arxiv: 2605.14413 · v1 · submitted 2026-05-14 · 💻 cs.LG · cs.AI

Recognition: 2 theorem links

· Lean Theorem

MahaVar: OOD Detection via Class-wise Mahalanobis Distance Variance under Neural Collapse

Donghwan Kim , Hyunsoo Yoon

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:14 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords OOD detectionMahalanobis distanceNeural Collapseout-of-distributiondistance variancepost-hoc methodimage classification

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The pith

Class-wise Mahalanobis distance variance distinguishes in-distribution from out-of-distribution samples under Neural Collapse geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that in-distribution samples produce a sharp minimum in class-wise Mahalanobis distances, with small distance to the nearest class and large distances to others, yielding high variance across classes. Out-of-distribution samples lack this structure and show lower variance. This pattern is derived from relaxed Neural Collapse assumptions on within-class compactness and inter-class separation, providing a geometric basis for variance as an OOD indicator. The authors introduce MahaVar, which augments standard Mahalanobis scoring with the variance term, and report improved AUROC and FPR@95 on CIFAR-100 and ImageNet under the OpenOOD protocol.

Core claim

Under relaxed Neural Collapse assumptions on within-class compactness and inter-class separation, in-distribution samples structurally exhibit high class-wise Mahalanobis distance variance due to a pronounced sharp minimum structure, whereas out-of-distribution samples exhibit lower variance. This difference supplies a theoretical basis for the class-wise variance term, which MahaVar adds to the Mahalanobis distance to form an OOD score that achieves state-of-the-art results on standard image benchmarks.

What carries the argument

The class-wise Mahalanobis distance variance term, which measures the sharp minimum structure across distances to different class means.

If this is right

MahaVar yields consistent gains in both AUROC and FPR@95 over prior Mahalanobis-based detectors across all tested benchmarks.
The method remains a simple post-hoc addition that follows the OpenOOD v1.5 evaluation protocol.
The variance signal is grounded in Neural Collapse geometry rather than dataset-specific tuning.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same variance signal could be tested as an add-on to other prototype-based OOD scores that compute distances to class centers.
If Neural Collapse geometry weakens on non-image data, the variance advantage may shrink and require separate validation.
A controlled ablation that varies the degree of collapse could map the exact compactness threshold where the method loses effectiveness.

Load-bearing premise

In-distribution samples must satisfy relaxed Neural Collapse conditions of within-class compactness and inter-class separation so that high variance appears.

What would settle it

Direct computation on CIFAR-100 or ImageNet showing that in-distribution samples do not produce reliably higher class-wise Mahalanobis variance than out-of-distribution samples.

Figures

Figures reproduced from arXiv: 2605.14413 by Donghwan Kim, Hyunsoo Yoon.

**Figure 1.** Figure 1: Sorted class-wise dMaha,c(x) of Mahalanobis++ [22] on CIFAR-10 and CIFAR-100 as ID datasets, evaluated on OOD datasets following the OpenOOD v1.5 benchmark protocol [38] with ResNet-18 as the backbone. The x-axis represents the Mahalanobis distance rank, where rank 0 corresponds to the nearest class mean. Shaded regions indicate ±0.5σ across samples at each rank. where a higher score indicates a greater li… view at source ↗

**Figure 2.** Figure 2: Distribution of Varc[dMaha,c(x)] for ID and OOD samples on CIFAR-10 and CIFAR-100, evaluated on OOD datasets following the OpenOOD v1.5 benchmark protocol with ResNet-18 as the backbone. The dashed blue line indicates the mean variance of ID samples. Black bars indicate the median of each distribution. While existing Mahalanobis-based methods either focus solely on the nearest class distance [15, 22] or co… view at source ↗

**Figure 3.** Figure 3: Sorted class-wise dMaha,c(x) (without L2 normalization) on CIFAR-10 and CIFAR-100 as ID datasets, evaluated on OOD datasets following the OpenOOD v1.5 benchmark protocol with ResNet-18 as the backbone. The x-axis represents the Mahalanobis distance rank, where rank 0 corresponds to the nearest class mean. Shaded regions indicate ±0.5σ across samples at each rank [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Sorted class-wise dMaha,c(x) of Mahalanobis++ on ImageNet as the ID dataset, evaluated on five OOD datasets following the OpenOOD v1.5 benchmark protocol across three backbone architectures (ResNet-50, Swin-B, and ViT-B). The x-axis represents the Mahalanobis distance rank, where rank 0 corresponds to the nearest class mean. Shaded regions indicate ±0.5σ across samples at each rank. 13 [PITH_FULL_IMAGE:fi… view at source ↗

