Dissecting Mahalanobis: How Feature Geometry and Normalization Shape OOD Detection

## Abstract

Abstract Out-of-distribution (OOD) detection is critical for the reliable deployment of deep learning models. hile Mahalanobis distance methods are widely used, the impact of representation geometry and normalization on their performance is not fully understood, which may limit their downstream application. To address this gap, we conducted a comprehensive empirical study across diverse image foundation models, datasets, and distance normalization schemes. First, our analysis shows that Mahalanobis-based methods aren’t universally reliable. Second, we define the ideal geometry for data representations and demonstrate that spectral and intrinsic-dimensionality metrics can accurately predict a model’s OOD performance. Finally, we analyze how normalization impacts OOD performance. Building upon these studies, we propose radially scaled ℓ 2 \ell_{2} normalization, a method that generalizes the standard ℓ 2 \ell_{2} normalization recently applied to Mahalanobis-based OOD detection. Our approach introduces a tunable parameter to directly control the radial geometry of the feature space, systematically contracting or expanding representations to significantly improve OOD detection performance. By bridging the gap between representation geometry, normalization, and OOD performance, our findings offer new insights into the design of more effective and reliable deep learning models.

## 1 Introduction

Out-of-distribution (OOD) detection is foundational for building reliable, open-world vision systems, yet consistent evaluation at scale—especially with modern foundation models—remains challenging and essential for practice. Mahalanobis-based detectors (Lee et al., 2018) are surprisingly simple yet powerful baselines that often achieve state-of-the-art performance (Mueller & Hein, 2025; 2024). At its core, this approach models the feature distribution of in-distribution data—typically by fitting class-conditional multivariate Gaussians—and flags an input as OOD if its feature representation is far from all class centroids. While effective, it is not fully understood why this simple metric works so well or how the complex geometry of high-dimensional representations contributes to its success. This paper systematically investigates this question, revealing that representation geometry and feature normalization are the primary drivers of Mahalanobis-based OOD detection performance and providing a practical method to optimize them.

Figure: Figure 1: Effect of $\beta$-scaled $\ell_{2}$ normalization effect 2D feature geometry and Mahalanobis decision surfaces. Optimal $\beta$ improve OOD detection performance. Gray arrows indicate the mapping from the original to the transformed space. Increasing $\beta$ contracts norms and tightens clusters, yielding smoother, more localized decision regions; negative $\beta$ spreads points and broadens regions. Choosing an appropriate $\beta$ improves class separation and downstream OOD detection.
Refer to caption: figures/decision_boundary_col.png

We begin our work by benchmarking a diverse set of self-supervised models, revealing significant variance in the inherent OOD detection capabilities of their representations. We then demonstrate that this variance is not random, but correlates strongly with measurable geometric properties of the in-distribution feature space, such as its intrinsic dimensionality and spectral structure. Unlike prior works Ren et al. (2021); Mueller & Hein (2025) focused on refining the distance metric itself, we introduce a direct control knob for the geometry of the representation: a $\beta$-scaled $\ell_{2}$ normalization that radially contracts or expands the feature space. This allows us to reshape the representation to better suit the detector. Finally, we demonstrate that a simple regression model can predict this optimal $\beta$, without access to target OOD data, achieving nearly the same performance as an oracle with access to OOD samples.
Our main contributions are:

- 1.
A comparative study of several Mahalanobis-based OOD detectors across diverse models, including a per-dimension analysis of its variants.
- 2.
An analysis revealing that OOD performance is strongly predicted by the in-distribution geometry of features, such as spectral decay and intrinsic rank.
- 3.
The introduction of $\beta$-scaled $\ell_{2}$ normalization and a method to predict the optimal $\beta$ using only in-distribution data.

## 2 Related Work

Out-of-distribution (OOD) detection is essential for ensuring the reliability of machine learning systems in real-world deployment (Fort et al., 2021). Its goal is to identify whether inputs stem from the training distribution, thus preventing overconfident predictions on unexpected data (Yang et al., 2024). Post-hoc, training-free methods are particularly effective, as they combine efficiency with robustness without altering the model (Xu et al., 2023). Among OOD detection methods, Mahalanobis distance has become a cornerstone (Lee et al., 2018), with several refinements improving its robustness and performance. The standard Mahalanobis distance (MD) uses class-conditional covariance estimates to measure the distance of a sample from each class mean. In contrast, the Relative Mahalanobis distance (RMD) (Ren et al., 2021) compares each class-specific distance to a single global Gaussian fitted to all in-distribution (ID) data, effectively normalizing class distances against a global reference. Mahalanobis++ (Mueller & Hein, 2025) further improves performance by L2-normalizing features, making them adhere more closely to the Gaussian assumptions underlying the Mahalanobis distance.
However, our sudy reveals broader insight on the influence of normalization while computing Mahalanobis distance, particularly in the context of vision models.

Vision OOD detection has shifted toward leveraging large-scale pretraining and contrastive objectives, where vision transformers Dosovitskiy et al. (2021) and CLIP Radford et al. (2021) show strong near-OOD performance and benefit markedly from few-shot outlier exposure and even label-only supervision for outlier classes (Fort et al., 2021). However, full fine-tuning can distort pretrained representations and harm OOD generalization relative to linear probing, with similar cautions for vision–language models; recent work also explores training-time scaling and post-hoc enhancements, and revisits detector design in vision foundation models (Fort et al., 2021; Ming & Li, 2024; Xu et al., 2023; Zhao et al., 2024b). Evaluation rigor has improved through ImageNet-scale suites like NINCO that mitigate in-distribution leakage, while theory and diagnostics connect feature separability to OOD error and delineate when OOD detection is learnable (Bitterwolf et al., 2023; XIE et al., 2023).

Representation geometry and normalization have attracted increased attention for their role in OOD generalization. Analyses of contrastive learning and normalization approaches (Le-Gia & Ahn, 2023; Tan et al., 2025) show that geometric priors, such as hyperspherical projection or $\ell_{2}$ normalization, can yield more robust representation spaces. Studies like (Zhao et al., 2024a) and (XIE et al., 2023) link improved feature separability and lower intrinsic dimensionality to higher OOD detection performance.

## 3 Comparative Study of Self-Supervised Models

### 3.1 Background

Let $z^{\prime}=f(x^{\prime})\in\mathbb{R}^{d}$ be the feature representation of a test image input $x^{\prime}$, and let $\{\mathcal{N}(\mu_{k},\Sigma)\}_{k=1}^{K}$ denote the $K$ class–conditional Gaussian distributions fitted to in–distribution (ID) training data.

Mahalanobis distance (MD) (Lee et al., 2018) measures the squared distance of $z^{\prime}$ from
the mean of each class:
$\operatorname{MD}_{k}(z^{\prime})\;=\;(z^{\prime}-\mu_{k})^{\top}\Sigma^{-1}(z^{\prime}-\mu_{k}).$
A confidence score is obtained as the negative minimum distance,
$\mathcal{C}(x^{\prime})=-\min_{k}\operatorname{MD}_{k}(z^{\prime}).$

Relative Mahalanobis distance (RMD) (Ren et al., 2021) compares the class-specific distance to a single global Gaussian fitted to all ID data, the marginal Mahalanobis distance (MMD):
$\operatorname{MD}_{0}(z^{\prime})=(z^{\prime}-\mu_{0})^{\top}\Sigma_{0}^{-1}(z^{\prime}-\mu_{0}),$
and defines the RMD score as:
$\operatorname{RMD}_{k}(z^{\prime})=\operatorname{MD}_{k}(z^{\prime})-\operatorname{MD}_{0}(z^{\prime}).$
This effectively normalizes class distances against a global reference.

#### Eigenvalue Decomposition.

Let $\Sigma=U\Lambda U^{\top}$ be the eigendecomposition of the shared
covariance matrix, where
$\Lambda=\operatorname{diag}(\lambda_{1},\ldots,\lambda_{d})$ and $U$ is
orthonormal. As shown in (Mueller & Hein, 2025), in this basis the regular Mahalanobis distance decomposes as

$$ $\operatorname{MD}_{k}(z^{\prime})=\sum_{i=1}^{d}\frac{\bigl[u_{i}^{\top}(z^{\prime}-\mu_{k})\bigr]^{2}}{\lambda_{i}},$ (1) $$

revealing the contribution of each principal component.
The eigenvalues $\{\lambda_{i}\}$ quantify the spread of in-distribution features
along each direction; small $\lambda_{i}$ correspond to directions of small
variance and thus high discriminative power for OOD detection.

### 3.2 Cross-model OOD Detection Performance

Figure: Figure 2: OOD detection performance across model families on the NINCO. RMD consistently outperforms the standard MD, especially for models pretrained but not fine-tuned on ImageNet.
Refer to caption: x1.png

Understanding how different representation learning strategies affect OOD detection is a critical first step in our study.
Mahalanobis-based detectors are widely used, but their sensitivity to model architecture, pretraining data, and fine-tuning is not well characterized. We therefore begin with a broad, model-agnostic comparison to answer a simple question:
*Which modern self-supervised or pretrained vision models produce representations that naturally lend themselves to Mahalanobis-style OOD detection?*

To this end, we gathered publicly available checkpoints from timm (Wightman, 2019) and huggingface-transformers (Wolf et al., 2020), spanning a range of architectures, sizes, and pretraining objectives. Following the OpenOOD protocol (Yang et al., 2022), we evaluate on five standard benchmarks—NINCO (Bitterwolf et al., 2023), iNaturalist (Van Horn et al., 2018), SSB-Hard (Bitterwolf et al., 2023), OpenImages-O (Krasin et al., 2017), and Textures (Cimpoi et al., 2014).
Performance is reported as the false positive rate at 95% true positive rate (FPR@95 or FPR). Additional implementation details and a full model list appear in the Appendix.

