EDL Prob
2.55/99.19/99
3.79978.82/99.5
2.39/9.00899 79
33.87/92.577988 57/98. 16
44,47/90.08/97.5
33.42/99.0W/98 42
Uncertainty
2.30/99. 13/99.80
2.91/98.72999 72/99.74
006/99.08/99
31,2093 21.20099..//..33 32
39.58/91 23/91.83
33.42/93.00 00.98.0988
VPC Score
2.27/99.19/99.81
2.84/8..3//9967
2.10998.66/99
32.04/93.65/98.53
38.96/92.0398822
31.17/94.01/98.71
4.5 ABLATION RESULTS
4.5.1 SCALE OF ORTHOGONAL. EQUIANGULLAR FEATURE SPACE
In a fixed ID dimensionality (CIFAR-10 10 with K=10, CIFAR- 100 with K=100), we conduct an ablation on the VEBV subspace size V (see Table (). On CIFAR-10, as V increases from 10 to 2000, performance improves monotonically, and indicates that larger VEBV subspace provides more fine-grained geometric anchors, enabling more precise resolution of diverse OOD modes and strengthening the subspace's L2 activation. On CIFAR-100, performance peaks at V=1000, suggesting that when K is large, excessively increasing V causes subspace dispersion and more diffuse gradients, thereby weakening optimization pressure toward that subspace.
Table Ablation on OEFS subspace size V (number of VEBVs) with WideRessNer40-22 backbone.
Metrics are reported FPRSS//AUROC/AAPR::
OEFS
CIFAR-10
OEFS
CIFAR-100
10
96/99.75
100
34.89/92.78998.39
500
2.61/99.1//99.77
500
34.65/93.01/98
1000
2.27/99. 1819999.119998../ 81
1000
32.04/93.65/98..
2000
2.21/99.21/99885
2000
32.56/93.31/98.44
4.5.2
ABLATION ON TRAINING Loss
This section analyzes different loss combinations to validate the geometric logic of OEFS. As shown in Table 7 compare: (i) Baseline (CE OE); (ii) Neural Collapse orthogonality constraint (see

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Published as a conference paper at ICLR 2026
Appendix $16); (iii) Evidential Neural Collapse (CENc) + orthogonality constraint (Coc); and (iv) the full method adding LVEBV. The progressive improvements across these configurations clearly indicate that OEFS's core value lies in constructing two orthogonal subspaces, which in turn relies on the synergy of the loss: LENC stabilizes the convergence of IDD features toward EBV prototypes, Loc strengthens subspace orthogonality, and LyEBV guides OOD features activate the orthogonal VEBVs, thereby explicitly characterizing their complex intrinsic variability. Each component is indispensable. Notably, LVEBV is pivotal for breaking the performance bottleneck: its introduction elevates the OEFS geometric separation from ID-side only optimization to dual-sided ID-OOD constraint, ultimately yielding superior OOD detection performance.
Table 7: Ablation Training Loss. Metries reported as FPR95/AURRCC/AUPR]
Training Loss
CIFAR-10
CIFAR-100
LCE Loe
3.44/99.05/99 005999.79
36.14/92.76/98.38
LNC + Loc
2.7 2.71/98.97799.74
34.65/93.05/98
LENC Loc
2.68/98. 96/99.75
44.89/92.78/98.3
LENC + Loc LVEBV
2.27//9.18/99.88
32.04/93.65/98.53
4.6 VISUALIZATION
We visualize ID/OOD activations on a unified VPC model under three scoring functions: VPC Score. Uncertainty, and MSP (Figure 2); additional examples appear in Appendix B.4] On CIFAR-10/100
VPC Score produces two sharply separated modes: ID samples primarily excite EBVs, whereas OOD samples activate VsBVs. When challenging OOD data partially align with an ID EBV, MSP readily confounded. Uncertainty and our class-agnostic VPC Score suppress this bias, with VPC Score achieving superior discriminability.
A.S.A
201
61
1000 CTPAR
000 CTPAARR
000 CHRMB-30
Figure 2: Visualization and OOD detection using different scoring functions in CIFAR-10. VPC Score maintains distinguishability even when MSP and Uncertainty perform poorly.
5 CONCLUSION
In this paper, we present VPC, geometry-driven framework that predefines an OEFS to explicitly separate ID and OOD representations. Beyond IIDonll classifiers, VPC injects a training-time time prior using an ENC loss for ID-EBV alignment, an orthogonality loss to shield the ID subspace from OOD, and a VEBV exploration loss to enrich the OOD subspace. The resulting VPC Score measures subspace L2 activation, offering class-agnostic, interpretable, and robust discrimination. On CIFAR- 10/100 and large scale ImageNet- Ik benchmark, and multiple architectures, VPC achieves strong and consistent performance. These findings highlight geometric priors plus evidential modeling as an effective path to feature-separation OOD detection with auxiliary OOD data.
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ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Nos. 62476165, 62472315, 62406182) Shanghai Science and Technology Innovation Action Plan (No. 25511102102).
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A THEORETICAL ANALYSIS AND METHODOLOGICAL DISCUSSIONS
A.1 NEURAL COLLAPSE
Recent studies have shown that, in the late stage of training, classification networks exhibit the Neural Collapse geometry: within-class collapse, inter-class equiangular separation, and alignment between class means and classifier prototypes. This geometry achieves simultaneous intra-class compactness and maximal inter-class separation, and is regarded as the natural convergent state of classification models (Papyan et al. 2020; Seo et al. 2024 Yang et al. 2023 Markou et al., 2024).
In OE-based OOD detection, relying only on output-layer uniformization or simple orthogonality constraints is insufficient to maintain controllable and stable representation geometry during joint ID and OE fine-tuming We therefore treat NC as an explicit alignment target. We fix the subspace to simplex ETF using preset Equiangular Basic Vectors (EBVs) and expand a bank of Vast EBVs (VEBVs) to characterize diverse OOD features, forming an Orthogonal Equiangular Feature Space (OEFS). We next introduce the simplex ETF induced by EBVs, summarize the core properties of NC, and present the corresponding optimization.
Simplex Equiangular Tight Frame Let K 2 and d 2 K-1, where is the feature embedding dimension. A simplex equiangular tight frame (ETF) is matrix E [W1,.... [ wkl Rd K whose columns satisfy
K
WKy Wkk
kk
1 S k1 h2 S K,
(10)
K - 1'
so that all columns have the same squared norm K/(K- - 1) and any two distinct columns have inner product -1/(K - 1). A constructive form is
K
1 K 1k1k). (lK
E E= K K
E U
UTU= Ik
(II)
which yields zero-sum columns E 1lx- among K equiangular vectors in Fd our setting, these K columns serve as preset EBVs that fix the ID subspace. For OOD representation we append large bank of VEBVs constructed analogously in the same ambient space with d K + +V which together instantiate OEFS and provide preset classifiers for both ID and OOD.
and achieves the minimum possible cosine similarity
Basic properties of Neural Collapse four coupled properties.
Neural Collapse at the end of training is characterized by
NC1 (within-class collapse):
W Avg, [(AK, (A)( m4x) - 0,
(12)
where μk, is the penultimate feature of sample in class k and p4 is the class mean.
NC2 (simplex-ETF class means):
11g
HG
1 ISkS
(13)
14
Hall"
where PG is the global mean. The centered and normalized means {px} approach the vertices of simplex ETF, that is, their pairwise inner products converge to -1/(K* - 1).
NC3 (moan--rototype alignment):
We
1sk5 K,
(14)
namely the centered class means align with their classifier prototypes.
NC4 (nearest-center decision):
arg max(u wx) = arg min ||
mell,
(15)
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Point Estination
Deta Uistriburiin Extimation
Dirlchle UNstribation Extimattio
NIG Distributinn Extimatiin
Unvertaioty
Deswbooks
(00) 4n)"
of
Over Coutleer
(2,8)
Coufilent
Confliet
Cantidaz
1'
1b
0.
[0
715
O)" R
ML ( Ogger Oug] E R
B PT..a, R
Prman
{+. Dunin
(r,0)
First-order Minimization Model
Second-order Risk Minimication Model
Figure 3: Overview of First-Orde Risk Minimization Models and Evidential Deep Learning Based on Second-Orde Risk Minimization.