**Figure 5.** Figure 5: Sorted class-wise dMaha,c(x) (without L2 normalization) on ImageNet as the ID dataset, evaluated on five OOD datasets following the OpenOOD v1.5 benchmark protocol across three backbone architectures (ResNet-50, Swin-B, and ViT-B). The x-axis represents the Mahalanobis distance rank, where rank 0 corresponds to the nearest class mean. Shaded regions indicate ±0.5σ across samples at each rank [PITH_FULL_IM… view at source ↗

**Figure 6.** Figure 6: Distribution of Varc[dMaha,c(x)] for ID and OOD samples on CIFAR-10 and CIFAR-100 without L2 normalization, evaluated on OOD datasets following the OpenOOD v1.5 benchmark protocol with ResNet-18 as the backbone. The dashed blue line indicates the mean variance of ID samples. Black bars indicate the median of each distribution. 14 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗

**Figure 7.** Figure 7: Distribution of Varc[dMaha,c(x)] for ID and OOD samples on ImageNet with L2 normalization following Mahalanobis++, evaluated on five OOD datasets following the OpenOOD v1.5 benchmark protocol across three backbone architectures (ResNet-50, Swin-B, and ViT-B). The dashed blue line indicates the mean variance of ID samples. Black bars indicate the median of each distribution [PITH_FULL_IMAGE:figures/full_f… view at source ↗

**Figure 8.** Figure 8: Distribution of Varc[dMaha,c(x)] for ID and OOD samples on ImageNet without L2 normalization, evaluated on five OOD datasets following the OpenOOD v1.5 benchmark protocol across three backbone architectures (ResNet-50, Swin-B, and ViT-B). The dashed blue line indicates the mean variance of ID samples. Black bars indicate the median of each distribution. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

read the original abstract

Out-of-distribution (OOD) detection is a critical component for ensuring the reliability of deep neural networks in safety-critical applications. In this work, we present a key empirical observation: for in-distribution (ID) samples, class-wise Mahalanobis distances exhibit a pronounced sharp minimum structure, where the distance to the nearest class is small while distances to all other classes remain large, resulting in high variance across classes. In contrast, OOD samples tend to exhibit a less pronounced sharp minimum structure, producing comparatively lower variance across classes. We further provide a theoretical analysis grounding this observation in Neural Collapse geometry: under relaxed Neural Collapse assumptions on within-class compactness and inter-class separation, ID samples are shown to structurally exhibit high class-wise distance variance, offering a theoretical basis for its use as an OOD score. Motivated by this observation and its theoretical backing, we propose MahaVar, a simple and effective post-hoc OOD detector that augments the Mahalanobis distance with a class-wise distance variance term. Following the OpenOOD v1.5 benchmark protocol, MahaVar achieves state-of-the-art performance on CIFAR-100 and ImageNet, with consistent improvements in both AUROC and FPR@95 over existing Mahalanobis-based methods across all benchmarks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

MahaVar adds a variance term to the standard Mahalanobis OOD score and reports consistent gains on CIFAR-100 and ImageNet, but the Neural Collapse justification rests on unverified assumptions about the actual trained features.

read the letter

The main point is that MahaVar takes the usual class-wise Mahalanobis distances, notes that ID samples produce high variance across them while OOD samples produce lower variance, and adds that variance as an extra scoring term. The paper shows this simple tweak improves AUROC and FPR@95 over plain Mahalanobis on the OpenOOD benchmarks for CIFAR-100 and ImageNet. That is the concrete advance: an easy post-hoc adjustment that builds directly on an existing detector without new training or parameters beyond the usual class means and covariance. The empirical observation is clear and the numbers are reported under a public protocol, which makes the gains straightforward to check. The method stays within the Mahalanobis framework rather than inventing a new one, so the lift is incremental but reproducible. The soft spot is the theoretical part. The argument invokes relaxed Neural Collapse to claim that ID points must show a sharp minimum in the distance vector and therefore high variance. Yet the paper does not appear to verify whether the ResNet features actually satisfy the required within-class compactness and inter-class separation at the layer used for the detector. If those conditions do not hold tightly, the performance edge becomes an empirical observation rather than a geometric prediction. The stress-test note correctly flags this gap. Readers working on practical OOD detection for vision models will find the experiments useful and the code easy to try. The work is coherent enough on its own terms to merit referee time, even if the theory needs tightening and the gains remain modest. I would send it for peer review.

