arxiv: 2605.01063 · v1 · submitted 2026-05-01 · 💻 cs.LG · cs.CV

Recognition: unknown

GEODE: Angle-Adaptive OOD Detection with Universal Scorer Compatibility

Bruno Abrahao

Authors on Pith no claims yet

Pith reviewed 2026-05-09 19:22 UTC · model grok-4.3

classification 💻 cs.LG cs.CV

keywords out-of-distribution detectionoutlier exposureneural collapseangle-adaptive norm lossfeature geometryscorer compatibilityboundary calibration

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The pith

GEODE replicates outlier exposure's boundary calibration using an angle-adaptive norm loss to achieve consistent out-of-distribution detection performance across all standard scorers.

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

The paper shows that outlier exposure succeeds mainly by positioning its features at the geometric boundary between in-distribution classes, specifically in the boundary-adjacent quartile. GEODE achieves the same calibration effect without any auxiliary data by using an angle-adaptive norm loss that scales each sample's target according to its angle to the nearest class mean. This approach preserves the feature geometry needed by distance-based scorers while also benefiting probability-based ones. Readers should care because deployment-time OOD types are unknown, making scorer-independent performance essential for reliable detection.

Core claim

Outlier exposure works by boundary calibration rather than broad OOD coverage, with its gain coming from features in the boundary-adjacent quartile. GEODE replicates this synthetically with an angle-adaptive norm loss whose targets scale per-sample with cosine similarity to the nearest class mean. Four theorems based on neural collapse justify the design, leading to strong performance across all scorers.

What carries the argument

Angle-adaptive norm loss that sets targets scaling with cosine similarity to the nearest class mean to replicate boundary-adjacent quartile effects.

If this is right

GEODE delivers consistent AUROC improvements across all seven standard scorers on CIFAR-10 without catastrophic drops on any one.
It outperforms standard cross-entropy training when trained for the same number of epochs.
When combined with outlier exposure, it reaches top results on both MSP and KNN scorers.
The gains extend to CIFAR-100 and larger models like WRN-28-10.
It avoids the damage to distance-based scorers caused by methods that push OOD into null spaces.

Where Pith is reading between the lines

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

The geometric explanation suggests that any OOD method should be evaluated on its effect on the full range of scorers rather than just one or two.
Angle adaptation could be applied to other loss functions to improve compatibility in multi-scorer environments.
If the neural collapse theorems hold, similar adaptive mechanisms might help in related tasks like domain adaptation.
Future work might test whether the method scales to high-resolution images where feature geometry differs.

Load-bearing premise

The angle-adaptive norm loss exactly replicates the boundary-adjacent quartile effect of real OE data without introducing new distortions to feature geometry that could harm certain scorers or datasets.

What would settle it

Observing that GEODE underperforms vanilla training on a particular scorer or that the synthetic features do not match the geometric positions of real near-OOD samples would disprove the replication claim.

Figures

Figures reproduced from arXiv: 2605.01063 by Bruno Abrahao.

**Figure 1.** Figure 1: (a) Distance of each candidate source from real near-OOD geometry (ResNet-18, CIFAR10), measured as Euclidean distance in the (∥h∥/rid, maxc cos) plane. OE alone matches (distance 0.006; the next closest source is 13× further). Including the projection ratio and perpendicular norm columns of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: Theorem 4 in pictures. (a) For each method, the fraction of OOD feature energy that lies in the classifier subspace col(W⊤) (green; productive for both logit and feature scorers) versus the null space nul(W) (gray; invisible to logit scorers, but still contributes norm). PFS pushes 92.7% of OOD energy into nul(W); GEODE preserves 93.0% in col(W ⊤). (b) Resulting nearOOD KNN AUROC. PFS collapses to 14.38 A… view at source ↗

**Figure 3.** Figure 3: Feature norm distributions for ID (CIFAR-10, blue) and near-OOD (CIFAR-100, red) [PITH_FULL_IMAGE:figures/full_fig_p029_3.png] view at source ↗