Misinization
Stati
eusi ande adjat
equal angle adjar
Clan Thear
(a) Neural Collapse
(h) Evidential Prior Guided Neural Collupse
Figure 4: Overview of Evidential Prior Guided Neural Collapse. (a) Training drives final-layer features, class means, and classifier vectors towards a Simplex ETF (b) Our Evidential Prior Guided Neural Collapse further regularises this collapse.
which holds when the equal-norm alignment in NC2 and NC3 satisfied. Together, NC1-NC4 describe optimally discriminative geometry with minimal within-class dispersion and maximal inter-class separation.
Neural Collapse Loss To operationalize this geometry, Neural Collapse Loss loss fix the ID subspace to simplex ETF via preset EBVs and optimize features to align angularly with these EBV
exp(mi idT wobo// /r)
(xc(al)) log K Σ5 exp (m nggby /T)
(16)
This objective is the discriminative counterpart of our ENC likelihood, both share the same cosine- based targets; ENC further fuses a uniform evidential prior to temper over-confidence and stabilize alignment.
A.2 EVIDENTIAL PRIOR GUIDED NEURAL COLLAPSE
Traditional first order risk minimization fits point estimate of p(u æ)) and directly learns the parameters of the data generating distribution according to the task, for example the Bernoulli parameter for binary classification, the multinomial parameter for multiclass classification, or the
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Gaussian parameter for regression. This paradigm struggles provide reliable uncertainty under distribution shift and hard examples. In contrast, second order models predict the parameters of conjugate priors, such Beta for Bermoulli, Dirichlet for multinomial, or Normal Inverse Gamma (NIG) for Gaussian, and compute the final predictive probability or regression value from these distributional parameters. Evidential Deep Learning (EDL) (Sensoy al. 2018 Jøsang [016] [ennz & Ferson 2002) adopts the second order view. The network first predicts nonnegative evidence and maps it to Dirichlet parameters a Ek which defines a prior posterior family Dir(a) over class probabilities
E A A--1, The Dirichlet parameters can be interpreted as pseudocounts for class and the total evidence S Σx ar quantifies overall confidence. When is small, the model should remain unconfident. Figure 3 summarizes the differences between first order risk minimization and the second order evidential formulation.
Our ENC loss builds explicit bridge between NC's geometric alignment and EDL's second- order evidence modeling. Rather than deriving evidence from learnable classifier activations, ENC computes cosine similarities between features and OEFS EBVs, applies the evidence activation (Eq. (4)), and injects a uniform evidence prior, forming a unified angular adjustment that is further made learnable via Bayesian inference. In joint ID+OE training, ENC concurrently regulates direction and scale, stabilizes class-wing wise collapse, suppresses unsupported high evidence, and prevents hard OE samples from eliciting large, non-uniform responses on the ID classifier. Figure compares ENC with standard Neural Collapse.
A.2.1 OPTIMALITY ANALYSIS
We first analyze the optimality of the ID subspace optimization. We adopt the unconstrained feature model [Papyan et al. 22020), omitting backbone architecture to focus on the convergence properties of the final-layer features.
Problem Setup. Consider ID samples distributed across K classes, with ng samples per class Let WEIF- = [u....... ux] E R K denote the fixed, predefined EBVs forming a Simplex ETF. The optimization objective for the feature matrix M under the ENC loss is defined as:
K
min LENG
(log S; - log ai,)):
N
(17)
k=1
s.t. ||mx,e12 <
Ak {l.....,
Ji....,KK,i. K}iE (I....,
where ef, exp(1ē] m/,a/T) az. = 63 and S - 03.0-
Theorem (Global Optimality of ENC). For any class fl.....
.a....,,, K} and sample i, the global
minimizer vu of the optimization problem in Eq. 17)) satisfies mhgi wk. This implies that
minimizing LENC strictly enforces the Neural Collapse geometry where features collapse to their corresponding class prototypes.
Proof. Based on the definition of the EBVs (W.T)), we have:
K
w6 Do =
Sk.K
Wk, E [].
(18)
K-1
K 1
Furthermore, WETF 1x Og,, which implies:
K Σ
K Σ
0.,
w -
(19)
jz
When WETF is fixed, LENG is convex with respect to my... and the constraint || < is also convex. Therefore, use the Kuhn-Tuccer conditions (KKT) conditions to determine its global
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optimality. The Lagrangian function is constructed as:
K
K
L
=
logl
exp[10 mx.//T) + K)
k
i=1
log exp(eix
mk.e/T) + 1)]
(20)
K
(llma,a1l 1). -1).
k=1==1 1=1
The gradient form with respect to my,s is (using the definitions of S and of;;):
Σ ビ G.a0
m =
+ mu,i
(21)
NT
S
ak,i
When Ap.5 = 0, the gradient equation simplifies to:
K j-11
(22)
S
That is:
Ejzeh mm.m W
1
mmall Wek
(23)
S
a
Multiplying both sides by wh, we obtain:
1 Ejpn t mus/t
1
cot mxs/
(24)
K-1
S
S
The left-hand side is always S 0. Since S > o the term
( is 0, making the
right-hand side > 0. Therefore, the condition VL- = cannot be satisfied when AR. = 0.
When AR. > 0, by KKT conditions, the global optimal solution my,3 satisfies the active constraint:
m. 11 1.
(25)
The gradient equation V C 0 then:
Je S
+ 2N DeNTA,,mE, en Wx
(26)
Si
ST
Multiplying both sides by 00 (j # k) gives:
Eitk
1
+
Cheb + NNN 2N = =0.
(27)
S
0 S:
K-1
Based the definition S Therefore:
(E
Cg,)+ we have Σjtt Cj. -
K - 6%
5 K c
1
+
K - 1 ck.
8
07,i S
(28)
+ +22T m 1y
= 0.
Multiplying the gradient equatio by W gives:
S K Ch.
1
ch,. Wh
S
0
S
K
(29)
2N NNT Mk,
K
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Let 3 (K (K-1) 2N 5;
( S
9i-K-ei,
(11 [1-
}
1 k-1 (k.). Since AR, and S > ap, >
E > 0.
we get 0:
ma.1 = BB0k.
(30)
Introducing the feature unit norm constraint:
m = 1.
(31)
And the ETF column vectors (EBVs) satisfy:
| || = 1.
(32)
Substituting |mx,ill = 1 into ma, = Bw0 gives:
= | ||
||
[ [ 3.
(33)
Thus:
3=1,
mgg. vik-
(34)
This shows that the ID features are optimally aligned with the corresponding class prototypes EBVs. Vs. reaching the theoretical Neural Collapse state at the global minimum.
A.2.2 STABILITY ANALYSIS
We explicitly analyze the gradient properties of ENC to demonstrate its robustness against interference from optimization interference arising from the massive auxiliary data. We formulate this as a bound on the sensitivity of the evidential prior mechanism.
Consider a single sample m with true class k. We define the logits sj = -wj m/T, the evidence of = exp(s;), and the Dirichlet parameters az = 1. The total strength S* ンッz 0z. The per-sample LENG term (denoted as (ENC) is defined as:
LEN = log S - log a
(35)
We formally state the smoothing mechanism in the following proposition.
Proposition (Evidential Prior Smoothing). The evidential prior mechanism imposes strict bound on the gradient sensitivity. Specifically, the magnitude of the gradient of the log Dirichlet parameter ratio R with respect to the evidence et is strictly bounded by I, acting as geometric rate-limiter.
Proof. We prove this mechanism by analyzing the partial derivatives. First, the partial derivative of the loss tNNO with respect to the evidence e₃ is derived as:
DEENC θez
gr:
j i k.
(36)
j k.
8
Next, to quantify the angular separation stability, for any j + kt we define the log Dirichlet parameter ratio:
Rej log O a;
= log
(37)
e +1
We examine how this ratio changes with respect to the target evidence. Because Reg depends on ex only through aj,, the partial derivative is:
(logaū log a;)
(38)
deg
der
Oi
+1
Since the evidence is non-negative (e) 2 0), we have a 2 1. Consequently:
dR < 1.
(39)
dej
This bound (< 1) represents the smoothing effect. It proves that the gradient step's 's effect on the angular separation (RR,)) is strictly limited. This prevents uncontrolled, rapidly diverging attraction of m toward an EBV, rendering the alignment process resilient to optimization interference induced by massive OE samples.