#### Key Insights (Figure 2 )

*(1) RMD consistently improves performance* over standard Mahalanobis distance, with the largest gains in models pretrained—but not fine-tuned—on ImageNet.
For example, RMD markedly boosts the OOD detection of EVA02-In21k and ViT-In21k, matching or surpassing their fine-tuned counterparts. This weakens the usual correlation between in-distribution accuracy and FPR and yields more uniform score distributions. *(2) Classification accuracy is not a reliable proxy for OOD performance*. Substantial accuracy gaps (often $>10\%$) do not necessarily translate into improved detection, though we observe a mild correlation along the fine-tuning sequence In1k $\rightarrow$ In22k-In1k $\rightarrow$ large In22k-In1k models (complete results in Appendix [D.1](https://arxiv.org/html/2510.15202v2#A4.SS1)).

### 3.3 Mahalanobis Variants and Per-Dimension Analysis

Having established cross-model trends, we next ask:
*Which aspects of the Mahalanobis representation space actually drive OOD discrimination?*
Beyond aggregate scores, the structure of individual feature dimensions may reveal why certain models excel while others falter.
We therefore conduct a detailed *per-dimension* investigation using three Mahalanobis variants—regular, marginal, and relative.
Using the decomposition in Eq. equation [1](https://arxiv.org/html/2510.15202v2#S3.E1), we define the
*OOD separation* of the $i$-th eigenvector direction as the difference
between its mean contribution for out-of-distribution samples
and for in-distribution samples:
$S_{i}=\mathbb{E}_{x^{\prime}\sim\mathcal{D}_{\text{OOD}}}\!\left[\frac{\bigl[u_{i}^{\top}(z^{\prime}-\mu_{k})\bigr]^{2}}{\lambda_{i}}\right]-\mathbb{E}_{x\sim\mathcal{D}_{\text{ID}}}\!\left[\frac{\bigl[u_{i}^{\top}(z-\mu_{k})\bigr]^{2}}{\lambda_{i}}\right].$
Here $S_{i}$ quantifies how strongly dimension $i$ separates OOD from
in-distribution data; positive values indicate greater OOD spread along that eigen-direction. The eigenvalues $\{\lambda_{i}\}$ are sorted in descending order ($\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{d}$), so dimension $i$ on
subsequent plots corresponds to the $i$-th largest eigenvalue.

Figure [3](https://arxiv.org/html/2510.15202v2#S3.F3) presents two complementary analyses. The top row reports OOD separation, exposing which latent directions contribute most to detection.
The bottom row shows a dimension-ablation study: we incrementally compute FPR using the first $K$ principal components (forward) or start from the least-variant dimensions (backward).

Figure: Figure 3: Dimension-wise analysis of OOD separation (top) and FPR under progressive dimension ablation (bottom). Large embedding-space separation does not necessarily guarantee superior detection.
Refer to caption: x2.png

#### Key Insights

*(1) Large OOD separation $S_{i}$ do not always yield lower FPR:* for example, BEiTV2 FT In1k exhibits stronger separation across all three distance metrics yet performs on par with—or worse than—BEiTV2 FT In21k. *(2) The number of dimensions required for optimal detection varies widely:* some models saturate quickly, whereas others need nearly the full spectrum. Backward ablation reveals that ViT In21k achieves its best FPR using only the second half of the spectrum, indicating that directions of *smaller explained variance* can be more discriminative for OOD-ness.
*(3) MMD results show that certain models rely heavily on class-discriminative features*, while others—such as CLIP—spread OOD samples far from ID data regardless of their classes, consistently achieving low MMD scores.

## 4 Geometry of Representations

In the previous section we showed that no single OOD method yield consistent performance and behavior across multiple models. In fact, different SSL models and pretraining regimes produce representations with distinct geometric properties, indicating that OOD performance depends on the intrinsic structure of the representation space. To understand these effects, we analyze the internal geometry of model representations, seeking to answer: *What internal characteristics of a model’s feature space predict strong OOD detection, and how do pretraining and fine-tuning shape these characteristics?*

### 4.1 Spectral analysis

To understand how the intrinsic geometry of representations affects OOD performance, we begin by examining the spectral properties of three key matrices: the feature covariance $C$, the within-class scatter $S_{w}$, and the between-class scatter $S_{b}$. These matrices capture complementary aspects of the feature space: $C$ reflects overall variance, $S_{w}$ measures intra-class dispersion, and $S_{b}$ quantifies inter-class separation (more details in Appendix [A](https://arxiv.org/html/2510.15202v2#A1)).
Our first analysis focuses on the eigenvalue spectra of these matrices. The magnitude and decay of eigenvalues reveal how variance is distributed across dimensions, providing insight into the richness and anisotropy of the feature space. For instance, a steep decay in $S_{w}$ eigenvalues indicates that intra-class variability is concentrated along a few directions, resulting in tight clusters, whereas a slower decay suggests more diffuse intra-class variation. Similarly, large eigenvalues in $S_{b}$ correspond to well-separated class means, signaling strong discriminability.

Figure: Figure 4: Spectral ratios across models. Higher ratios indicate richer within-class variation and more expressive feature spaces. Fine-tuning tends to increase $S_{b}$ while preserving $S_{w}$.
Refer to caption: x3.png

#### Spectral ratios

To systematically compare models, we compute ratios between eigenvalues of $S_{b}$, $S_{w}$, and $C$. These ratios serve as compact summaries of representation geometry. Higher $S_{b}/S_{w}$ ratios indicate representations with greater between-class separation relative to intra-class spread, which generally favors OOD detection, while lower ratios may signal overlapping clusters or limited discriminative power. A higher $C$ ratio indicates that variance is distributed along multiple directions, reflecting a richer and more expressive representation that can better accommodate novel OOD inputs without major distortion. As illustrated in Figure [4](https://arxiv.org/html/2510.15202v2#S4.F4), models pretrained on large, diverse datasets (e.g., In21k) exhibit larger $C$ and $S_{w}$ ratios, capturing richer intra-class variations and producing more expressive feature spaces. Fine-tuning tends to increase $S_{b}$ ratios while preserving $S_{w}$, enhancing class separability without sacrificing cluster compactness. Models trained on smaller datasets exhibit smaller ratios, reflecting less expressive representations with weaker discriminability.

#### Eigenvalue shifts

Beyond static spectra, we are interested in how stable the representation geometry is under distributional shifts. To capture this, we define a *spectral shift metric*, which measures the relative change in eigenvalues from the training set to validation or OOD data (see Appendix [A](https://arxiv.org/html/2510.15202v2#A1)). A small shift indicates that the representation preserves its structure across data splits, signaling robustness. Large positive shifts reveal that features are spreading along new directions, while large negative shifts indicate compression. Figure [4](https://arxiv.org/html/2510.15202v2#S4.F4) shows that OOD samples induce larger spectral shifts in models trained on small datasets, reflecting lower generalization and brittle feature structures. Large-scale pretrained models show smaller shifts, indicating more stable, robust representations under distributional change. Fine-tuning generally maintains small shifts while increasing $S_{b}$, improving class separation without compromising intra-class compactness.

### 4.2 Geometric trade-offs

To systematically identify what makes a representation “good” for OOD detection, we correlate spectral and manifold-based metrics with detection performance. The geometry of the representations is assessed with two complementary families of measures: manifold-geometry metrics—including *Intrinsic Dimensionality (ID)* and *Local Intrinsic Dimensionality (LID)* (Ma et al., 2018)—and eigenvalue-based metrics (e.g., *Entropy*, *Slope*, *Fisher Ratio*) computed on the covariance matrix $C$ and the Fisher scatter matrices $S_{w}$ and $S_{b}$. A complete description of all metrics is provided in Appendix [B](https://arxiv.org/html/2510.15202v2#A2). Correlation analysis, visualized in Figure [5](https://arxiv.org/html/2510.15202v2#S4.F5), highlights several metrics that correlate strongly with OOD performance.

Figure: Figure 5: Spearman correlations between representation metrics and OOD performance across Mahalanobis variants. The three Mahalanobis-based detectors exploit different geometric cues, which explains their distinct correlation patterns. These correlations also hold for Pearson correlations (see Appendix [F](https://arxiv.org/html/2510.15202v2#A6)).
Refer to caption: x4.png

#### Key insights

*Mahalanobis distance* shows moderate correlation with individual metrics; its sensitivity to the global eigen-spectrum can make it less reliable for ambiguous OOD regions.
*Relative Mahalanobis* emphasizes how well features fit their class cluster by normalizing out global variance. It correlates strongly with $S_{w}$ metrics, such as entropy and eigenvalue decay, reflecting the importance of compact, well-separated clusters.
*Marginal Mahalanobis* ignores class structure and correlates primarily with global metrics ($C$ and $S_{b}$), indicating that its success depends on overall manifold shape rather than per-class separation.

### 4.3 Ideal geometry

OOD detection performance depends on both the magnitude of separation between ID and OOD features and the internal structure of the representation. Ideally, an effective representation exhibits a balance between local manifold complexity and intra-class compactness: low-dimensional manifolds require tighter clusters to separate OOD data, while high-dimensional manifolds allow looser clusters, as extra directions naturally push OOD samples away. This “ideal geometry” reflects a compensatory relationship between these two principles. However, as we observed in Sections [3.3](https://arxiv.org/html/2510.15202v2#S3.SS3) and [4.2](https://arxiv.org/html/2510.15202v2#S4.SS2), single metrics fail to satisfy both principles simultaneously. Models with larger raw feature separation can perform worse than models with smaller separation, indicating that additional geometric properties beyond simple separation are critical for reliable detection. While individual metrics provide insight, they capture only a single aspect of the geometry.

