Wasserstein Mahalanobis Distances for Recovering Latent Geometry

The Mahalanobis distance is a fundamental covariance-adapted metric for multivariate data and plays a central role in recovering latent geometry from nonlinear observations. We extend this principle from vector-valued data to probability measures by introducing a Wasserstein Mahalanobis distance. Our construction replaces Euclidean displacement vectors with optimal transport displacement fields and local covariance matrices with covariance operators defined on Wasserstein tangent spaces. We show that this construction inherits the geometry-recovery property underlying nonlinear independent component analysis. In particular, for Gaussian measures with common covariance transformed by a smooth nonlinear pushforward, the proposed Wasserstein Mahalanobis distance approximates the classical Mahalanobis distance between the transformed latent means. The correspondence is exact for affine transformations and holds up to controlled higher-order error terms for general smooth transformations. These results establish a distribution-valued analog of classical Mahalanobis geometry and provide theoretical support for covariance-adapted learning directly in Wasserstein space. Numerical experiments confirm the theoretical predictions and demonstrate accurate recovery of latent geometric structure.

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.

Reliable Modeling of Distribution Shifts via Displacement-Reshaped Optimal Transport

Optimal transport (OT) is a central framework for modeling distribution shifts. Because OT compares distributions directly in input space, a well-designed ground metric between observations is essential to ensure that the optimizer does not violate the true geometry of change. We propose Displacement-Reshaped Optimal Transport (ReshapeOT), a method that reshapes the ground metric by integrating observed sample displacements as an additional source of knowledge. Technically, ReshapeOT replaces the Euclidean metric with a Mahalanobis distance estimated from displacement second moments. This effectively carves expressways through the input space, inviting transport solutions that better align with observed displacements. Our method is computationally lightweight, integrates seamlessly into any OT solver that operates on a cost matrix, and can be kernelized for further flexibility. Experiments on synthetic and real-world data show that ReshapeOT achieves substantial gains in transport reliability. We further demonstrate our method's usefulness in two practical use cases.