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A.3 THEORETICAL JUSTIFICATION OF ORTHOGONALITY CONSTRAINT
In Section 3.4 we introduced the Orthogonality Constraint loss Loc to regulate the alignment of OE features. Formally, this loss minimizes the KL divergence between the predicted probability distribution poe-eby and the uniform distribution. In this section, we provide a rigorous proof demonstrating that, governed by the zero-sum property of the Simplex Equiangular Tight Frame (ETF), achieving the global minimum of this loss is mathematically equivalent to imposing a hard orthogonality constraint between the OE feature and all ID prototypes.
Proposition 2 (Geometric Equivalence of Uniformity). Let the ID prototypes (uigby 13=1 form centered Simplex ETF IR., An OE feature vector me E Bd yields a exactly uniform probability distribution p pe over these prototypes if and only if it is orthogonal to every prototype, i.e. in gebv 0, Vj E E (1)....K).
Proof. Let oF denote the normalized feature vector of an OE sample, and (1ābb J==1 be the set of normalized Equiangular Basic Vectors (EBVs) representing the ID classes. By definition, these EBVs constitute a Simplex ETF, which inherently satisfies the structural zero-sum property:
Di Od-
(40)
The objective of Loc is to induce a uniform probability distribution over the K classes. The probability for the j-th class, denoted as (p"*>eer),; )j, is derived via the Softmax function applied to the scaled cosine similarities. To achieve a exactly uniform distribution (i.e., (p = 1/K, Vj), the logits (cosine similarities) must be invariant across all classes. Let E IR denote this constant scalar:
ToeT Beee c, Vj E KK....,KK.
(41)
We establish the proof by evaluating the aggregate projection of ne onto the set of ID prototypes, denoted as S Σソ= K
(m (ighy).
Imposing the uniformity condition from Eq. 41, the summation algebraically becomes:
S = > c= K.c.
(42)
Conversely, invoking the linearity of the inner product and the ETF zero-sum property (Eq. 40 the summation must geometrically satisfy:
K Σ
= m
m To
00, = O.
(43)
Equating Eq. 422 and Eq. 43 yields K c O. Since K > 0. it follows that 0. Substituting c = 0 back into Eq. (4T). we arrive at the necessary condition for optimality:
m Te cbv = 0, Vj (J)
(44)
This concludes the proof that within the OEFS geometry, the only state in which OE feature can minimize Loc is when it lies in the null space of the ID prototypes. Thus, the uniformity loss effectively functions as geometric orthogonality constraint.
A.4 FROM LEARNABLE CLASSIFIERS TO VAST PREDEFINED CLASSIFIERS
To elucidate the design motivations and advantages of the VPC framework, we review the architectural evolution of OOD detection paradigms (Figure 5). Vanilla DNN-based OOD detection (Figure 5p) employs learnable ID classifier optimized via first-order logits. Training with Cor enforces output uniformity, while detection relies on post-hoc confidence- based measures like MSP. Limitations include: (1) representation drift from optimization interference of OE data during joint ID/OE optimization; (2) non-uniform OOD probabilities due to ID-centric classification. PFS exploits Neural Collapse (ETF formation between ID features and classifiers) to orthogonalize OOD features
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Dg DNN
Exidential
Separ Date
Aoxe
Data
Arns Ape
Closdes
Eomaabe Legh
DSN EBVs
(4) Evidantial DSN wah OEPS
t
Buts
Buts
Exp Apo
Egve Aezar-
Treddted Siclark: Nedtorts
Clasifors with 181
Figure 5: The architectural evolution from vanilla DNNs to our VPC. (a) Vanilla DNN: relies on learnable classifiers and first-order logits, using Softmax (S) and Maximum Softmax Probability (M) for OOD detection under LCE + LOE- (b) Evidential DNN: introduces second-order uncertainty (U) based on geometric evidence (F) and Dirichlet priors. (c) DNN with EBVs: aligns features with predefined Equiangular Basic Vectors (EBVs) to achieve Neural Collapse. (d) Proposed VPC: constructs an Orthogonal Equiangular Feature Space (OEFS) where ID and OE features are jointly aligned with EBVs and Vast EBVs (VEBVs), respectively. The L2 symbol denotes activation intensity in each subspace, and represents the final discriminative VPC Score.
against ID weights, yet the classifier's limited capacity hinders adaptation to diverse OOD patterns. Evidential DNNs (Figure(5b) substitute logits with second-order evidence, mitigating confidenee and enabling explicit OOD uncertainty quantification, thus avoiding non- onnuumfoomm outputs. The uniform evidence prior reduces drift in ID/OE optimization but fails to ensure low OOD evidence activation via the ID) classifier. VBB-based DNNs (Figure(5b) replace learnable classifiers with fixed, predefined EBVs Shen et al. 2025). optimizing Neural Collapse's ETF prior via first-order Lx to align ID features with EBVs. This stabilizes ID representations but remains ID-centric.
VPC (Figure (l) integrates these advances: adopting EBV priors from (c), incorporating second-order evidence from (b) via LENG (replacing LNc), which ensures stable alignment of ID features with EBVs under OOD perturbations. VPC extends to a bilateral framework with predefined VEBVs creating orthogonal subspace provided with distinct geometric anchors to resolve diverse OOD patterns. Loc and LVEBY guide OOD activation therein while minimizing ID interference.
We next compare VPC's parameter and computational overhead against learnable classifier DNNs. Parameter-wise, Vanilla DNNs (Figure [5, b) incur Oddea X K)) for the final classifier. VPC eliminates this, adding a projection layer (ddeat doefs) meet dimensional constraints Shen et al. (2025) Papyan et al. (2020). The (K + V) EBVs/VEBVs are fixed constants, excluding them from trainable parameters.
As quantified in Table 8 this change is modest. For instance, on CIFAR-100 (K = 100) with ResNet-18 (drat 512). VPC replaces the 0.05LM parameter classifier (the vanilla head) with su 0.524M parameter projection layer. This results in a net increase of only ze 0.47M parameters, which constitutes just = 4. 4.2% of the 11.2M total model parameters. This modest overhead the necessary cost to instantiate the structured geometric space. We provide fair comparisons by adding this same projection layer to baselines Appendix B.3
LENG and Loc are computed in the low- dimensional K-space, while LVEBV operates in the V-space. The gradients for LVEBY are diagonal, leading to backpropagation complexity of O(V). In contrast, naive frameworks with (K + V)- dimensional outputs require dense softmax Jacobians, resulting in computational cost that scales as O(K + 2 which becomes prohibitive for large V.
Computationally, VPC decouples the loss functions:
At inference time, VPC is equally efficient. Our VPC score computes similarities against all prototypes (EBVs and VEBVs) once by merging them into single weight matrix and performing one unified matrix- vector multiplication (daue x (K + +V)). By comparison, baseline methods like MSP/Uncertannty perform smaller O K) multiplication, but due to the parallel nature of
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Table 8: Comparison of learnable parameter overhead. All units are in Millions (M). dieat is the penultimate feature dimension. The projection dimension doets for CIFAR is 1024; for ImageNet- Ik, is set 2048 per analysis.
Dataset
Model
Total (M)
dreat
Vanilla Head (M)
doets
VPC Proj. (M)
Added Params (M)
WRN-40-2
2.20
128
0.00129
1024
0.13107
+0.13
CIFAR-10
ResNet-18
11.20
512
0.00513
1024
0.52429
+0.52
DenseNet- 121
8.00
1024
0.01025
1024
1,04588
+1.04
WRN-40-2
2.20
128
0.01290
1024
0.13107
+0.12
CIFAR- 100
ResNet-18
11.20
512
0.05130
1024
0.52429
+0.47
DenseNet-
8.00
1024
0.10250
1024
1,04858
+0.95
ImageNer- Ik
ResNet-50 VTT-B-16
25.56
2048
2,049
2048
4.194
+2.15
86.56
768
0.769
2048
1.573
+0.80
GPUs, the practical wall-clock time difference between the two is negligible. This efficiency stands in sharp contrast to neighborhood- methods like KNN (Sun et al. 2022), which require computing distances against the entire training set (N samples) at inference time (O(N x dram)) and are thus prohibitively slow on large datasets. Therefore, VPC exchanges a modest, linear computational overhead for the significant performance benefit of explicitly modeling the OOD space, capability the baselines lack.
A.5 WHY DOES THE VPC SCORE YIELD ADDITIONAL DISCRIMINATIVE POWER?