Figure: Figure 6: Correlation of the product of LID and $S_{w}$ slope with OOD detection performance. Minimizing the absolute value indicates an optimal balance between manifold richness and cluster compactness.
Refer to caption: figures/metric_combination_correlation.png

To address this limitation, we consider combinations of metrics across different geometric aspects. In particular, the product of LID and the slope of $S_{w}$ strongly predicts OOD performance, which is shown in Figure [6](https://arxiv.org/html/2510.15202v2#S4.F6). For the standard MD, this combined metric achieves a Spearman correlation of $\rho_{s}=0.85$, and it remains highly predictive for RMD and MMD ($\rho_{s}=0.72$ and $0.74$, respectively).

In practice, when the feature manifold is locally simple (low *LID*), the model has fewer dimensions to separate ID and OOD data, requiring extremely tight and compact class clusters (a steep, negative *slope*). The detector thus relies on strict confinement of ID data to identify outliers. Conversely, when the manifold is locally rich and complex (high *LID*), the space itself helps isolate OOD samples, allowing less compact clusters (a shallower *slope*), as high dimensionality drives OOD separation. The product of LID and slope reaches its ideal when minimized in absolute magnitude, reflecting the optimal balance between local manifold richness and cluster compactness.

## 5 Normalization Effects

Our preceding analysis has established that the efficacy of Mahalanobis-based OOD detectors is intrinsically linked to the geometric structure of a model’s feature space. Yet, even when the geometry is favorable, standard Mahalanobis distance assumes Gaussian-distributed features with tied covariance, an assumption often violated in practice.
Empirical studies, such as Mahalanobis++ (Mueller & Hein, 2025), show that deep neural network features frequently exhibit heavy-tailed distributions and large variations in feature norms.
These deviations imply that ID features do not occupy a single, globally Euclidean space; instead, they lie on a collection of low-dimensional, non-Euclidean submanifolds. Consequently, the raw Mahalanobis metric can misestimate distances, undermining OOD detection.
These observations motivate explicit control over feature geometry: by adjusting feature magnitudes we can either better satisfy the Mahalanobis assumptions or deliberately reshape the space to improve OOD separation.

Table: Table 1: False-positive rate (FPR) across models (averaged over datasets) using different Mahalanobis variants: MD* uses the empirically optimal $\beta$, $\hat{\textbf{MD}}$ uses the regression-predicted $\hat{\beta}$, MD (standard) fixes $\beta=0$, and MD++ (Mahalanobis++) fixes $\beta=1$. The regression-guided detector generally outperforms fixed-$\beta$ settings.

### 5.1 Radially Scaled ℓ 2 \ell_{2} Normalization

To control the geometry of the representation space and mitigate sensitivity to raw feature norms, we introduce a tunable radial $\ell_{2}$ normalization that applies a radially symmetric map to each feature vector. For a feature $z\in\mathbb{R}^{d}\setminus\{0\}$, the transformation is defined as:

$$ $\phi_{\beta}(z)=\frac{z}{\|z\|^{\beta}},$ (2) $$

where the scalar parameter $\beta\in\mathbb{R}$ governs the amount of radial scaling. This mapping transforms the feature space by altering the radial distances from the origin while preserving the direction of each vector. The new radius of a feature is given by $\|\phi_{\beta}(z)\|=\|z\|^{1-\beta}$. This transformation induces a new Riemannian metric on the feature space (see Appendix [C](https://arxiv.org/html/2510.15202v2#A3) for details).

Figure: Figure 7: Distribution of empirically optimal $\beta$ for MD and RMD detectors across OOD datasets. High variability indicates model- and dataset-specific tuning is needed.
Refer to caption: x5.png

Intuitively, adjusting $\beta$ allows us to compress or expand the feature space radially, controlling how tightly features cluster near the origin or how much “empty space” surrounds them:

- •
$\beta<0$: Radii are expanded, aggressively emphasizing differences between norms.
- •
$\beta=0$: Radii are unchanged, corresponding to the standard Mahalanobis distance.
- •
$0<\beta<1$: Radii are compressed, while preserving relative ordering.
- •
$\beta=1$: All points are projected onto the unit sphere, equivalent to the MD++ normalization.
- •
$\beta>1$: Features farther from the origin are pulled inward more strongly than those closer, creating strong radial contraction.

By tuning $\beta$, we obtain a flexible mechanism to shape the feature geometry, providing a simple yet powerful knob for improving OOD separation while maintaining compact ID clusters.

### 5.2 Predicting the Optimal β \beta

Our experiments show that the optimal radial parameter $\beta$ is highly model- and dataset-dependent, reflecting the intrinsic geometry of the learned representations. Moderate positive values often align the feature distribution with the Gaussian, tied-covariance assumptions of the Mahalanobis detector, yet in some cases larger or even negative values yield stronger in/out-of-distribution separation. Consequently, a fixed choice of $\beta$ is rarely optimal. Figure [7](https://arxiv.org/html/2510.15202v2#S5.F7) shows the empirically optimal $\beta$ values (searched over $[-2,3]$ in $0.25$ steps) for MD and RMD detectors across different OOD datasets. The wide spread of optimal values underscores that a one-size-fits-all approach is ineffective.

#### Regression Framework.

To eliminate the need for tuning on the target OOD dataset, we train a regression model that predicts $\beta$ using in-distribution geometry metrics, while allowing it to learn from other OOD datasets. We adopt a Leave-One-Dataset-Out scheme: for each target OOD dataset, the regression model is trained on all other OOD datasets and their corresponding ID features, ensuring that the target OOD samples are never seen during training. This setup is conceptually similar to outlier exposure (Hendrycks et al., 2018), where access to auxiliary OOD data helps guide the detector, but crucially, here the model generalizes to completely unseen OOD distributions.
Candidate predictors include spectral properties of the feature covariance, intrinsic dimensionality estimates, and other representation-geometry statistics; highly collinear features ($\rho>0.9$) are removed. The target variable is the $\beta$ value
minimizing the false-positive rate (FPR) for each model–dataset pair.
Let $\hat{\beta}$ denote the predicted value.

Figure: Figure 8: Predicted vs. optimal $\beta$ for MD under Leave-One-Dataset-Out validation. The diagonal indicates perfect prediction.
Refer to caption: x6.png

#### Results.

Table [1](https://arxiv.org/html/2510.15202v2#S5.T1) reports FPR across models and datasets.
The regression-predicted $\hat{\beta}$ consistently improves OOD detection compared to fixed baselines ($\beta=0$ for standard MD, $\beta=1$ for MD++), for both MD and RMD detectors.
For MD, the regression achieves a MAE of $0.72$ and $R^{2}=0.25$ in predicting the optimal $\beta$; for RMD, the MAE is $0.89$ with $R^{2}=0.47$. While the regression does not perfectly recover the empirically optimal $\beta$, it captures sufficient geometric information from in-distribution features to meaningfully improve detection.
Figure [8](https://arxiv.org/html/2510.15202v2#S5.F8) visualizes predicted vs. optimal $\beta$ for MD. Points close to the diagonal indicate that the regression captures key trends even for previously unseen OOD datasets. This demonstrates a practical path for tuning radial $\ell_{2}$ scaling without access to target OOD data, leveraging ID feature structure and prior knowledge from other auxiliary OOD datasets. We present the complete results and comparisons with baseline OOD detectors in Appendix [D.2](https://arxiv.org/html/2510.15202v2#A4.SS2).

## 6 Conclusion

In this work, we conducted a comprehensive empirical study across diverse image foundation models, datasets, and distance normalization schemes to understand how representation geometry shape Mahalanobis-based OOD detection performance. Our comparative analysis revealed that these detectors are not universally reliable, with significant variance in inherent OOD detection capabilities across different self-supervised models. We demonstrated that this variance correlates strongly with measurable geometric properties of the in-distribution feature space. In particular, the product of Local Intrinsic Dimensionality and within-class scatter slope achieves strong correlations with detection effectiveness. The proposed measure reflects the balance between local manifold complexity and cluster compactness, crucial for effective OOD.

Building on these geometric insights, we introduced radially scaled $\ell_{2}$ Normalization, a transformation that provides direct control over the feature space’s radial geometry. This allows to reshape feature distributions to better satisfy the assumptions of the Mahalanobis distance or to improve out-of-distribution (OOD) separation. We also developed a regression framework capable of predicting the optimal $\beta$ value using only in-distribution data, achieving performance comparable to an oracle. Future work should explore fully model-free methods for selecting the optimal $\beta$.

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## Appendix A Computation and Intuition for Covariance and Scatter Matrices

Our spectral analysis relies on three core matrices derived from the feature
embeddings $\{z_{i},y_{i}\}_{i=1}^{N}$ of a trained model: the overall feature covariance $C$, the within-class scatter $S_{w}$, and the between-class scatter $S_{b}$. All eigenvalues reported in the main text are sorted in descending order.

#### Feature Covariance.

The global covariance is

$$ $C=\frac{1}{N}\sum_{i=1}^{N}(z_{i}-\mu)(z_{i}-\mu)^{\top},\qquad\mu=\frac{1}{N}\sum_{i=1}^{N}z_{i}.$ (3) $$

This matrix captures the overall spread of representations in feature space. Large eigenvalues correspond to directions of high variance, indicating axes along which the model representation varies the most across all samples.

