A Probabilistic Circuit-Induced Pseudo-Metric for Out-of-Distribution Detection

Probabilistic Circuits (PCs) are tractable generative models whose internal nodes encode a hierarchy of probabilistic sum- maries over different variable scopes. Existing PC-based out- of-distribution (OOD) detection methods ignore this hierar- chy, reducing the entire circuit to the scalar likelihood (or its uncertainty) computed at the root. We introduce Hierar- chical Likelihood Vector (HLV), a representation whose en- tries are the likelihoods associated with selected PC nodes and define the Hierarchical Likelihood Distance (HLD), a PC-induced pseudo-metric that compares the probability dis- tributions through the expectations of their HLVs. We show that HLD is an integral probability metric over a function class naturally induced by the PC and develop a principled goodness-of-fit hypothesis test for unsupervised OOD detec- tion. Unlike existing approaches, the trained PC alone serves as the representation of the in-distribution: no held-out in- distribution data are required at deployment. We further show that the quantities required by the hypothesis test can be com- puted exactly, directly from the trained circuit, yielding an ap- proximate analytic decision threshold. Experiments on tabular and MNIST datasets demonstrate that exploiting the hierarchi- cal probabilistic summaries encoded through the PC improve OOD detection over root-likelihood, uncertainty-, typicality- and kernel-based baselines, while naturally localizing distri- bution shifts to the PC nodes responsible for the shift.

From Uncertainty to Failure Attribution: Self-Diagnosing Models for Failure Attribution under Distribution Shift

Distribution shift poses a significant challenge to the robustness of machine learning models, but the current solutions only aim to detect out-of-distribution (OOD) samples and predict uncertainty levels. We introduce a problem setting for failure attribution under distribution shift, which enables the models not only to detect OOD samples, but also to find out the reason for their failure. The solution we propose is called self-diagnosing models, which are capable of jointly learning predictive output, predictive uncertainty, and a failure attribution signal. In particular, we use the failure attribution vector, produced by a neural network, which provides a structured representation of predictive unreliability by distinguishing four different types of failures: covariance shift, semantic shift, noise corruption, and adversarial perturbation. In other words, we move from scalar uncertainty towards failure identification. For training the model, we introduce a consistency regularizer that encourages consistency between uncertainty and failure attribution predictions. Moreover, to be able to evaluate the model on its ability to find the reasons for failure, we construct several distribution shift benchmarks with predefined mechanisms for generating distribution shifts.

Canonical Joint Energy-Based Model on CIFAR-10: failure modes and practical indistinguishability of Predictor-Corrector and SGLD samplers

Joint Energy-Based Models (JEM) unify classification and generation within a single network and support out-of-distribution (OOD) detection. Canonical JEM training relies on stochastic gradient Langevin dynamics (SGLD); a theoretically motivated alternative, the Predictor-Corrector (PC) sampler, has not previously undergone a systematic replication test on the canonical model. We reproduce canonical JEM on WideResNet-28-10 without normalisation layers on two independent runs and test whether PC retains its theoretical advantage without an annealed noise schedule, across three protocols: PC replacing SGLD throughout the roughly 130 training epochs; cold-start generation (FID); and refinement-style multi-OOD detection (AUROC). The reconstruction reaches 92.88% test accuracy and buffer-FID 44.46 (canonical: 92.9% and 38.40). We document two failure modes: catastrophic late-training divergence via the canonical outlier-buffer mechanism (both SGLD runs and, with the same signature, both PC runs), and run-dependent SVHN OOD-discrimination dynamics. No method-level advantage of PC over SGLD is observed on any protocol: at inference the absolute AUROC difference stays below 0.007 across all ten checkpoint-OOD pairs and the FID difference below 0.5; on the training protocol a hierarchical seed-by-image bootstrap gives a 95% confidence interval on the macro-averaged AUROC difference that contains zero, while a seed-level equivalence test with two runs per method cannot establish formal equivalence. The data are consistent both with equivalence and with a small directional effect. This practical indistinguishability is theoretically expected: under fixed noise the PC predictor step degenerates by construction, so its guarantees do not transfer to canonical JEM.