Prior methods are confined to ID-class features and the learnable ID-class classifier, building various scoring functions on top of that setup. Our approach steps outside this limitation. To our knowledge, we are the first to explicitly expand a large preset classifier for OOD features. We analyze the effect through cosine similarity, Euclidean distance, and L2 activation strength.
Note that all our metrics are computed between normalized features and the preset VBVs/VEBVs. By contrast, the metrics for PFS and OE are computed between normalized features and the normalized classifier weights of the ID classes. For Euclidean distance and cosine similarity, we apply a unified MSP (max over classes) reduction and then average the per-sample values. The final aggregated results under different settings are reported in Table ( If evaluation is restricted to metrics based solely on the ID classifiers or EBVs, our gains are comparable to those of the state-of- of-th-ofthheart PFS, as both methods impose orthogonality The additional benefit of our approach is the use of predefined VEBVs to generate supplementary L2 activation differences, which constitutes distinctive advantage over competing methods.
OOD features, while the latter emphasizes uniform output.
In addition to the above findings, Table IU provides a comprehensive comparison of the VPC Score under LI (Manhattan) and 1.2 (Fuclidean) norms, ablating the influence of the EBV subspace coefficient (a) and the VEBV subspace coefficient (3). The results first indicate that the I.2 norm generally yields superior or more stable performance than the LI norm. This superiority likely arises because the L2 norm, which squares activations, more faithfully captures the variations in activation magnitude that are highly informative for complex OOD samples. Under the LI norm, performance is highly sensitive to the coefficient choice. Notably, relying solely on the VEBV 100) consistently and significantly outperforms relying only on the EBV subspace (a = 1 B = 0), especially on the more complex 11FAR-100 dataset. This demonstrates the critical role of the VEBV subspace in capturing OOD variability.
subspace (a 0,
The most striking finding, however, comes from the L2 norm results. Performance is remarkably stable and remains near-optimal across all tested combinations of a and B. Whether using only the ID-EBV subspace (a 3 = 0), only the OOD-VEBV subspace (a 0. 3 = 100), our main combined score (a = 1, 8 100), the detection metrics are nearly indistinguishable. This strongly implies that OEFS training successfully creates geometric separation where the L2 activation strength in either subspace alone becomes a sufficient and robust signal for OOD detection, capability rarely observed in previous methods.
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Table 9: Statistical results for ID/OOD under different metries on CIFAR-IO, cosine similarity, Euclidean distance, and L2 activation strength. Our method obtains additional discriminative scores by using the L.2 activation strength within the subspaces spanned by EBVs and VEBVs. We report ID/OOD averages and the gap (Difff).
Fuclidean Distance
Cosine Similarity
1.2
(Ingits/wcigh/EBBVs)
1.2 (VEBVs) OOD
Method
ID
0OD
Diff
ID
OOD
Difft
ID
0OD
Diff'
ID
Diff
OE
0.62
1.08
0.46
0.80
0.40
0.40
0.87
0.55
0.32
PFS
0.67
1.38
0.71
0.75
0.05
0.70
0.87
0.06
0.82
Ours: VPC
0.70
1.39
0.69
0.73
0.03
0.70
0.77
0.04
0.72
0.63
1.03
0.40
Table 10: Norm-aware comparison of VPC scores on CIFAR-10 and CIFAR-100 across backbones 401d R8sNet-18, DenseNet-121) 121). Metries are reported as FPRO5//AUROCT/AURR:
Score
CIFAR-10 10
CIFAR-100
Norm
a
B
WRN--102
ResNot-118
DeuseNet-121
WRN-40-2
ReaNet-18
DenseNet-121
0
9-41/9904499) 71
2.95/98.24999.6
2.01.9..1/999700
39.57788,00 01//8880//9.3. 39
40.01/89 26/9
28.63/89, 08/66 84
0
100
2.29/99. 00/99.78 78
2.86/97.
2,10/96.53/99
32. 22..2999.52/9855
38.71/92.000 08/28.2111811111
31.01/9.00098.77
L1
-1
1
10/99779
2.88/98.
66999.6
08/98. 34/99 68
38. (1.66999.1/999 .12/97.59 .99
38.39/22.1099 20
32.41/93. 78/98 65
2.28/999 02/99/79 78
2.87/97.58//99.
2.10/66.84// 84/99.43
34.90/93.26/998. .46
38.48/92.00
31.22//9.996//9,
-1
100
2,29//9.0009 00/99.88
1.86/97.32/99950
2.1009658/99.08
$2.40/93.50/98.55
08.744/2.08/98.22
11.03/99.00998.70
0
0127/99.19/999.1
955998550999.0
1.8/98.95/99977
32.04/93.65/98.55 555
39. 29.10/92.03//9 20
31.19/94.01/9%
0
100
2.27/99. 18/99.81
2.84/98 32/99.67 677
2. 70/9977
$2.03/93.65/99
38. 38.99.992.03/98.20
31.17/94.01/99
L2
1
2.27/99. 19/99.81
2.85/98.50.999 50.99.77
2. 95/99.78
$2.07/99.65/996 .53
39. 90.11192.03/98.222 98.20
31.19/94.01
2.27/99. 18/99.81
2.84/98. 49/99.70
94/99.78
32.05/99.65/98.5 53
38. 28.98/92.03/998200
31.17/94..0/9%%.
100
2.27/99 18 18/99.81
1884/8.32//99.6
2.1099886/99977
32.04/93.66/98.55
38.96.92.03.98.22
31.17.99..0/998.7
A.6 ACTIVATION STRENGTH IS A BETTER OOD SCORING FUNCTION
Building on the previous section, we introduce a new 1.2 activation-strength score for both PFS and OE Specifically, for PFS the activation matrix is the cosine similarity between normalized features and the normalized classifier weights; for OE we directly use the model's activation outputs. As shown in Table IT incorporating the L2 strength yields substantial improvements, further validating the benefit of class- score design for OOD detection.
Prior approaches typically rely on constraining OOD features to be orthogonal to the ID classifier or enforcing uear-uniform outputs. However, MSP tends to exaggerate the non-uniform outputs of hard samples, often leading to inseparable cases. In contrast, computing the overall activation strength (I.2) naturally benefits from both orthogonality and uniformity regularization, producing a more discriminative score.
Table 11: 1.2 Activation Strength Driven Improvements in 0OD Detection
Far-OOD Dulietts
Score Function
Method
SVHN
LSUN
Testues
Placsa365
Average
1PR095
AUROCT
199995
AUROCT
EPRO5
AUROCT
EPRUS
ROUY
EPR95]
AUROCT
AUROYY
CIFAR-10
MSP PES Score
00
1.95
99.23
0.80
99.67
1.95
99.36
3.70
99.23
8.80
97.76
3.44
99.05
PBS
0.75
98.95
0.35
99.51
1.30
99.26
2.85
98.49
7.25
97.02
2.50
98.65
L2 Score
0.70
1.45
99.36
0.90
99:43
7.10
7.20
97.90
0.91)
0.05)
CHEAK-100
MSP
06
34.95
93.75
14,90
97.23
49.50
85.16
43.35
90.63
$2.50
$7.68
39.04
91.49
PRS Scane
PP8
24.75
95.81
12.65
97.78
38:40
91.44
44.20
91.92
$1.85
90.33
$4.37
03.83
L.2 Score
06
PRS
23.60
25.00
94.97
95.78
10.10
13.00
97.59
95.08
53.50
91.97
85.03
43.05
91.99
90.45
49.40
32.25
90.64
90.27
34,03
36,93
(1
(1 (14)
92.43 (1994)
93.40 0.07)
A.7 CONVERGENCE VS. DIVERGENCE OF OOD FEATURES
PFS is the first to leverage the properties of Neural Collapse to explore the subspace spanned by ID-class classifier weights way to represent richly varying OOD features. However, it has several limitations. It aligns ID features with ID classifiers in stages and then imposes an orthogonality constraint that encourages OOD features to explore the representation space formed by the ID classifiers. This staged scheme cannot guarantee a stable representation space while ID/OE data are being jointly fine-tuned: ID features and ID classifiers may not be well aligned, and OOD outputs may
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remain non-uniform. In such dynamically changing and unstable space, enforcing orthogonality OOD features is unlikely to deliver the desired effect.