**Figure 4.** Figure 4: Training dynamics on CIFAR-10 (ResNet-18, 200 epochs) for the GEODE [PITH_FULL_IMAGE:figures/full_fig_p030_4.png] view at source ↗

read the original abstract

Outlier Exposure (OE) is among the strongest training-based OOD detectors on standard benchmarks but exhibits scorer-dependent tradeoffs (e.g., strong on MSP, weak on KNN) and requires curated auxiliary data. We show why OE works: its features sit at the same geometric locus as real near-OOD data, with the boundary-adjacent quartile driving nearly all of OE's gain. OE is boundary calibration, not OOD coverage. GEODE (GEOmetry-preserving DEtection) replicates this calibration synthetically through an angle-adaptive norm loss in which targets scale per-sample with cosine similarity to the nearest class mean, preserving feature geometry where boundary structure matters. Four theorems grounded in neural collapse justify the design. GEODE works across all seven standard scorers on CIFAR-10 (near-OOD AUROC 89.0-92.3, far-OOD reaching 93.05; no catastrophic failure on any scorer). Since the OOD regime is unknown at deployment, this is the test that matters. GEODE outperforms vanilla CE at matched epoch counts. Combined with OE, GEODE reaches 95.0 MSP / 94.8 KNN on CIFAR-10 and beats OE on every scorer on CIFAR-100. The gains hold on WRN-28-10 (+4.5 Energy, 3 seeds). Unlike methods that push OOD into the classifier null space (e.g., PFS, 14.38 KNN AUROC, worse than random), GEODE's adaptive target preserves the geometry that distance-based scorers depend on.

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

GEODE gives a workable angle-adaptive loss that delivers consistent OOD AUROC across seven scorers on CIFAR without auxiliary data, but the claim that it exactly copies OE's boundary quartile geometry still needs the theorem derivations and direct geometric checks to hold.

read the letter

The main point is that this paper introduces GEODE, a norm loss where the target for each sample scales with its cosine similarity to the nearest class mean. The authors argue this synthetically places features at the same boundary-adjacent locus that real outlier exposure data occupies, which they trace to neural collapse results. They report that this produces AUROC values of 89.0-92.3 on near-OOD and up to 93.05 on far-OOD across all seven standard scorers on CIFAR-10, with no scorer collapsing, and further gains when combined with OE on CIFAR-100. It also beats plain cross-entropy at matched training length and holds on WRN-28-10 with three seeds. The contrast with PFS, which tanks KNN to 14 AUROC by pushing mass into the null space, is a clear practical win for preserving distance-based scoring. That consistency matters when the deployment scorer is not fixed in advance. The geometric framing of OE as boundary calibration rather than broad coverage is a useful reframing of existing results. The four theorems are presented as justification for the per-sample scaling rule. On the downside, the abstract states the theorems and the quartile replication claim without showing derivations or error bars, and there is no direct comparison of quartile-masked feature distributions between GEODE and actual OE. If the adaptive scaling compresses norms or shifts angles differently than real OE, distance scorers could still see uneven effects on other datasets even if MSP stays strong. The work stays within the OOD subfield and relies on standard benchmarks. It is aimed at researchers who need scorer-robust training methods for safety or deployment settings. The empirical consistency and the attempt to derive the loss from geometry rather than post-hoc tuning make it worth a referee's time, though the reviewers will have to verify the proofs and run geometric ablations themselves. I would send it to review.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes GEODE, a geometry-preserving OOD detection method using an angle-adaptive norm loss (targets scaled per-sample by cosine similarity to the nearest class mean) to synthetically replicate the boundary-adjacent quartile calibration effect of Outlier Exposure (OE) without auxiliary data. Grounded in four neural collapse theorems, it claims universal compatibility with seven standard scorers, reporting near-OOD AUROC 89.0-92.3 and far-OOD up to 93.05 on CIFAR-10 (no catastrophic failures), gains over vanilla CE at matched epochs, and further improvements when combined with OE on CIFAR-10/100 and WRN-28-10.

Significance. If the geometric replication claim holds, this would be a meaningful contribution to OOD detection by addressing OE's scorer-dependent tradeoffs and auxiliary-data requirement while preserving feature geometry needed for distance-based methods (KNN, Energy). The explicit grounding in neural collapse results and the multi-scorer, multi-dataset empirical evaluation (including 3-seed WRN results) are strengths; the work offers a practical alternative to methods that distort geometry (e.g., PFS).

major comments (3)

[Theoretical Analysis] Theoretical Analysis (theorems section): The four neural collapse theorems are load-bearing for justifying the angle-adaptive scaling, yet the manuscript provides no explicit derivation showing how per-sample cosine-based target adjustment preserves inter-class angles and intra-class norm distributions equivalently to real OE data; without this, the claim that distance-based scorers remain undistorted is unsupported.
[Experimental Evaluation] Experimental Evaluation (CIFAR-10 results): The universal scorer compatibility claim (89.0-92.3 near-OOD AUROC across all seven scorers) rests on the assumption that the adaptive loss replicates OE's boundary-adjacent quartile locus without new distortions, but no direct geometric verification (e.g., quartile-masked AUROC deltas, norm histograms, or inter-class angle comparisons between GEODE and OE features) is reported; this is required to rule out scorer-specific effects.
[Results tables] Results tables (CIFAR-10/100): While AUROCs and the +4.5 Energy gain on WRN-28-10 (3 seeds) are given, the absence of full error bars, quartile-effect ablations, and matched-epoch controls for all baselines leaves the 'no catastrophic failure' and 'outperforms vanilla CE' claims difficult to assess for robustness.