Covariance Last-Layer Ensembles: Function-Space Diversity for Efficient Uncertainty Quantification

A Last-Layer Ensemble (LLE), $K$ linear units on one shared frozen feature map, is an efficient single-pass approach to the disagreement-based epistemic uncertainty for out-of-distribution (OOD) detection. Its weakness is that members share the backbone gradient and can converge toward the same function, collapsing the inter-member diversity the signal depends on. Whether last-layer diversity can be restored, and what mitigates the collapse, is an open question. The weight-orthonormality defining Orthonormal Certificates (OC), the weight-orthonormal special case of the LLE, is only an indirect correction; it decorrelates the weights of the members, not their predictions. Here, we instead target the collapse directly in function space, with a Covariance Last-Layer Ensemble (cov-LLE) that places a direct covariance penalty on member activations. Cov-LLE restores the function-space diversity that weight-orthonormality cannot, and at matched $K$ recovers much of the diversity and calibration of a deep ensemble at $1\times$ backbone cost (in-distribution prediction variance $0.05\!\to\!9.3$ vs. $22.1$ ($\times10^{-3}$), and ECE $0.135\!\to\!0.090$ vs. $0.035$, for a $K\times$-cost deep ensemble), at no cost to accuracy. Viewing OC as a last-layer ensemble also organizes detectors into a two-axis taxonomy (by how their units are trained and how their outputs are scored) and exposes the OC score as a magnitude, motivating a scale-invariant, label-free direction score that repairs its near-OOD failure, adding $+0.16$ to $+0.18$ ROC AUC on every backbone.

Sparse Autoencoders for Interpretable Out-of-Distribution Detection

Reliable detection of out-of-distribution (OOD) samples is crucial for the safe deployment of machine learning models. Neural networks often produce overconfident predictions for inputs that deviate from their training data, leading to significant degradation in performance. While many OOD detection methods focus on the final output layer, they neglect the rich hierarchical information present in intermediate network layers. This paper introduces a novel approach that leverages sparse autoencoders (SAEs) to learn interpretable features from these intermediate activations. We find that in-distribution (ID) and OOD data activate distinct sets of these sparse features. We propose a new OOD score derived from the cosine similarity between the sparse feature activations of a test sample and the mean activations of ID classes. Our post-hoc detection method not only achieves state-of-the-art performance on standard OOD detection benchmarks, but yields interpretable insights into how distribution shift affects learned representations.

Decodable but Not Detectable: A Leakage Fingerprint for Near-OOD Benchmarks

While auditing a perturbation-based OOD detector on a document benchmark, we recorded an AUROC of 0.326 -- well below the 0.5 chance level. The cause is a benchmark leak: the designated "OOD" class is one the model was trained on, so its examples sit inside the in-distribution fit set and the detector is penalized for correctly ranking them as familiar. Deleting the class and retraining 35 models across two domains raises the score to 0.911. We distill the contamination into a leak fingerprint -- near-perfect supervised decodability (AUROC approximately 1) coupled with unsupervised detection collapsed below 0.65 -- and validate it on a controlled battery of 52 settings (20 leaked, 32 clean) across ResNet-50 and ViT-B/16 on CIFAR-10/100, achieving sensitivity 18/20 and specificity 31/32 in embedding space; the matched fit-set-exclusion controls are perfect at 20/20. An in-the-wild audit of 24 standard near/far OOD benchmark pairs fires on exactly one (the intrinsically hard CIFAR-100 vs CIFAR-10 pair) and on no far-OOD pair, confirming specificity and that standard cross-dataset construction is clean. Under the corrected protocol, perturbation signals are decodable but not detectable: a supervised reader recovers the OOD signal (AUROC 0.87-1.00) while no unsupervised detector does, and the perturbation method does not improve on plain Mahalanobis distance. We provide a theoretical account of why and, for transparency, retract an earlier circular correlation. The contributions are a corrected protocol and a validated leak diagnostic, not a new OOD method.