Our method effectively addresses this issue by directly presetting an approximately orthogonal, optimal classifier. It requires no staged constraints and can even achieve strong ID/OOD separability with single training run. Building this well-behaved geometric constraint, investigate whether OOD features should diverge or converge. Beyond the divergence loss proposed in the main text, which induces OOD features explore the VEBV subspace, we additionally propose VEBV convergence loss that encourages OOD features to collapse onto any one VEBV, defined as follows:
Ccon VEBV VEBV(<; ( =1 - max ISISV
m v
T vyyeb.
(45)
All other settings follow the main experiments. As shown Table 12 under our OEFS space constraint, forcing OOD features to collapse onto single VEBV yields reasonably good OOD detection, though the performance weakens as the number of ID classes increases. This is because when all OOD features collapse onto one VEBV, maintaining orthogonality all ID classes becomes increasingly difficult.
Table 12: Divergence vs. Convergence Loss the VEBVs Subspace.
Optimization Loss
SVIN
LSUN
iSUN
Textures
Places365
Average
FPR951
AUROCT
FPR95
AUROC1
EPR95L
AUROC1
FTR95L
AUROC1
FPR95]
AUROCT
FTR95
AUROCT
CIFAR-10
CLERY VEDY
0.85
99.62
0:45
99.50
1.10
99.50
2.25
99.38
6.70
97.91
2.27
99.18
CIPAR-100
Collyy Viaav
24.10
95.89
18,50
96.79
55.35
88.01
$2.90
89.70
29.40
91.96
38.07
92.47
97.98
26.25
94.97
32.04
We further provide visualizations that clearly depiet the starkly different activation patterns produced by the two optimization regimes. As shown in Figure 6land Figure analysis.
7
see Section B for detailed
B ADDITIONAL EXPERIMENTAL RESULTS AND VISUALIZATION
B. HARD OOD DETECTION
We further assess the generalization of VPC in a Hard OOD setting. We follow TTack et al. 2020 Sun ct al. 2022 Wang et al. 2024a) and use CIFAR-IO as ID while evaluating on LSUN-Fix, ImageNet- Resize, CIFAR-I00 100, and Lnn-emmgeNee, comparing VPC with OE, Energy-OE, DAL, and PFS (3). Across different architectures, VPC consistently improves OOD detection. Notably, the mean FPR95 on WideResNeet40022 and DenseNet-121 1 11.97% and 10.65% (lowest among baselines), and DenseNet-122 121 achieves the best mean AUROC = 97.55%, with leading subset performance on LSUN-Fix (99.52%) and ImaseNet-Resize (99.35%) ). These outcomes are attributable VPC's explicit OEFS design and enforced geometric separation, which jointly curtail false positives and preserve discriminative power in near-OOD regimes.
B.2 MORE RESULTS OF ONE-STAGE TRAINING AND TWO-STAGE TRAINING
Most existing OOD detection methods follow a two-stage training paradigm: pretrain on ID data, then jointly fine-tune on ID and OOD (OE) data. Although this paradigm appears closer to real- world deployment, in essence it relies on large OE distribution that is mismatched with both the ID distribution and the target test OOD distribution to cue the model to output discriminatively low-confidence predictions whenever inputs deviate from ID. Prior work has leveraged auxiliary OE datasets at cens-of-millionn of subsequent research. However, this approach entails substantial engineering complexity and reproducibility cost, and high sensitivity to numerous hyperparameters (e.g., OE ratio, loss weights, thresholding, staged leamnnng rates/schedules) and dependence on OE data selection bias and coverage.
scale to push models away from non-ID samples, spurring a line
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Table 13: Hard OOD detection CIFAR-10 benchmark.
Near-ODD
Model
Method
LSUN-Fix
InageNet-Resii Resize
CIFAR-100
Inn-llngeNNe
Average
FPR95]
AUROC1
FPR95]
AUROCT
FPR95]
AUROC1
[PR95]
AUROCT
FPR95L
AUROCT
CIFAR-10
With auxiliary ODD daza
OR
1.30
99.49
7.30
58 48
24.80
94.74
18.15
95.53
12.79
97.46
Energy-OE
8.75
91.57
94.03
WRNes-80-2
DAL
1:25
4.55
19:40
13:05
96
PFS
0.70
98, 59
5.35
98, 20
25. 15.15
88.88
17.45
92.41
12.16
91.52
Ours: VPC
1.20
99.46
5.40
98.59
25. 25
94.24
16.05
95.559
11.97
96,97
OB
1.45
26
4.45
29
24.
94.78
17.70
61
11.97
96.99
Energy OE
1.55
98.08
9:05
97.96
30.35
92.
21.10
94.64
15.51
95.78
ResNet-18
DAL
1.10
99.08
5.65
98.43
25. 20
95.07
16.95
$0.90
12.22
97.12
Ours: VPC
1.20
99.45
6.40
98 40
23.65
94-72
17.10
93.95
12.09
97.15
OE
1.35
99.31
1.85
98.99
24.30
94.75
15.65
95.95
10.79
97.25
Energy-OE
DemseNet-121
DAL
0.70
98.88
2.35
99.06
25.65
94.13
16.75
95.46
11.36
96.88
PES
0.95
99.21
3.55
98 47
25.15
92.21
16.75
94.02
11.60
95.98
97.55
To address these issues, we propose a one-stage joint training scheme that uses ID and OOD data within single optimization process under unified objective, thereby markedly reducing hyperparameter burden and training overhead. the main text we systematically compare two-stage and one-stage training: in most settings, the one-stage approach achieves superior comparable OOD performance. Morcover, Table reportss ID classification accuracy (Acc), showing that one- stage training does not incur significant drop in ID accuracy, further corroborating its effectiveness and practical deployability in improving OOD detection robustness while preserving ID performance.
Table 14: one-stage training vs. two-stage training on CIFAR-IO with WideResNer-40-22 We report average FPR95 AUROC1, AUPR1, and ID Acct.
one-stage (ID OE 150 epoch)
two-stage (ID 200 epoch + OE 50 epoch) FPR95
Method
FPR95
AUROCT
AUPR
Acc
AUROC1
AUPR
Acet
OE
2.74
99.01
99.79
95.39
3.44
99.05
99.79
95.67
Energy-OE
2.29
98.79
99.72
93.55
3.75
98.66
99.69
90.85
DAL
2.95
98.88
99.75
94.99
3.17
98.84
99.74
94.96
PFS
2.44
98.87
99.68
94.68
2.68
98.66
99.65
94.65
Ours: VPC
2.01
99.19
99.82
95.32
2.27
99.18
99.81
95.74
B.3 FAIRNESS EVALUATION UNDER OEFS DIMENSIONAL. CONSTRAINTS
OEFS requires scaling to large bank of predefined classifiers (VEBVs) sufficiently capture OOD variability. Since the number of EBV Vs is limited by dimensionality. the OEFS dimensionality must be greater than the combined count of EBVs and VEBVs. We therefore replace the final linear head in WideRessNer-40-2. ResNet-18, and DenseNet-122 with learnable projector that lifts features higher-dimensional space. For OEFS sizes 500, 1000, 2000, the projector outputs 512, 1024, 2048; its input equals each backbone's 's penultimate feature dimension. This design increases trainable parameters. To maintain fairness, we mirror the same projector in all compared baselines and also report the original backbones' detailed results (See Table 15 and Table 16)
Results reveal that than ours exhibit inconsistent performance across methods/backbones. We hypothesize that the extra dimensional transform exacerbates classifier learning, especially for PFS whose objective hinges enforcing weight orthogonality, thereby amplifying sensitivity to optimization and data idiosyncrasies. Our approach circumvents this fragility: classifiers are pre-instantiated as optimal prototypes and remain fixed; training focuses solely aligning features to these prototypes. This avoids the instability induced by learning classifier weights after projection and, consistently, delivers superior OOD detection. The of projector is principled in VPC: it provides the necessary ambient dimensionality to host a large set equiangular prototypes without compromising ID geometry.
baselines despite having strictly more trainable parameters
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Table 15: Fair Comparison by adding Unified Projection layer with different architectures.
Far-OOD Datasets
Model
Method
LSUN
Places365
Average
FPRM
FPRMS AUROOC
FPROS
AUROC
FPRUS
AUROC
FPROS
AURROC
With mustilrry
DE
0.70
99.71
0.80
99.64
3.60
$9.03
9.30
97.65
2.60
99.25
3.40
99.06
WidkResNes-40-33
Esergy-Ol
0.50
99:58
0.60
99.10
4.30
59. 03.