#### Within-Class Scatter.

For $K$ classes with means $\mu_{k}$ and $n_{k}$ samples each,

$$ $S_{w}=\frac{1}{N}\sum_{k=1}^{K}\;\sum_{i:y_{i}=k}(z_{i}-\mu_{k})(z_{i}-\mu_{k})^{\top}.$ (4) $$

$S_{w}$ measures the average dispersion of features around their class means, capturing intra-class variability. Intuitively, if $S_{w}$ has large eigenvalues, samples of a class are more spread out in feature space; small eigenvalues indicate tight, compact clusters. Notably, $S_{w}$ is *equivalent to the tied covariance* $\Sigma$ used in Mahalanobis distance (Eq. [1](https://arxiv.org/html/2510.15202v2#S3.E1)) when all classes share a common covariance estimate: $\Sigma=S_{w}$. Thus the Mahalanobis detector implicitly measures distances with respect
to the within-class scatter of the training distribution.

#### Between-Class Scatter.

The between-class scatter quantifies the variability of class means:

$$ $S_{b}=\frac{1}{N}\sum_{k=1}^{K}n_{k}(\mu_{k}-\mu)(\mu_{k}-\mu)^{\top}.$ (5) $$

This matrix captures inter-class separation: directions with large eigenvalues indicate axes along which class centroids are widely separated, while small eigenvalues correspond to directions where classes overlap. The spectrum of $S_{b}$ provides insight into the model’s ability to linearly discriminate between classes.

#### Spectral Shift Metric.

To study how representations change under distributional shifts, for each matrix $M\in\{C,S_{w},S_{b}\}$ we compute its eigenvalues
$\{\lambda_{i}^{\text{train}}\}$ on the training set
and $\{\lambda_{i}^{\text{eval}}\}$ on a validation or OOD set.
The relative eigenvalue shift is defined as

$$ $\Delta_{i}(M)=\frac{\lambda_{i}^{\text{eval}}-\lambda_{i}^{\text{train}}}{\lambda_{i}^{\text{train}}}.$ (6) $$

This spectrum of shifts highlights how the geometry of the representation
changes under distributional shift, providing a fine-grained indicator of
robustness or overfitting.

#### Intuition Behind the Shift Metric.

- •
Zero shift ($\Delta_{i}\approx 0$): The corresponding direction in feature space is stable across data splits.
- •
Positive shift ($\Delta_{i}>0$): The representation spreads out along this eigenvector in the new data, increasing variance.
- •
Negative shift ($\Delta_{i}<0$): The representation compresses along this eigenvector, reducing variance.
- •
Magnitude: Reflects the relative degree of expansion or contraction. For example, $\Delta_{i}=0.5$ indicates a 50% increase in variance, while $\Delta_{i}=-0.2$ indicates a 20% decrease.

#### Interpretation in Model Analysis.

- •
Small shifts across all eigenvectors: Robust and stable representations that generalize well.
- •
Large positive shifts: Features become more variable on new data, potentially indicating under-regularization or sensitivity to OOD inputs.
- •
Large negative shifts: Features compress on new data, potentially indicating overfitting.
- •
Consistent shift patterns: Systematic changes in representation geometry, revealing overfitting or robustness issues.

#### Types of Shifts.

- •
Validation covariance shift: Change in global covariance from training to validation data.
- •
OOD covariance shift: Change in global covariance from training to out-of-distribution data.
- •
Validation within-class shift: Change in within-class scatter from training to validation data.
- •
Validation between-class shift: Change in between-class scatter from training to validation data.

## Appendix B Detailed Description of Spectral and Manifold Metrics

Let $X\in\mathbb{R}^{N\times d}$ be the feature matrix and
$C=\tfrac{1}{N}(X-\bar{X})^{\top}(X-\bar{X})$ its covariance.
All metrics below operate on the sorted non-negative eigenvalues of $C$,
denoted $\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{d}$.

#### Intrinsic Dimensionality (ID).

Global estimate of the manifold dimension using
maximum-likelihood methods of Ma et al. (2018).

#### Local Intrinsic Dimensionality (LID).

For a point $z$,

$$ $\operatorname{LID}_{k}(z)=-\!\left[\frac{1}{k}\sum_{j=1}^{k}\log\frac{r_{j}(z)}{r_{k}(z)}\right]^{-1},$ $$

where $r_{j}$ is the $j$-th nearest-neighbor distance.
We report the dataset mean for $k\!\in\!\{10,25,50,100\}$.

#### Total Variance.

$$ $\text{Total Variance}=\sum_{i=1}^{d}\lambda_{i}.$ $$

#### Effective Rank.

Implements the code’s definition,

$$ $\text{Effective Rank}=\frac{\sum_{i}\lambda_{i}}{\lambda_{1}},$ $$

the ratio of total variance to the largest eigenvalue.

#### Participation Ratio.

$$ $\text{PR}=\frac{\bigl(\sum_{i}\lambda_{i}\bigr)^{2}}{\sum_{i}\lambda_{i}^{2}}.$ $$

#### Condition Number.

$$ $\kappa=\frac{\lambda_{1}}{\lambda_{d}},$ $$

where $\lambda_{d}$ is the smallest positive eigenvalue.

#### Spectral Gap.

Difference between the largest and *sixth* largest eigenvalue,

$$ $\text{Gap}=\lambda_{1}-\lambda_{6}.$ $$

#### Entropy.

Shannon entropy of the normalized spectrum,

$$ $\text{Entropy}=-\sum_{i}p_{i}\log p_{i},\qquad p_{i}=\frac{\lambda_{i}}{\sum_{j}\lambda_{j}}.$ $$

#### Average Log Decay Rate (Top-20).

Mean forward difference of the first 20 log-eigenvalues,

$$ $\frac{1}{19}\sum_{i=1}^{19}\bigl[\log\lambda_{i}-\log\lambda_{i+1}\bigr].$ $$

#### Slope.

Slope of the least-squares fit

$$ $\log\lambda_{i}=a+b\,i,$ $$

i.e., regression of $\log\lambda_{i}$ on the linear index $i$.

#### Beta Power Law.

Exponent $\beta$ of a power law
$\lambda_{i}\propto i^{-\beta}$,
computed as the negative slope of
$\log\lambda_{i}$ versus $\log i$.

#### Dim. 90% Var.

Minimum $k$ such that
$\sum_{i=1}^{k}\lambda_{i}\big/\sum_{j}\lambda_{j}\geq 0.9$.

These definitions ensure exact reproducibility of the results reported
in Section [4.3](https://arxiv.org/html/2510.15202v2#S4.SS3).

## Appendix C Geometry Induced by Radially Scaled ℓ 2 \ell_{2} Normalization

This appendix provides a formal derivation of the Riemannian metric induced by the tunable radial $\ell_{2}$ normalization used in our Mahalanobis-based OOD detector.

### C.1 Setup and Derivation

Let $z\in\mathbb{R}^{d}\setminus\{0\}$ be a feature vector with the standard Euclidean metric $g_{\mathrm{Euc}}$. We apply a radially symmetric map, $\phi_{\beta}$, to the feature space:

$$ $\phi_{\beta}(z)=\frac{z}{\|z\|^{\beta}},$ $$

where the scalar parameter $\beta\in\mathbb{R}$ controls the degree of radial scaling. This smooth map induces a new Riemannian metric on the domain via the *pullback* of the Euclidean metric, $g_{\beta}:=\phi_{\beta}^{*}g_{\mathrm{Euc}}$.

The Euclidean metric in polar coordinates $(r,u)$ where $z=ru$, $r=\|z\|$, and $u=z/r\in S^{d-1}$ is $g_{\mathrm{Euc}}=\mathrm{d}r^{2}+r^{2}g_{S^{d-1}}$. Our map transforms the radius to $R=\|\phi_{\beta}(z)\|=r^{1-\beta}$. The differential of the new radius is $\mathrm{d}R=(1-\beta)r^{-\beta}\mathrm{d}r$. By pulling back the Euclidean metric in terms of the new coordinates, we obtain the induced metric $g_{\beta}$:

$$ $g_{\beta}=\mathrm{d}R^{2}+R^{2}g_{S^{d-1}}=\left((1-\beta)r^{-\beta}\mathrm{d}r\right)^{2}+\left(r^{1-\beta}\right)^{2}g_{S^{d-1}}.$ $$

This simplifies to:

$$ $\boxed{g_{\beta}=(1-\beta)^{2}r^{-2\beta}\,\mathrm{d}r^{2}+r^{2(1-\beta)}g_{S^{d-1}},}$ $$

where $g_{S^{d-1}}$ is the standard round metric on the unit sphere. Thus, $\beta$ continuously interpolates between different geometries by re-weighting the radial and angular components of the metric.

### C.2 Geometric Interpretation of the Parameter β \beta

The exponent $\beta$ is a single parameter that continuously interpolates between different geometries by controlling the radial-angular trade-off.

- •
Case 1: $\beta=0$ (Euclidean Geometry). The map $\phi_{0}$ is the identity. The induced metric becomes $g_{0}=(1-0)^{2}r^{0}\mathrm{d}r^{2}+r^{2}g_{S^{d-1}}=\mathrm{d}r^{2}+r^{2}g_{S^{d-1}}$, which is precisely the Euclidean metric in polar coordinates.
- •
Case 2: $\beta>0$ (Contractive Geometries). This mapping pulls points towards the unit hypersphere. If $0<\beta<1$, radii are compressed but their order is preserved. For $\beta=1$, the radial component of the metric vanishes since $(1-1)^{2}=0$, and the metric reduces to $g_{1}=g_{S^{d-1}}$. This confirms that the space geometrically collapses onto the unit sphere. When $\beta>1$, points that were far from the origin are pulled inward even more strongly, inducing a strong radial contraction. This contractive effect can make the ID distribution more compact.
- •
Case 3: $\beta<0$ (Expansive Geometries). Let $\beta=-\gamma$ for $\gamma>0$. The mapping becomes $\phi_{-\gamma}(z)=z\cdot\|z\|^{\gamma}$. This transformation pushes points with large norms even further from the origin, radially stretching the space. This may be optimal when ID class manifolds are already well-separated and situated far from the origin, as it further increases inter-cluster distances while creating a large void around the origin where OOD samples can be easily detected.