Referee Report

1 major / 1 minor

Summary. The paper introduces MahaVar, a post-hoc OOD detector that augments the class-conditional Mahalanobis distance with a variance term computed over the vector of class-wise distances. It reports the empirical observation that ID samples produce a sharp minimum structure (small distance to nearest class, large to others) yielding high variance, while OOD samples produce flatter distance vectors and lower variance. This observation is theoretically motivated by relaxed Neural Collapse assumptions of within-class compactness and inter-class separation. Experiments on OpenOOD v1.5 benchmarks claim state-of-the-art AUROC and FPR@95 on CIFAR-100 and ImageNet, with consistent gains over prior Mahalanobis-based baselines.

Significance. If the reported gains are reproducible and the variance term can be shown to be a direct geometric consequence of Neural Collapse rather than an empirical tweak, the method supplies a lightweight, training-free improvement to an established OOD baseline. The absence of additional fitted parameters and the use of existing class means and covariances are practical strengths. The work would be more significant if the theoretical section verified that the feature representations of the evaluated ResNets actually satisfy the compactness and separation conditions invoked.

major comments (1)

[Theoretical analysis] Theoretical analysis section: the claim that relaxed Neural Collapse (within-class compactness plus inter-class separation) implies high class-wise Mahalanobis distance variance for ID points is not accompanied by any verification that the actual ResNet-18/50 feature layers on CIFAR-100 and ImageNet satisfy these conditions. Without measuring within-class covariance relative to mean separation at the layer used for Mahalanobis, the geometric argument remains unanchored to the experimental models and does not establish that the variance term follows from NC geometry rather than post-hoc tuning.

minor comments (1)

[Method] The exact algebraic form of the MahaVar score (how the variance term is normalized and combined with the Mahalanobis distance) should be stated explicitly in the main text rather than deferred to the appendix.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the single major comment below.

read point-by-point responses

Referee: [Theoretical analysis] Theoretical analysis section: the claim that relaxed Neural Collapse (within-class compactness plus inter-class separation) implies high class-wise Mahalanobis distance variance for ID points is not accompanied by any verification that the actual ResNet-18/50 feature layers on CIFAR-100 and ImageNet satisfy these conditions. Without measuring within-class covariance relative to mean separation at the layer used for Mahalanobis, the geometric argument remains unanchored to the experimental models and does not establish that the variance term follows from NC geometry rather than post-hoc tuning.

Authors: We agree that direct verification of the relaxed Neural Collapse conditions on the specific feature representations would strengthen the link between theory and experiments. In the revision we will add quantitative measurements of within-class compactness (trace of class-conditional covariance) and inter-class separation (distances between class means) at the penultimate layer for the ResNet-18/50 models on both CIFAR-100 and ImageNet. These measurements will be reported alongside the existing results to confirm that the observed high variance for ID samples is consistent with the geometric assumptions rather than an empirical adjustment. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation rests on external NC assumptions and direct computation

full rationale

The paper grounds its key claim—that ID samples exhibit high class-wise Mahalanobis distance variance under relaxed Neural Collapse assumptions on within-class compactness and inter-class separation—directly in prior NC literature rather than self-referential definitions or fits. The MahaVar score is formed by augmenting the standard Mahalanobis distance with the variance of the same class-wise distances, without any parameter estimation that reduces the output to the input by construction. No self-citation load-bearing steps, uniqueness theorems imported from the authors, or ansatz smuggling appear in the derivation chain. The empirical observation and theoretical analysis are independent of the current paper's fitted values, rendering the result self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard Mahalanobis parameter estimation from ID data and on relaxed Neural Collapse assumptions; no new entities are postulated.

free parameters (1)

Class means and covariance matrix
Estimated from in-distribution training data exactly as in the baseline Mahalanobis detector.

axioms (1)

domain assumption Relaxed Neural Collapse assumptions on within-class compactness and inter-class separation
Invoked to prove that ID samples must exhibit high class-wise Mahalanobis distance variance.

pith-pipeline@v0.9.0 · 5526 in / 1332 out tokens · 67635 ms · 2026-05-15T02:14:03.966574+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

under relaxed Neural Collapse assumptions on within-class compactness and inter-class separation, ID samples are shown to structurally exhibit high class-wise distance variance
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

class-wise Mahalanobis distances exhibit a pronounced sharp minimum structure... resulting in high variance across classes

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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