7.10
97.885
2.20
99.16
2.96
99.90
Projection Layer
PRS
DAL
0.75
0.90
99.56
99
0.70
0.50
99. 50
99
3.05
3.40
98.
99..03
8.00
2.85
99.21
86
8.30
2.55
98. 74
97.55
3.12
3.08
98. 70
96.97
Ours: VPC
0.85
99.50
99.50
2.25
99.38
6.70
97.91
2.27
99.18
1.50
10:55
ResNec-18
Enengy
Projectio Layer
DAL
1.20
1:10
3.35
97.79
Ours: VPC
0.93
98.56
1.60
98.81
2.00
98.61
2.55
98.30
7.10
97.34
2.84
98.32
DR
1.40
90:13
0.65
99 52
1.80
99:
8.10
97.51
2.20
99.06
2.83
98.91
DenseNet-121 121
Energy OE
0.85
99.52
0.45
99 18
1.75
98.88
3.00
97.58
3.55
98, 98.70
2.92
96.77
Projection Layer
PRS
DAL
1.30
1.15
991.1
98.76
0.35
0.40
99. 50
99.55
0.60
0.20
99
19. 19:12
2.45
7.55
99899
96.87
7.30
1.70
98,
97.05
2.37
2.19
98.92
98.61
VPC
WideResNes-4BU
Energy OC
$2.75
18.05
96.96
61, 61.90
65.38
50.75
39.70
42.95
91.27
41.28
91.67
Projection Layer
PPS
DAL
21.45
14.45
96.17
97.13
17:45
13.30
97. 38
96.89
43, 05
56 80
92.88
89.27
50.45
40.05
91.
50.11
29.15
.99 60
92.09
88.51
36.84
35.31
92.91
91.
Ours: VPC
9.95
97.08
26.25
95.75
26,
94.97
45.05
29.67
52.45
89.85
32.04
93.65
47.55
91.28
37.05
92.81
43.55
91.71
56.85
$5.96
53.85
$7.48
47.77
$9.85
ResNec-18
DAL
Flergy-OE
29.05
95.03
30.85
95.00
$4.90
45.35
37.37
92.68
Projection Layer
90.94
22.30
30.80
94.09
$8.52
92.03
DB
20.75
96.24
24.10
95.30
23.55
95.78
00.30
36.28
55.80
86.59
36.90
92.08
DemseNirt121
Fsergy-IF
17.40
97.00
18.35
96.75
56.40
89:37
56.50
80.3 34
45.05
92.00
38.74
92.92
Projection Layer
PFS
DAL
22.45
16.85
96.33
96.62
16.25
18.50
96.92
96.89
49 80
56.80
88.73
$8.28
41.40
56.20
55. 69
90.52
59,50
45.60
91.06
56.72
36.76
39.91
91.81
92.34
Ours: VPC
12.50
96.67
22.70
88.89
94.01
Table 16: Detailed Results with different architectures. The best result in bold.
Far-ODD Dutusets ISUN
Mode
Method
SVIEN
LSUN
Tesuuess
Places365
Averaze
FPROS]
AUROCT
FPROSL
AUROCC
FFROSL
AUROC
FPRUSI
AUROCT
FPRSS]
AUROCT
FPRUS]
AUROCT
CIFAR-10
Energy
WoldResNa-40022
98.74
09.20
VPC
OE
3.25
96.37
1.25
96.97
1.30
95.75
3.05
95.35
5.45
97.34
3.45
98.36
Eac
1.40
96.30
2.55
98.44
2.50
98.85
3.40
95.55
9.15
97.17
3.85
98.26
R8
DAL
1.00
99.70
1.15
99
1.90
98.95
2.70
99.11
5,35
97.31
3.02
98.96
PES
1.45
99.29
0.50
99
1.95
95.85
2.45
95.83
5,80
97.26
3.05
99.72
Dure: VPC
0.95
06.56
1.60
08.81
2.00
08.61
2.55
95.50
7.10
97.34
2.54
96.32
OE
2.05
99.18
0.55
0.45
99.33
3.20
95.77
7.65
97.56
2.84
98.86
Eneggy DE
1.55
99.18
1.05
2.05
2.85
93.94
9.75
97.18
3.65
98.71
DenoNNas121
Cun:
99.33
98.95
95.87
ausiliary 0OD Jata
95.08
97.98
19:55
89.32
91.57
59.57
36.14
23.80
96.18
96.88
85.00
50.50
91.09
DAL
19.30
95.75
16.20
96
30.70
93.85
43.11
91.36
$5.11
55.330
32.80
93.21
PFS
24.70
95.81
12.65
97.75
38.35
91.44
44.20
91.32
51.85
90..3
34.35
93.33
Ours: VPC
9.95
97.98
26.25
95.75
26.50
94.97
45.01
89:67
$2.45
$9.58
32,04
93.65
OE
52.40
00.28
34.90
05.25
45.00
90.92
52.75
85.47
53.70
83.91
43.75
59.36
Eac
33.50
$4.29
41.50
93.12
56.30
88.55
45.60
89.46
$2.50
$8.28
46.34
90.70
18
DAL
36.50
93.94
33.85
93.39
49.50
90.15
43.95
90.10
59.75
56.74
44.57
90.57
PFS
24.30
95.33
19.35
96.79
46.55
91.70
$1.10
90.85
59.45
$8.51
40.15
92.64
Oure: VPC.
95.03
95.11
Energy
91.35
DeneNaa-121
94.04
96-43
Ours:
97.71
91.20
B.4 VISUALIZATION RESULTS AND ANALYSIS
This section presents a suite of visualizations to illustrate how VPC improves OOD detection on CIFAR-10 and CIFAR-100. The figures highlight OEFS activations, feature orthogonality, ID/OOD>
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score distributions, cross-man messic comparisons, and failure analyses, providing intuitive view of the advantages of our approach.
B.4.1 OEFS ACTIVATION ANALYSIS
Figure Soureeaane and show the mean OEFS activations after optimizing VPC with the divergence loss and the convergence loss CIFAR-10 and CIFAR-100, respectively. The divergence loss induces canonical activation pattern across EBVs (i.e., activation shape is consistent across EBVs), reflecting the scale and coverage of the OE data. Despite this, OOD samples still exhibit distinct L.2 activation magnitudes that our VPC Score can capture. Note that the seemingly higher activations along EBVs arise because per-sample activations are dispersed and then averaged to approximately 1/C over classes. In contrast, the convergence loss collapses OOD features toward a single EBV and also influences the ID features.
B.4.2 ORTHOGONALITY OF OOD FEATURES
Figure g visualizes, on CIFAR-10, the projections of the first three class features, OOD features, and the corresponding classifier weights/EBBs. Both PFS and our method enforce weiuurr-eiigh/EEB orthogonality, which leads to stronger OOD separability. We also observe class-wise drift under PFS, whereas our ENC loss via a uniform evidence prior that suppresses such drift.
B.4.3 IMPACT OF SCORING FUNCTIONS ON ID/OOD DISTRIBUTIONS
Figure 9 and 10 compare ID/OOD score distributions under different scoring functions CIFAR-10 and CIFAR-ION 100. The VPC Score consistently reduces the overlap between ID and OOD distributions and produces a sharper OOD peak, which is crucial for effective detection.
B.4.4 VISUAL PERFORMANCE ACROSS SCORING FUNCTIONS
Figures II and 12 illustrate that, CECCI--O 100, when MSP and Uncertainty perform poorly, the VPC Score remains discriminative; the figures show paired ID/OOD examples. Additional qualitative results on CIFAR- are given in Figures 13 18 (ID and OOD datasets including SVHN, LSUN, iSUN, Texture, and Places365), and the corresponding CIFAR-10 results appear Figures In each figure, the left three panels depict cases where all metries perform well, while the right three highlight challenging cases where the VPC Score performs better.
19 24
B.4.5 FAILURE CASE ANALYSIS
Figure 25 focuses on failures detection: the left panels show representative failures on CIFAR- 100, and the right panels show those on CIFAR-10 10, helping reveal model weaknesses and guide improvements. Figure 26 presents failures in OOD detection, again split by dataset (left: CIFAR-100, right: CIFAR-100 exposing the limitations of the methods under edge conditions.