### C.3 Mahalanobis Distance with Radially Scaled ℓ 2 \ell_{2} Normalization

The final OOD detector combines this geometric transformation with the standard Mahalanobis distance calculation. The procedure is as follows: first, a whitening transformation is applied to the features. Then, we apply the radially scaled $\ell_{2}$ normalization map $\phi_{\beta}$ to the whitened features. Finally, we compute the standard Mahalanobis distance (which, in the whitened space, is equivalent to the squared Euclidean distance) between the mapped test feature and the mapped class mean. The final OOD score is the minimum distance to any class mean:

$$ $\text{Score}(z)=\min_{c}D_{M}(\phi_{\beta}(z^{\prime}),\phi_{\beta}(\mu^{\prime}_{c})),$ $$

where $z^{\prime}$ and $\mu^{\prime}_{c}$ are the whitened versions of the feature and class mean, and $D_{M}$ is the Mahalanobis distance. By tuning $\beta$, we directly control the geometry of the space where the distance is measured, providing a principled way to align the feature distribution with the assumptions underlying Mahalanobis-based OOD detection.

## Appendix D Full results

### D.1 Cross-model performance

This section reports the complete cross-model results corresponding to Figure [2](https://arxiv.org/html/2510.15202v2#S3.F2). Table [2](https://arxiv.org/html/2510.15202v2#A4.T2) provides detailed false positive rates (FPR@95) across all evaluated architectures and OOD datasets for both Mahalanobis distance (MD) and its regularized variant (RMD). These results complement the trends discussed in the main text: RMD typically outperforms MD across architectures and training regimes, with particularly pronounced gains for models pretrained on large-scale datasets without ImageNet-1k fine-tuning.

Table: Table 2: False positive rate across different models and datasets using two Mahalanobis distance variants (MD and RMD)

### D.2 Detailed OOD performance

The results demonstrate that conformal normalization significantly improves OOD detection performance across diverse vision model architectures. Among the tested methods, the optimal variants of Mahalanobis Distance (MD*) and Relative Mahalanobis Distance (RMD*) consistently achieve the lowest false positive rates, with RMD* performing particularly well on transformer-based models like ViT and DeiT3. The Mahalanobis-based variants are defined as follows: MD* uses the empirically optimal $\beta$; $\hat{\textbf{MD}}$ uses the regression-predicted $\hat{\beta}$; MD (standard) fixes $\beta=0$; and MD++ (Mahalanobis++) fixes $\beta=1$. The performance gains are most pronounced on larger models (e.g., ViT-L, DeiT3-L) and those pre-trained on larger datasets (In21k, In22k), suggesting that conformal normalization is especially beneficial for models with richer feature representations. Traditional methods like MSP and MLS show relatively poor performance, while the proposed conformal variants provide substantial improvements in OOD detection reliability.

Table: Table 3: False positive rate (FPR, $\downarrow$) and validation accuracy (Acc) across different models and detectors. In most cases, the optimal MD* or RMD* achieves the best overall performance. Note that higher accuracy does not necessarily correspond to a lower FPR.

**Table 4: False-positive rate (FPR, $\downarrow$) across models for NINCO dataset using different Mahalanobis variants: MD* uses the empirically optimal $\beta$, $\hat{\textbf{MD}}$ uses the regression-predicted $\hat{\beta}$, MD (standard) fixes $\beta=0$, and MD++ (Mahalanobis++) fixes $\beta=1$.**
| Model | MD* | $\hat{\text{MD}}$ | MD | MD++ | RMD* | $\hat{\text{RMD}}$ | RMD | RMD++ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| BEiTV2 FT In1k | 0.446 | 0.463 | 0.506 | 0.470 | 0.414 | 0.418 | 0.457 | 0.443 |
| BEiTV2 FT In21k | 0.281 | 0.281 | 0.475 | 0.364 | 0.311 | 0.322 | 0.339 | 0.331 |
| DINOV2 | 0.403 | 0.415 | 0.424 | 0.445 | 0.573 | 0.573 | 0.577 | 0.574 |
| DeiT3 | 0.496 | 0.496 | 0.505 | 0.500 | 0.422 | 0.429 | 0.437 | 0.433 |
| DeiT3 FB In22k In1k | 0.455 | 0.464 | 0.480 | 0.455 | 0.445 | 0.455 | 0.457 | 0.445 |
| DeiT3 In21k In1k | 0.423 | 0.426 | 0.432 | 0.423 | 0.349 | 0.349 | 0.407 | 0.387 |
| DeiT3-L In22k In1k | 0.386 | 0.391 | 0.402 | 0.388 | 0.368 | 0.368 | 0.405 | 0.385 |
| EVA02 | 0.535 | 0.630 | 0.691 | 0.630 | 0.499 | 0.515 | 0.515 | 0.523 |
| EVA02 FT In1k | 0.409 | 0.412 | 0.418 | 0.413 | 0.418 | 0.418 | 0.431 | 0.425 |
| EVA02 FT In21k | 0.538 | 0.546 | 0.638 | 0.563 | 0.385 | 0.393 | 0.393 | 0.400 |
| EVA02 FT In21k In1k | 0.406 | 0.406 | 0.424 | 0.407 | 0.387 | 0.388 | 0.425 | 0.406 |
| EVA02-L FT In22k In1k | 0.349 | 0.358 | 0.356 | 0.388 | 0.311 | 0.312 | 0.321 | 0.316 |
| EVA02-S FT In22k In1k | 0.494 | 0.499 | 0.526 | 0.504 | 0.506 | 0.506 | 0.517 | 0.513 |
| MAE FT In1k | 0.481 | 0.485 | 0.516 | 0.488 | 0.428 | 0.433 | 0.463 | 0.450 |
| ViT | 0.553 | 0.557 | 0.557 | 0.556 | 0.511 | 0.513 | 0.516 | 0.514 |
| ViT CLIP In12k In1k | 0.302 | 0.317 | 0.379 | 0.336 | 0.301 | 0.301 | 0.373 | 0.352 |
| ViT CLIP In1k | 0.436 | 0.439 | 0.476 | 0.455 | 0.409 | 0.411 | 0.446 | 0.433 |
| ViT In21K | 0.630 | 0.659 | 0.641 | 0.658 | 0.442 | 0.454 | 0.451 | 0.453 |
| ViT In21K In1k | 0.394 | 0.394 | 0.405 | 0.481 | 0.415 | 0.422 | 0.425 | 0.425 |
| ViT-L CLIP In12k In1k | 0.309 | 0.309 | 0.320 | 0.310 | 0.272 | 0.282 | 0.299 | 0.290 |
| ViT-L In21K In1k | 0.303 | 0.306 | 0.322 | 0.358 | 0.264 | 0.272 | 0.272 | 0.277 |
| ViT-S In21K In1k | 0.499 | 0.500 | 0.515 | 0.505 | 0.512 | 0.514 | 0.512 | 0.514 |

## Appendix E Geometry of Eigvenvalues

Figure: Figure 9: BEiTV2 eigenspectra and their respective shifts: top—eigenvalues of covariance C, within-class $S_{w}$, and between-class $S_{b}$ across train (solid), val (dashed), and OOD (dotted); bottom—corresponding OOD-induced eigenvalue shifts relative to train.
Refer to caption: x7.png

Figure: Figure 10: CLIP eigenspectra and their respective shifts: top—eigenvalues of covariance C, within-class $S_{w}$, and between-class $S_{b}$ across train (solid), val (dashed), and OOD (dotted); bottom—corresponding OOD-induced eigenvalue shifts relative to train.
Refer to caption: x8.png

Figure: Figure 11: EVA02 eigenspectra and their respective shifts: top—eigenvalues of covariance C, within-class $S_{w}$, and between-class $S_{b}$ across train (solid), val (dashed), and OOD (dotted); bottom—corresponding OOD-induced eigenvalue shifts relative to train.
Refer to caption: x9.png

Figure: Figure 12: ViT eigenspectra and their respective shifts: top—eigenvalues of covariance C, within-class $S_{w}$, and between-class $S_{b}$ across train (solid), val (dashed), and OOD (dotted); bottom—corresponding OOD-induced eigenvalue shifts relative to train.
Refer to caption: x10.png

Figure: Figure 13: Eigenspectrum of covariance shift between train and OOD data (NINCO) for ViT variants: left—ViT In21K vs ViT; right—ViT In21K In1k vs ViT In21K.
Refer to caption: x11.png

Figure: Figure 14: Eigenspectrum differences by model pair (BEiTV2, ViT, EVA02/CLIP): for each pair, we plot ID vs OOD covariance $C$, ID within‑class $S_{w}$ and between‑class $S_{b}$, and covariance‑shift curves (OOD and ID), showing relative eigenvalue changes between the first and second model.
Refer to caption: x12.png

## Appendix F Pearson correlations

We replicate the correlation analysis from the main paper using Pearson correlations instead of Spearman correlations (Figure [15](https://arxiv.org/html/2510.15202v2#A6.F15)). The trends remain consistent: manifold-geometry and eigenvalue-based metrics show similar relationships with OOD performance across the three Mahalanobis variants, confirming that the observed patterns are not sensitive to the choice of correlation metric.