minor comments (2)

[Abstract] Abstract: The seven standard scorers should be named explicitly rather than referenced generically; the GEODE acronym expansion appears late.
[Method] Notation and loss definition: The angle-adaptive target scaling would benefit from an equation number and a short pseudocode block for reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback on our manuscript. We address each major comment below with clarifications and indicate the revisions we will make to strengthen the presentation of the theoretical and empirical claims.

read point-by-point responses

Referee: [Theoretical Analysis] Theoretical Analysis (theorems section): The four neural collapse theorems are load-bearing for justifying the angle-adaptive scaling, yet the manuscript provides no explicit derivation showing how per-sample cosine-based target adjustment preserves inter-class angles and intra-class norm distributions equivalently to real OE data; without this, the claim that distance-based scorers remain undistorted is unsupported.

Authors: We agree that an explicit derivation connecting the per-sample cosine scaling to the preservation of inter-class angles and intra-class norm distributions (as achieved by real OE) would make the link to neural collapse more direct. The four theorems establish the NC regime conditions under which geometry is preserved, and the adaptive loss is constructed to target the boundary-adjacent quartile locus without pushing features into the null space. To address the gap, we will add a dedicated derivation subsection in the revised theorems section that step-by-step shows how the cosine-based target adjustment maintains the required angle and norm statistics equivalently to OE, thereby supporting undistorted performance for distance-based scorers. revision: yes
Referee: [Experimental Evaluation] Experimental Evaluation (CIFAR-10 results): The universal scorer compatibility claim (89.0-92.3 near-OOD AUROC across all seven scorers) rests on the assumption that the adaptive loss replicates OE's boundary-adjacent quartile locus without new distortions, but no direct geometric verification (e.g., quartile-masked AUROC deltas, norm histograms, or inter-class angle comparisons between GEODE and OE features) is reported; this is required to rule out scorer-specific effects.

Authors: The consistent AUROC range across all seven scorers provides indirect support for the absence of new distortions, but we acknowledge that direct geometric verification would more rigorously confirm replication of OE's quartile locus. In the revision we will add norm histograms, inter-class angle distributions, and quartile-masked AUROC delta comparisons between GEODE, OE, and vanilla CE features on CIFAR-10. These analyses will explicitly demonstrate that the adaptive loss replicates the boundary calibration geometry without introducing scorer-specific artifacts. revision: yes
Referee: [Results tables] Results tables (CIFAR-10/100): While AUROCs and the +4.5 Energy gain on WRN-28-10 (3 seeds) are given, the absence of full error bars, quartile-effect ablations, and matched-epoch controls for all baselines leaves the 'no catastrophic failure' and 'outperforms vanilla CE' claims difficult to assess for robustness.

Authors: The WRN-28-10 results already report 3-seed averages for the +4.5 Energy gain, but we agree that fuller statistical reporting and controls would improve assessment of robustness. We will revise the results tables and experimental section to include standard-deviation error bars for all reported AUROCs, add targeted quartile-effect ablations, and explicitly present matched-epoch comparisons against all baselines. This will better substantiate the 'no catastrophic failure' and 'outperforms vanilla CE' statements. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation grounded in external neural collapse analysis and empirical validation

full rationale

The paper derives the angle-adaptive norm loss from geometric analysis of OE features (boundary-adjacent quartile locus) and four theorems grounded in neural collapse, which are presented as independent prior results rather than self-referential. The universal scorer compatibility claim (AUROC ranges across seven scorers) is supported by direct experiments on CIFAR-10/CIFAR-100, not by any equation that forces the outcome from the loss definition itself. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided derivation chain; the central result remains falsifiable against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the central claim rests on four unspecified theorems from neural collapse theory and the geometric equivalence between OE features and near-OOD data; no explicit free parameters or invented entities are named.

axioms (1)

domain assumption Neural collapse organizes features such that boundary-adjacent quartiles dominate OE gains
Invoked to justify why synthetic angle-adaptive targets replicate real OE calibration

pith-pipeline@v0.9.0 · 5581 in / 1316 out tokens · 27150 ms · 2026-05-09T19:22:34.204383+00:00 · methodology

discussion (0)

Reference graph

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