Exploiting Local Flatness for Efficient Out-of-Distribution Detection

Detecting out-of-distribution (OOD) data is crucial for reliable machine learning deployment. Among detection strategies, post-hoc methods are particularly attractive due to their efficiency, as they operate directly on pre-trained networks without requiring retraining. Within this paradigm, one promising direction exploits loss-landscape curvature to estimate model uncertainty; however, such methods incur substantial computational cost and rely on implicit assumptions about how landscape flatness differs between in-distribution (ID) and OOD data. In this work, we provide the first systematic investigation of this curvature discrepancy and show that OOD inputs exhibit larger Hessian curvature than ID data, with the gap widening under stronger distributional shifts. Motivated by these observations, we propose Fold, a lightweight flatness-modulated OOD detector that leverages the feature Hessian and partial feature normalization to improve ID-OOD separability while avoiding costly parameter-space curvature approximations. To optimally adapt this normalization across diverse datasets, we further introduce AutoFold, a self-supervised tuning scheme that synthesizes pseudo-OOD samples via ID logit masking for automatic calibration without requiring external data. Experiments on OOD benchmarks show that Fold outperforms prior methods, improving the average AUROC by 1.63% and reducing FPR95 by 2.30%, while maintaining computational efficiency comparable to a standard forward pass. Supported by theoretical analysis and extensive ablations, Fold provides a principled and practical solution for robust real-world deployment.

Theoretical Grounding of Out-Of-Distribution Detection With Reinforcement Learning Optimizer

Out-of-distribution (OOD) detection in dynamic open-world environments requires a model to continually adapt to evolving data distributions while generalizing to covariate-shifted inputs and rejecting semantic-shifted OOD examples. Most existing OOD detection methods optimize only the current-step objective and do not explicitly account for how post-deployment environment changes affect future OOD behavior. In this paper, we establish a theoretical grounding for dynamic OOD detection using a reinforcement learning (RL)-guided optimizer that explicitly favors updates that reduce the semantic OOD false positive rate over time. We develop a novel augmented optimizer that uses an RL-guided correction term on top of standard gradient descent (GD) and show its improvement over both future-domain generalization and semantic-OOD rejection. We analyze temporal error decomposition in terms of model-change and environment-change generalization errors and develop a new theoretical framework for comparing the generalization errors under both GD and RL-guided optimizers.

MM++: Post-Hoc Scale-Invariant Multilayer OOD Detection via Top-K Gated Feature Fusion

We introduce MM++ (Multilayer Mahalanobis++), a strictly post-hoc, and scale-invariant framework for Out-of-Distribution (OOD) detection. To address the trade-off between scale invariance and hierarchical expressivity, MM++ constructs a principled joint feature space. It first identifies discriminative intermediate layers by measuring entropy density drops, which mark the boundaries of sharp semantic compression. By fusing these selected layers with the terminal representation, the framework captures latent cross-layer correlations while mitigating early-layer noise. Crucially, a Ledoit-Wolf regularized tied covariance matrix stabilizes this unified space, enabling reliable distance estimation. Requiring no auxiliary OOD data, classifier fine-tuning, or architectural modifications, MM++ delivers robust performance across distinct architectures for both near- and far-OOD detection.

The Signal in the Noise: OOD Detection Through Goodness-of-Fit Testing in Factorised Latent Spaces

Deep generative models offer a natural foundation for out-of-distribution (OOD) detection, yet prior work has shown that their assigned likelihoods are notoriously unreliable indicators for in- vs out-of-distribution data. In this paper, we address this problem by leveraging the diffeomorphic and mass-preserving properties of continuous normalising flows. Our analysis shows that OOD samples are mapped to noise samples that are highly atypical under the noise prior in ways not captured by the likelihood. Based on this observation, we propose a new method -- Signal in the Noise (SITN) -- for OOD detection on the single-sample level. SITN requires no access to OOD data, incurs minimal computational overhead, and provides strict control of false positive rates. Comprehensive evaluations through standard benchmarks and synthetic perturbations highlight the method's effectiveness and the absence of the complexity bias inherent to likelihood-based methods.