Figure: Figure 15: Pearson correlations between representation metrics and OOD performance across Mahalanobis variants. The three Mahalanobis-based detectors exploit different geometric cues, leading to distinct correlation patterns consistent with the Spearman results in Figure [5](https://arxiv.org/html/2510.15202v2#S4.F5).
Refer to caption: x13.png

## Appendix G Full model names

Table: Table 5: Mapping of model names to checkpoints and sources.

## Appendix H Use of AI Assistance

AI assistants, such as ChatGPT, were utilized in various aspects of the research, including coding, data analysis, and writing tasks. These tools helped automate repetitive tasks, generate initial drafts, and assist in exploring potential solutions. However, all AI-generated outputs were reviewed and refined by researchers to ensure accuracy and coherence.

## Peer Reviews

confidence: 4
contribution: 2
initial_rating: 4
presentation: 3
questions: - How does the $\beta$-scaled Mahalanobis detector compare to other common OOD detection methods (e.g., MSP, ODIN, energy-based scores) on the same benchmarks?
- What data and procedure were used to train the $\beta$ prediction regression model, and how confident can we be in its generalization to arbitrary new OOD scenarios? In particular, if one encounters a novel OOD setting without any related auxiliary datasets, can the regression still be applied (perhaps using your pre-trained model)?
soundness: 3
strengths: - The study is comprehensive in scope, evaluating Mahalanobis-type OOD detectors across a wide range of foundation vision models, training regimes, and datasets. The authors follow established protocols (e.g. OpenOOD) and test on multiple OOD benchmark suites (NINCO, iNaturalist, OpenImages-O, Textures, etc.), lending credibility to the findings. This broad evaluation reveals interesting trends, e.g., the Relative Mahalanobis Distance (RMD) consistently outperforms the standard Mahalanobis scoring, especially for models that were pretrained on large data but not fine-tuned on the target domain.
- The paper provides a deep analysis connecting representation geometry to OOD performance, which appears original and insightful. The authors identify measurable statistics of the feature space (covariance eigenvalue distribution, intrinsic dimensionality (ID), within-class scatter, etc.) and show that these correlate strongly with how well OOD examples can be detected. Notably, they find that a compound metric, the product of Local Intrinsic Dimensionality and scatter slope, is highly predictive of a model’s OOD detection effectiveness.
- Building on the above geometric insights, the paper introduces $\beta$-scaled $\ell_2$ normalization as a novel extension of prior Mahalanobis-based methods. Whereas earlier work (e.g., Mahalanobis++ by Mueller & Hein, 2025) simply applied unit-length normalization to features (equivalent to a fixed $\beta = 1$ case), here the authors generalize this idea by allowing a continuous tuning of the feature norms via the $\beta$ parameter. This conformal transformation of the feature space is an original contribution: it provides a direct “knob” to control the radial spread of feature vectors, effectively interpolating between the original Euclidean feature space ($\beta = 0$, no normalization) and a fully normalized hypersphere ($\beta = 1$), and even beyond ($\beta > 1$ for extra contraction, or $\beta < 0$ for expansion). The paper demonstrates that by choosing an appropriate $\beta$ for a given model, one can significantly improve OOD detection performance (by tightening clusters to better fit Gaussian assumptions or loosening them if needed).
- The paper is well-written and structured in a logical manner. It clearly lays out the problems and questions in early sections, follows up with corresponding analyses, and then transitions into the proposed method. The use of visuals (plots of OOD separation per dimension, conceptual diagram of $\beta$-scaling effects, etc.) is effective in reinforcing the arguments. Important claims are either theoretically explained or empirically demonstrated. The authors do a commendable job connecting the dots between observations (e.g. a model’s spectral properties) and the design of their method ($\beta$ normalization), which makes the paper easy to follow.
summary: This paper presents a comprehensive empirical study of Mahalanobis distance-based out-of-distribution (OOD) detection methods, with a focus on understanding how the feature representation geometry and normalization techniques influence OOD detection performance. The authors evaluate a diverse collection of modern vision models (self-supervised and pretrained transformers) across multiple benchmarks, revealing that Mahalanobis-style detectors exhibit highly variable effectiveness across models and settings. A key finding is that intrinsic properties of the in-distribution feature space, such as the spectral decay of covariance eigenvalues and estimates of intrinsic dimensionality, strongly correlate with OOD detection success. Building on these insights, the paper introduces a conformal $\ell_2$ normalization technique parameterized by $\beta$, which allows one to contract or expand the radial distribution of feature vectors. By tuning $\beta$, the representation’s geometry can be adjusted to better align with the Mahalanobis detector’s Gaussian assumptions, thereby improving OOD detection performance. Importantly, the paper proposes a simple regression-based approach to predict the optimal $\beta$ using only in-distribution data, without requiring any OOD samples. This learned predictor uses metrics derived from the training features (e.g. intrinsic dimensionality, within-class scatter) to estimate the best $\beta$, and achieves nearly the same OOD detection performance as an oracle that had access to OOD validation data.
weaknesses: - **Limited comparison to non-mahalanobis baselines**: A notable limitation is that the study restricts its focus to Mahalanobis-based detectors (standard MD, Relative MD, etc.) and does not compare against other mainstream OOD detection approaches. Methods like the Maximum Softmax Probability (MSP) baseline or energy-based scores are standard references in OOD detection literature; however, the paper does not report results for these alternatives. This omission makes it difficult to gauge the absolute improvement or significance of Mahalanobis-based methods in comparison to simpler or better-known baselines. For example, if a simpler MSP or energy score performs on par with the tuned Mahalanobis detector, that would affect the paper’s claim of advancing the state-of-the-art. Originality and significance could be better established by demonstrating that the proposed techniques outperform not only variants of Mahalanobis, but also other competitive OOD detection methods.
- **Primarily empirical findings without deeper theoretical explanation**: The connection drawn between feature geometry metrics and OOD performance, while intriguing, is based on empirical correlation rather than a derived theory. The paper shows strong correlations (e.g., between the product of LID and scatter slope and the detector’s FPR@95), but it stops short of providing a theoretical understanding of why these particular metrics are predictive. As a result, the insight, though useful, can feel a bit post hoc. A reader might wonder: to what extent are these geometric properties causally influencing OOD detection capability, versus simply being correlated indicators?
- **Incremental methodological novelty**: The proposed $\beta$-scaled $\ell_2$ normalization, though useful, could be perceived as a relatively incremental extension over prior art. The idea of normalizing feature vectors ($\ell_2$ normalization) to improve Mahalanobis detection was already introduced in earlier work (referred to as Mahalanobis++, which corresponds to the special case $\beta = 1$). This paper extends that concept by making the normalization strength tunable via $\beta$, which is a natural generalization. While this generalization is well-motivated and yields empirical gains, it may not represent a fundamentally new direction in OOD detection but rather a refinement of an existing technique.
- **Dependency on auxiliary OOD data for $\beta$ tuning**: Although the authors stress that their regression model selects $\beta$ “without access to target OOD data,” in practice the training of this regression does rely on OOD datasets (just not the target one). The experimental setup uses a meta-dataset of OOD experiences: the regression is trained using several auxiliary OOD datasets (and corresponding in-distribution features) to learn a mapping from feature statistics to optimal $\beta$. This implies that the method’s success partly hinges on having a representative set of OOD scenarios available beforehand.
- **Presentation issues**: (1) Most of the figures and tables in the paper are a bit too small (the text are difficult to read). (2) Inconsistent capitalization in paragraph titles, e.g., “Key Insights” and “Spectral ratios”.

---

confidence: 5
contribution: 2
initial_rating: 2
presentation: 2
questions: The authors have started a good empirical study that sheds light on how feature geometry and normalisation influence Mahalanobis-based OOD detection. The findings are partially informative, but the paper’s principal claims exceed what is demonstrated. The regression model for predicting \beta using other data sets' statistics to predict a remaining one seems strange. Also it shows weak generalisation. To strengthen this work, future versions should (1) establish a formal link between the conformal transformation and probabilistic assumptions of the Mahalanobis metric, (2) provide robust and statistically validated evaluation protocols, and (3) strengthen the prediction model for \beta considering the real world scenarios for practitioners who may have a single data set at hand. With these additions, the paper could become a valuable contribution to the understanding of representation geometry in OOD detection.
soundness: 2
strengths: The paper presents an empirical analysis that covers a set of foundation models, datasets, and normalisation schemes. This breadth of experimentation offers comparative insights into how representation learning choices influence OOD detection ability. The geometric viewpoint that relates intrinsic dimensionality, eigenvalue spectra, and normalisation effects to OOD reliability is conceptually appealing and shows some analytical depth. The authors’ systematic investigation of covariance, within-class and between-class scatter, and their spectral properties demonstrates a commendable effort to interpret model behaviour beyond raw performance metrics. Moreover, the proposed \beta-scaled normalisation is a simple, easily reproducible idea that gives practitioners a tunable parameter, if handled well, for shaping feature geometry. The attempt to predict this parameter through regression gives some hint to step towards automating hyperparameter tuning for post-hoc OOD detectors.
summary: This paper explores how the geometry of learned representations and feature normalisation affect the performance of Mahalanobis-based out-of-distribution (OOD) detection methods. The authors conduct a comprehensive empirical study across a range of self-supervised and pretrained vision models, examining spectral and manifold properties of feature spaces. They introduce a \beta-scaled conformal \ell-2 normalisation to control radial feature geometry and propose a regression-based approach to predict the optimal \beta value using only in-distribution (ID) data. The stated goal is to provide both an explanatory framework linking representation geometry and OOD performance, and a practical technique for improving Mahalanobis-based detection.
weaknesses: Although the study is broad and the analyses are detailed, the technical depth of the contribution remains limited. Several major concerns are below.