C LIMITATION
Our method, Vast Predefined Classifiers (VPC), is founded on pre-specified Orthogonal Equiangular Feature Space (OEFS), which is populated using a training paradigm guided by our proposed ENC, VEBV, and OC losses. While this approach yields strong empirical results, its foundational design choices introduce several key limitations.
(i) Dependence on Dimensional Constraints. The reliance predefined OEFS imposes ar- chitectural prerequisite rooted in the properties of ETFs. Specifically, the existence of a Simplex ETF dictates that the feature dimension must satisfy strict lower bound determined by the total number of prototypes. This structural coupling introduces a scalability constraint, often necessitating higher-dimensional projections for large-scale applications.
(ii) The enforcement a rigid equiangular structure for in-distribution classes prioritizes maximal separability at the expense of potentially valuable information regarding inter-class semantic similarity. While our focus is on leveraging OE data for stable separation, incorporating hierarchical prototype structures into the OEFS framework remains promising avenue for future research.
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(iii) The multi-objective training paradigm presents its set of challenges. The stable convergence required to achieve an ideal Neural Collapse state for in-distribution data can be disrupted by the competing objectives of attracting OOD features to their subspace while enforcing orthogonality. This inherent tension makes the full materialization of the ideal geometry difficult.
D LLM USAGE STATEMENT
We utilized LLMs to assist the drafting and editing process, aiming to enhance the clarity, coherence, and quality of the text and technical presentations. All content generated refined by LLMs has been thoroughly reviewed, verified, and revised by the authors, who assume full responsibility for all scientific content.
sub-space
Nemill
(b) Pull OOD fentures closer to one collapted VEBVs
Figure 6: Mean activation after optimizing VPC with divergence and convergence loss on CIFAR-10.
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(4) Pull OOD features closer the full VEBV's subsspace
(b) Pull OOD features clower collapsed VEBV'
Figure 7: Mean activation after optimizing VPC with divergence and convergence loss CIFAR-100.
Ol-ttrltedd model
PFS-tramol
Figure 8: Orthogonality visualization of OOD features and top three class features CIFAR- 10. comparing projections of the top three class features, OOD features with classifier weights/EBVs. PFS and our method optimize OOD detection via orthogonal constraints.
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(N) Uxcerbinty
Figure 9: ID/OOD distribution plot on CIFAR-IO with various scoring functions for OOD detection.
VPC Score minimizes overlap and boosts OOD distribution peak sharpness.
do Enceriabity
Figure 10: ID/OOD distribution plot on CIFAR- 100 with various scoring functions for OOD detection. VPC Score minimizes overlap and boosts OOD distribution peak sharpness.
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Figure 11: Visualization of ID and OOD detection using different scoring functions in CIFAR-100. VPC Score maintains distinguishability even when MSP and Uncertainty perform poorly.
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Figure 12: Visualization ID and OOD detection using different scoring functions in CIFAR-10. VPC Score maintains distinguishability even when MSP and Uncertainty perform poorly.
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Figure 14: Visualization of 0OD:SVHN detection using different scoring functions in CIFAR-100.
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Published as a conference paper at ICLR 2026
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Figure 15: Visualization of OOD:LSUN detection using different scoring functions in CIFAR-100.
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Figure 16: Visualization of OOD:iSUN detection using different scoring functions in CIFAR-100.
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Figure 17: Visualization of OOD: detection using different scoring functions CIFAR-100.
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Figure 18: Visualization of 0OD:Place365 detection using different scoring functions in CIFAR-100.
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Figure 20: Visualization of OOD:SVHN detection using different scoring functions in CIFAR-10.
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Figure 21: Visualization of OOD:LSUN detection using different scoring functions in CIFAR-10.
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Figure 22: Visualization of OOD:iSUN detection using different scoring functions in CIFAR-10.
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Figure 23: Visualization of O0D:Texture detection using different scoring functions in CIFAR-10.
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Published as a conference paper at ICLR 2026
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Figure 24: Visualization of Place3ace365 detection using different scoring functions in CIFAR-10.
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Figure 25: Visualization of ID detection using different scoring functions in CIFAR- 100 and CIFAR- 10. The left three images show failure cases for ID detection in CIFAR-100; the right three images show failure cases in CIFAR-10.
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Figure 26: Visualization of OOD detection using different scoring functions in CIFAR- 100 and CIF The left three images show failure cases for OOD detection in CIFAR-100; the right three images show failure cases in CIFAR-10.
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Reviews:
[{"review_id": "rEqa3l62K5", "reviewer": "Reviewer_MkUp", "summary": "The paper proposes Vast Predefined Classifiers (VPC), a geometry-driven out-of-distribution (OOD) detection framework that leverages auxiliary outlier exposure (OE) data. VPC defines an Orthogonal Equiangular Feature Space (OEFS) with two sets of fixed, equiangular basis vectors:\n\n* EBVs (Equiangular Basic Vectors) for in-distribution (ID) classes,\n* VEBVs (Vast EBVs) to span a rich subspace for diverse OOD patterns.\n\nDuring training, ID features are aligned to EBVs using an evidential prior-guided loss (LENC), while OE features are attracted to the VEBV subspace via a VEBV loss (LVEBV) and kept orthogonal to ID prototypes via an orthogonality constraint (LOC). OOD detection is performed using the VPC Score, based on the L2 activation intensity over the two subspaces. Experiments on CIFAR-10/100 show state-of-the-art performance in both two-stage and one-stage training setups.", "questions": "Although PVC achieved SOTA results, the fact that the experiments are only conducted on CIFAR raises a few doubts about the strength of the approach and its broader applicability to real-word scenarios. Could the authors show some results on stronger/larger datasets such as (tiny-)ImageNet or apply their approach on domain-shift scenarios? This is the main limitation of the paper.\n\nCould the authors share some insights on the computational overhead of their approach? While PVC achieved SOTA results, the improvements seem rather marginal. It's important to know how the relative gains fare with the computational needs of the approach.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The key strengths of the proposed approach are:\n\n* Explicit geometric control: OEFS provides a principled, interpretable structure that cleanly separates ID and OOD representations.\n\n* Scalable OOD modeling: VEBVs offer a flexible, expandable subspace to capture diverse OOD modes, unlike methods that merely push OOD to uniform outputs.\n\n* Strong empirical results: VPC achieves SOTA on standard benchmarks across multiple architectures and training paradigms.\n\n* Robust scoring: The VPC Score is class-agnostic, geometry-based, and avoids pitfalls of softmax-based confidence.", "weaknesses": "The main limitations of the proposed PVC approach are:\n\n* Fixed classifier geometry: Predefined EBVs/VEBVs may limit adaptability to complex or evolving ID/OOD distributions, especially when class semantics shift.\n\n* Computational overhead: Large VEBV sets increase memory and computation (e.g., for projection and loss calculation), though not deeply analyzed. It's not clear what the computational overhead of PVC is since no ablation results are proposed. Showing how PVC scales as a function of the dataset size, or providing some asymptotic bounds with respect to $K + V$ could help understand the applicability of PVC to real-world scenarios.\n\n* Dependence on OE data: Like all OE-based methods, performance relies on the quality and relevance of auxiliary OOD data; may degrade if OE is unrepresentative.\n\n* Limited real-world validation: The experiments are mainly confined to CIFAR benchmarks and lack evaluation on more complex datasets (e.g., ImageNet-scale or domain-shift scenarios).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "NrT0PNoAoK", "reviewer": "Reviewer_m4Yu", "summary": "The paper “LET OOD FEATURE EXPLORE VAST PREDEFINED CLASSIFIERS (VPC)” proposes a new framework for out-of-distribution (OOD) detection that focuses on explicitly modeling the geometry of feature representations. Current outlier exposure (OE) methods improve robustness by fine-tuning networks on in-distribution (ID) and auxiliary OOD data, but they largely encourage low-confidence OOD predictions or post-hoc score regularization without directly shaping feature space structure. As a result, separability between ID and OOD representations is often unstable, and OOD features lack sufficient representational diversity.