The first major concern is that the theoretical foundation of the work is a bit shaky. There is an obvious disconnection between OOD and optimality. Each test, based on FPR@95 as a target and surrogate for OOD, appears to be a single trial without repeat. How to justify the findings are robust but not random? There are many hand-wavy statements indicating "better for OOD detection" without any source of evidence, including many tests that are highly training dependent (for example, suboptimal due to insufficient training) or sampling rate (sufficiency of the data). As an example of the arbitrary choice in this paper, the proposed transformation is introduced as a geometric intuition but is not supported by a rigorous derivation showing why it improves OOD separability. The Riemannian interpretation provided in Appendix C is ok but not connected to any probabilistic or optimisation principle that would justify its efficacy. As a result, the paper remains largely empirical rather than theoretically grounded.

Second is that the novelty of the work is modest. The \beta-scaling formulation extends the idea of \ell_2 normalisation in Mahalanobis++ (Mueller & Hein, 2025) but offers little conceptual advance beyond introducing a continuous scaling factor. Similarly, the regression model predicting \beta is only a simple mapping from empirical metrics to a hyperparameter and does not introduce new methodological insights. Consequently, the paper reads more as a diagnostic study than as a genuine algorithmic contribution.

A third issue is that the empirical results do not convincingly demonstrate causal relations between geometry and OOD performance. Most conclusions are based on correlations rather than controlled experiments. The claim that representation geometry “drives” OOD detection is not substantiated, as confounding factors such as model size, dataset diversity, or training objective are not accounted for. Without controlled ablations, it is impossible to determine whether geometry itself or these associated factors explain the observed trends.

Another major concern is the limited specification and weak predictive performance of the regression model for \beta. Important implementation details, including the regression type, feature preprocessing, and regularisation, are omitted. The reported R2 values (0.25 for MD and 0.47 for RMD) are low, indicating that the predictor explains little variance. Figure 8 shows large deviations from the diagonal, yet this is not critically discussed. The claim that the regression achieves near-oracle performance is therefore overstated.

The evaluation protocol lacks clarity and statistical rigour. The process for selecting the “empirical optimal” \beta is underdefined, and it is unclear whether results are averaged across seeds or if statistical confidence is reported. In Table 1, the differences between methods are small and presented without variance estimates. This weakens the validity of the claimed improvements.

The interpretability and necessity of the various geometric metrics also remain questionable. The study introduces a large set of spectral and manifold statistics, but many are highly correlated and redundant. The proposed composite metric (the product of Local Intrinsic Dimensionality and within-class scatter slope) appears to have been found post-hoc and is not compared systematically against simpler alternatives. The reported Spearman correlation of 0.85 is presented as strong evidence but is not verified across dataset splits.

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: Overall the paper needs to be more theoretically grounding. In the absence of that, could the author justify their focus on radial transformations and not on other geometric manipulations such as anisotropic scaling?
soundness: 3
strengths: The main strengths of the paper are listed below:

* Comprehensive empirical study: The paper evaluates a wide range of modern self-supervised and foundation models across multiple OOD benchmarks.

* Insightful geometric analysis: The paper identifies that the product of Local Intrinsic Dimensionality (LID) and within-class scatter slope strongly predicts OOD performance (Spearman $\rho$ up to 0.85).

* Practical contribution: The paper introduces $\beta$-scaled $\ell^2 normalization, a simple, effective, and theoretically grounded method to control feature geometry.

* Data-efficient tuning: The paper demonstrates that optimal $\beta$ can be predicted using only ID data and auxiliary OOD datasets, avoiding reliance on target OOD samples.
summary: This paper investigates why Mahalanobis-based out-of-distribution (OOD) detection methods perform inconsistently across different vision foundation models. Through extensive empirical analysis, the authors show that OOD performance strongly correlates with geometric properties of in-distribution (ID) feature spaces, in particular intrinsic dimensionality and within-class scatter structure. They propose a $\beta$-scaled $\ell^2$ normalization, a conformal transformation that radially reshapes feature geometry to better align with Mahalanobis assumptions or enhance OOD separation. A regression model trained on ID geometry metrics and auxiliary OOD data can predict a near-optimal $\beta$ without access to target OOD samples, improving detection across diverse models and datasets.
weaknesses: The paper can be improved along the following lines:

* Limited theoretical grounding: While empirical correlations are strong, the paper lacks formal theoretical justification for why the LID × slope product is predictive.

* Dependence on auxiliary OOD data: The regression-based $\beta prediction still requires access to other OOD datasets (via leave-one-out), which may not always be available in real-world settings.

* Narrow scope of normalization: The paper focuses only on radial (conformal) transformations. Could other geometric manipulations such as anisotropic scaling be explored?

* Computational overhead: Fitting class-conditional Gaussians and performing eigen-decompositions may be costly for very large models or high-dimensional features. This limits the applicability of this study to (only?) small models.

---

confidence: 4
contribution: 3
initial_rating: 4
presentation: 4
questions: - The per-dimension analysis in Section 3.3  appears quite similar to the analysis in Kamoi & Kobayashi (2020)[1], which is not cited. Could the authors clarify the novel insights from their analysis beyond this prior work?

-  The "ideal geometry" metric (LID $\times$ $S_w$ slope) is shown to be a strong correlational predictor of OOD performance. Have the authors considered or investigated a causal link? For example, have you explored using this metric as a regularization term during model training to actively enforce this "ideal geometry," and if so, does it improve downstream OOD detection?

References:
1- Kamoi, Ryo, and Kei Kobayashi. “Why Is the Mahalanobis Distance Effective for Anomaly Detection?” arXiv:2003.00402, arXiv, 30 Apr. 2020. arXiv.org, http://arxiv.org/abs/2003.00402.
soundness: 3
strengths: The paper introduces a novel $\beta$-scaled (conformal) $l_2$ normalization. This is a creative generalization of prior work, reframing standard Mahalanobis ($\beta=0$) and Mahalanobis++ ($\beta=1$) as specific points in a continuous space of geometric transformations. 

This "control knob" for feature geometry is a new and elegant contribution.The identification of the product of Local Intrinsic Dimensionality (LID) and the $S_w$ (within-class scatter) slope as a strong predictor of OOD performance is highly original . 

Most prior work correlates performance with single metrics, whereas this combined metric uniquely captures the trade-off between manifold complexity and cluster compactness, which the authors define as the "ideal geometry" .
The problem formulation of predicting the optimal normalization parameter $\hat{\beta}$ using a regression model trained only on in-distribution geometry and auxiliary OOD datasets is a novel and practical approach to the problem of hyperparameter tuning without target OOD data

The finding that OOD performance is predictable from in-distribution geometric properties (LID $\times$ $S_w$ slope) is a fundamental contribution that could guide future work in representation learning, suggesting that models can be explicitly trained to have an "ideal geometry" for OOD robustness.
summary: This paper investigates the factors that determine the effectiveness of **Mahalanobis-based Out-of-Distribution (OOD) detectors**. Through a comprehensive empirical study across various vision foundation models, the authors demonstrate that the performance of these detectors is not universal  and is often weakly correlated with a model's in-distribution (ID) classification accuracy.

The core argument is that OOD detection performance is primarily driven by the **geometric properties of the in-distribution feature space**. The study analyzes the spectral properties of the feature covariance ($C$), within-class scatter ($S_w$), and between-class scatter ($S_b$) matrices. A key finding is that an "ideal geometry" for OOD detection involves a balance between **local manifold complexity** (measured by Local Intrinsic Dimensionality, LID) and **intra-class compactness** (measured by the spectral slope of $S_w$). The authors show that the product of these two metrics serves as a strong predictor of a model's OOD performance.

Building on this, the paper examines the role of **feature normalization**. It argues that deep features often violate the Gaussian assumptions of the Mahalanobis distance  and introduces a novel **conformal $l_2$ normalization** to reshape the feature space geometry. This method provides a "control knob" that generalizes prior work and can be optimized to improve OOD detection.
weaknesses: - Missing some related work analyzing the mahalanobis distance too 

for example:

1-  Kamoi, Ryo, and Kei Kobayashi. “Why Is the Mahalanobis Distance Effective for Anomaly Detection?” arXiv:2003.00402, arXiv, 30 Apr. 2020. arXiv.org, http://arxiv.org/abs/2003.00402.

2- Tajwar, Fahim, et al. "No true state-of-the-art? ood detection methods are inconsistent across datasets." arXiv preprint arXiv:2109.05554 (2021).

- Per dimension analysis of Mahalanobis distance is similar to the analysis conducted in [1].

- Dependence on Auxiliary OOD Data: The paper's main practical solution, the regression framework for predicting the optimal $\beta$, is not fully "post-hoc" in the strictest sense. It relies on a "Leave-One-Dataset-Out" scheme, meaning it must be trained using a collection of other OOD datasets to learn the relationship between in-distribution geometry and the optimal $\beta$ 

- Limited Scope of Experimental Comparison: The evaluation of the proposed method is not benchmarked against the current state-of-the-art. While the paper uses the OpenOOD protocol , the results tables only compare variants of Mahalanobis distance (MD, RMD) and report a single metric (FPR@95). A more convincing evaluation would include a comprehensive table comparing the proposed optimized detector ($\phi_{\beta}(z^{\prime})$) against a wider range of non-Mahalanobis baselines on the OpenOOD benchmark, using standard metrics like AUROC in addition to FPR.

---

confidence: 3
contribution: 3
initial_rating: 6
presentation: 3
questions: Please refer to the Weaknesses.
soundness: 3
strengths: 1. A comparative study of several Mahalanobis-based OOD detectors across diverse models, including
a per-dimension analysis of its variants.
2. An analysis revealing that OOD performance is strongly predicted by the in-distribution geometry
of features, such as spectral decay and intrinsic rank.
3. The introduction of β-scaled ℓ2 normalization and a method to predict the optimal β using only
in-distribution data.
summary: This paper conducted a comprehensive empirical study
across diverse image foundation models, datasets, and distance normalization schemes.
First, the study shows that Mahalanobis-based methods aren’t universally reliable. Second, the authors define the ideal geometry for data representations and demonstrate that spectral
and intrinsic-dimensionality metrics can accurately predict a model’s out-of-distribution
(OOD) performance. Finally, the paper analyzes how normalization impacts OOD performance.
Building upon these studies, a conformal generalization of
ℓ2 normalization that allows to control the degree of radial expansion of the representations geometry, which in turn helps improve OOD detection, is proposed.
weaknesses: 1. Why does the combination of the LID and  S_w metrics outperform single metrics (e.g., using LID or entropy alone)? Visualizing the feature clustering distributions of different models  may intuitively illustrate the relationship between this combined metric and clustering quality.

2. Lack of comparison with current state-of-the-art (SOTA) methods. The paper only compares against traditional Mahalanobis-distance variants, but omits recent SOTA OOD detection approaches based on “feature calibration” or “contrastive learning.” This makes it difficult to position the proposed method within the current research landscape.

3. Experiments are limited to image classification models and do not validate the method’s effectiveness on downstream tasks such as object detection or segmentation. Adding OOD detection experiments on 1–2 object detection models (e.g., Faster R-CNN) would better demonstrate the method’s task adaptability.

---

confidence: 4
contribution: 2
initial_rating: 4
presentation: 1
questions: - Could the authors explicitly define what is meant by “features”, and how they expect the praticular choice of representations to affect the results?
- How does the proposed normalization affect the geometric properties that the paper identifies as important for OOD detection?
soundness: 2
strengths: **(S1) Analysis of factors underlying successful OOD detection:** The paper examines which properties of neural network representations geometry lead to successful OOD detection using Mahalanobis-based detectors. This is an important and timely topic, as OOD detection performance is known to vary considerably across settings, and concrete insights or recommendations on what drives success remain scarce in the literature. The paper provides empirical evidence on how various metrics characterizing the feature distribution influence the performance of Mahalanobis OOD detectors.


**(S2) Scope of empirics:** The empirical analysis covers a wide range of models and spectral metrics.
summary: The paper investigates the geometric properties of neural network feature representations that influence the success or failure of out-of-distribution (OOD) detection, focusing on variants of Mahalanobis-based detectors. The main contributions are: (1) empirical analyses demonstrating how the performance of several Mahalanobis detectors correlates with various spectral metrics in feature space, and (2) a proposed feature normalization method, given by $\phi(z) = z / \||z\||^\beta$, where the parameter $\beta$ is fitted to improve the performance of Mahalanobis detectors.
weaknesses: **(W1) Clarity and precision:** The paper’s precise contributions are difficult to understand due to clarity issues. For example:
- The term “features” is never clearly defined. In Mahalanobis OOD detectors, features typically correspond to activations from specific hidden layers, but performance can depend strongly on which layers are used and how they are aggregated. The paper does not specify the exact setup considered.  
- Several quantities presented in Figure 4 are not formally defined and cannot be easily inferred from the text. For instance, the “Fisher scatter ratio” usually involves both $S_w$ and $S_b$, yet the corresponding plot separates them into distinct lines without explanation.  
- Most figures lack x-axis labels, which makes them difficult to understand.  

**(W2) Method for fitting $\beta$:** The proposed normalization method requires fitting the parameter $\beta$. *Contrary to the claim made in the contributions summary*, the procedure for choosing $\beta$ *requires access to OOD data* (the authors propose to train on *all OOD datasets except the target one*). Moreover, the parameter must be refitted for each setting, as Figure 7 shows that its optimal value varies widely across models and datasets. Given this variability, the choice of OOD data used for fitting $\beta$ could substantially influence the results. This raises concerns about the practical applicability of the proposed method, since a comprehensive OOD dataset is usually not available in practical settings.  

**(W3) Insight and discussion:** To my mind, the biggest weakness is that the paper presents many isolated results that are not synthesized into a coherent framework. The first part of the contributions shows that Mahalanobis OOD detection depends on various spectral metrics of the feature space. The second part proposes a normalization technique with a tunable parameter $\beta$ to improve performance. However, the paper does not discuss how this normalization affects the spectral properties relevant to OOD detection, or why fitting $\beta$ should modify these properties in a favorable way. If I am not missing something, these questions are not addressed in the discussion, and no formal results are provided to support the claim that normalization makes the geometry “better.” Therefore, the proposed method’s connection to the geometric arguments presented in the paper seems unsupported.

**(W4, minor) Scope limited to Mahalanobis detectors:** Since Mahalanobis detectors rely on fitting Gaussian distributions to ID data, it is expected that spectral properties of the feature space correlate with OOD detection performance. It would be more interesting and impactful to investigate which factors influence the success of OOD detection methods more generally, beyond the Mahalanobis family.

---

confidence: 4
contribution: 2
initial_rating: 4
presentation: 4
questions: - **Clarification on contribution 3 (line 66).** "Predicting optimal $\beta$ using only ID data" is very misleading. Please explicitly state that auxiliary OOD datasets are used to train the $\beta$-regressor. Using only ID data implies you use nothing else than what model is trained.

- **Per-PC signal (line 186).** You observe smaller variance directions can be more discriminative for OOD. It feels very counterintuitive to me, do you have an intuition that explains this behavior in some models?

- **$S_b / S_w$ ratio (line 231).** Adding a side by side plot of the ratio vs OOD performance would support these claims. Right now it is not straightforward to me the conclusion that it would favor OOD detections directly. As written, the ratio alone only tells how the model assigns ID points to the representation space. Therefore, it doesn’t guarantee separability for OOD; an empirical correlation would help.

- **Benchmark breakdowns.** I would enjoy seeing near vs far OOD breakdowns to match the analyses done in the literature more easily. Moreover, the varying performances over near and far cases potentially reveal additional insights on the representation behaviors of these models.


### Minor Comments/Edits

- **Line 83.** The sentence beginning "However, our study reveals..." would read better if you immediately state how for a better reading experience.

- **Line 145 & 149.** These insights have been observed before; please cite [1] and [2] again at these points to orient readers.

- **Start of the Section 4 (line 194).** Up to here, RMD seems consistently ahead on FPR metrics. I already made my comment on how average separability metric could be misleading, I would temper the claim here. 


[1] Bitterwolf, J., Mueller, M., & Hein, M. (2023). In or out? fixing imagenet out-of-distribution detection evaluation. arXiv preprint arXiv:2306.00826. 

[2] Ren, J., Fort, S., Liu, J., Roy, A. G., Padhy, S., & Lakshminarayanan, B. (2021). A simple fix to mahalanobis distance for improving near-ood detection. arXiv preprint arXiv:2106.09022.
soundness: 3
strengths: - **Clarity.** The text is cleanly written and asks the right questions (I particularly enjoyed the questions on lines ~133, ~197), leading readers to an intuitive and informative analysis.

- **Per dimension contribution $S_i$.** The view of per-PC contributions is useful and well-motivated. Authors may consider calculating the FPR95 with the ith principal component. Aggregating the first k is in my opinion harder to read and $S_i$ as a metric could be misleading as it is a distributional metric. However, we do the OOD detection on the sample level, $S_i$ would be more appropriate if we were tackling set level OOD detection.

- **Predictive diagnostics.** Geometry-based metrics (e.g. $LID \cdot S_w$) correlate with OOD performance and feel like the right direction to explore when a backbone is suitable for post-hoc OOD.

- **Spectral-shift observation.** The link between OOD-induced spectral shift and pretraining dataset size is a practically important observation. Here one get insights on how robust we expect the models to be on the OOD representations.

Overall, the first half builds a strong understanding and experiments are coherent and in line with the insights.
summary: The paper benchmarks Mahalanobis-style OOD detectors across modern vision backbones and links OOD performance to representation geometry (using metrics like covariance, within/between class scatter, LID etc.). It then introduces a $\beta$-scaled L2 normalization and trains a leave-one-dataset-out regressor to predict $\beta$ using ID geometry plus auxiliary OOD datasets.
weaknesses: - **Fairness of comparisons.** This is the biggest concern I have. The $\beta$-regressor trained with auxiliary OOD which moves the method away from training-free post-hoc and makes direct comparisons to strictly post-hoc Mahalanobis++ less fair.
- **Baseline gap.** Natural follow up to my previous point is that if auxiliary OOD is allowed, a simple linear OOD classifier can be put as a baseline to let us understand the value add of $\beta$ scaling.
- **Section bridge.** The normalization section feels disconnected from the diagnostics. A natural bridge would be using the predictive metric ($LID \cdot S_w$) to set $\beta$ without auxiliary OOD, keeping the method post-hoc and strengthening the narrative.
- **Maha++ missing in sections 3 and 4.** Mahalanobis++ should be present in the core geometry analyses (not just final comparisons) to catch potential insights from normalized feature space.