\n\nTo address these issues, the authors introduce Vast Predefined Classifiers (VPC), which define an Orthogonal Equiangular Feature Space (OEFS) composed of fixed Equiangular Basic Vectors (EBVs) representing ID classes and a large pool of Vast EBVs (VEBVs) representing an extended OOD subspace. They design several complementary losses that guide feature alignment: an evidential prior–guided Neural Collapse (ENC) loss that stabilizes ID features toward their EBVs, a VEBV loss that encourages OOD features to explore the VEBV subspace, and an orthogonality constraint that prevents mixing of ID and OOD features. The framework introduces the VPC Score, based on the L2 activation intensity difference between the ID and OOD subspaces, serving as a class-agnostic and interpretable OOD detection metric. Experiments on CIFAR-10 and CIFAR-100 using multiple backbones (WideResNet, ResNet, DenseNet) show that VPC achieves state-of-the-art results, outperforming existing baselines such as OE, Energy-OE, DAL, and PFS. Ablation studies confirm that each component—particularly the geometric subspace design and the VEBV loss—contributes to improved performance.", "questions": "Kindly please see the weakness above.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "The advantages of this method lie in its geometric interpretability, stable training through evidential priors, scalability for modeling diverse OOD patterns, and class-agnostic scoring that avoids bias from classifier heads. \n\nEmpirical results demonstrate strong robustness and clear separability between ID and OOD features. \n\nEq 5 in preventing over-bias toward any single class and promoting stable convergence of ID feature is new to me, sounds interesting.", "weaknesses": "The authors may need to further consider the definition of OOD detection in Sec 3.1. Defining OOD data to be those out of the label space might be more suitable for OOD detection. The adopted definition based on support set may fail to consider the case of covariate shift, which should be considered as ID instead of OOD. \n\nA paragraph of intuition at the beginning of Sec 3.2 will improve the clarity of the manuscript, otherwise readers may understand what the authors want to do very late in Sec 3.3 – 3.4. The intuition should echo the drawbacks of previous works in OOD diversity and some other things like this. \n\nHat m in Eq 4 is not defined, I think it is the embedding feature, right? There are a lot of other notations that are not defined, such as \\tau in the same formulation. The authors should carefully check about them. \n\nThe proposed method is somehow like a variance of KNN, could the authors further highlight the improvement over KNN?\n\nCould Eq 6 ensure the diversity of OOD basis vectors? The models tend to map OOD data to the same region in the embedding space even at the very beginning,  optimizing via Eq 6 will exacerbate this case and in this case V=1 is enough, obviously violating the claimed “vast” features in Sec 3.2.\n\nThe authors define w in the unit hypersphere, but, from my view, none of the objective in Sec 3 can ensure this constraint. \n\nAs far as I know, many works conducted experiments on ImageNet benchmark or ViT family, such as [1,2]. I do not want to add the rebuttal burden of the authors, as many researchers lack sufficient computational resources. But you can consider it as a suggestion to improve the quality of this research in the future. \n\n[1] Watermarking for Out-of-distribution Detection\n\n[2] Out-of-distribution Detection Learning with Unreliable Out-of-distribution Sources", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "lko9Hj2rE8", "reviewer": "Reviewer_H2ze", "summary": "This paper focuses on the out-of-distribution detection problem. Specifically, this work proposes a geometry-driven framework for OOD detection, that distinguishes ID data from OOD data by embedding to an orthogonal equiangular feature space, which provides an evidential priors that mitigates the instability from outlier exposure data, and then a new activation-based OOD score is introduced which is demonstrated to be effective than traditional softmax and energy scores.", "questions": "1. What guarantees separability between unseen OOD samples and ID features, as the OOD distribution shifts outside the auxiliary OE coverage, with the orthogonal equiangular construction? \n2. What is the computational complexity for maintaining and updating thousands of supervised classifiers in high dimensionality? Can this approach extend to modern vision transformers or multimodal models? \n3. Is it ensured that the penultimate layer retains the intended geometry under nonlinear activations or the presence of normalization layers? \n4. Neural Collapse is known to predominantly occur asymptotically with cross-entropy, under certain conditions. In the context of fixed EBVs and with extra constraints, what guarantees converge to the intended geometric configuration?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The idea of this work that fixing the orthogonal equiangular basis builds on the neural collapse, to the reviewer's knowledge on the area of OOD detection, is new and interesting.\n2. The paper has provided comprehensive experimental verification on demonstrating the performance of the proposed method.", "weaknesses": "1. Although the proposal is empirically demonstrated to be effective, there is no theoretical justification that rigorously justify or explain why the newly introduced representation structure can help the OOD separability or generalize to the unseen OOD distributions.\n2. The computational cost or overhead seems to be much than previous OOD score based on softmax and energy, especially for large-scale ID dataset.\n3. Except for the common benchmark datasets, some large-scale dataset should also be considered in the verification. And more advanced OOD detection methods like ASH or other scoring functions are not considered in the comparison, which should carefully support the claim that current proposal is SOTA.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "fKkllL6r3G", "reviewer": "Reviewer_ioxr", "summary": "The paper proposes VPC (Vast Predefined Classifiers), a novel approach for Out-of-Distribution (OOD) detection that explicitly models OOD variability by allocating a large set of predefined optimal classifiers distinct from ID classifiers. Inspired by Neural Collapse and equiangular basis vectors (EBVs), the method aligns the feature space to an Orthogonal Equiangular Feature Space (OEFS) with predefined EBVs/VEBVs. The authors further propose VPC Score, based on the VEBVs, which measures the relative activation intensity across orthogonal subspaces to distinguish ID and OOD samples. The authors demonstrate that VPC achieves state-of-the-art performance on CIFAR-10/100 benchmarks, significantly outperforming existing OE-based methods.", "questions": "See the weaknesses above.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 4, "strengths": "1. Originality: The proposed method VPC can sufficiently address the problems that are brought by traditional OE methods, i.e., incomplete ID/OOD separation and limited representation for OOD samples\n2. Quality: The paper is written with clear motivation and logical support, making the proposed method more interpretable and theoretically justifiable.\n3. Handling the variability of OOD features: different from previous works, the proposed VPC can jointly separate the ID/OOD features, as well as preserving the diverse OOD features, making it possible for more representation tasks.", "weaknesses": "1. Limited implementation details: the paper lacks sufficient discussion on how the hyperparameters are determined for optimal performance, such as the $\\alpha$ and $\\beta$ for the VPC score.\n2. The experiment setting is limited to CIFAR datasets, which are relatively small. Given the empirical analysis in Table 5, the proposed method works by adopting a significantly larger V value, compared to the number of classes. The performance of VPC remains unclear when the number of ID classes is large (e.g., ImageNet-1K), which may require an even larger V value. This can potentially lead to unaffordable computational overhead or suboptimal OOD performance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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": "zmQtPclnvG", "reviewer": "Reviewer_wRrj", "summary": "The paper proposes Vast Predefined Classifiers for OE-based OOD detection by combining Neural Collapse and Evidential Deep Learning. extensive experiments demonstrate the effectiveness of Vast Predefined Classifiers.", "questions": "see weakness.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "1. This paper is well-written.\n2. This paper use 3 performance metrics\n3. This paper provides extensive visualization", "weaknesses": "1. lack of novelty: the use of ETF for OOD detection has been explored in [1]. Compared with [1], this paper additionally introduce V vectors, which is a nature extenstion in the setting of OE and also similarly explored in [2]. Eq. (5) is normal learning objective in Evidential Deep Learning without any modification.\n2. what is $f_j()$ in $\\mathcal{L}_{OC}$?\n3. the authors claim that their propsoed VPC score differs from softmax score and energy score in that the latter two are heuristic. However, there is thorectical justification for the VPC score with regards to the Separation between ID and OOD data.\n4. Given that this paper focuses on improve the feature discriminative. The mostly reent feature-based OOD detection methods [3-5] should be discussed.\n5. Lack of experiments on large-scale datasets (e.g. ImageNet-1k) and transformer-based backbones (e.g. ViT).\n6. the authors are encouraged to conduct experiments on the OpenOOD benchmark.\n\n[1]  Distributional Prototype Learning for Out-of-distribution Detection\n\n[2]  Learning Placeholders for Open-Set Recognition\n\n[3]  Conjnorm: Tractable density estimation for out-of-distribution detection\n\n[4]  Out-of-distribution detection based on in-distribution data patterns memorization with modern hopfield energy\n\n[5]  Learning with Mixture of Prototypes for Out-of-Distribution Detection", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "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:

Decision: Accept (Poster)