Invariance on Manifolds: Understanding Robust Visual Representations for Place Recognition

Visual Place Recognition (VPR) demands representations robust to drastic environmental and viewpoint shifts. Current aggregation paradigms, however, either rely on data-hungry supervision or simplistic first-order statistics, often neglecting intrinsic structural correlations. In this work, we propose a Second-Order Geometric Statistics framework that inherently captures geometric stability without training. We conceptualize scenes as covariance descriptors on the Symmetric Positive Definite (SPD) manifold, where perturbations manifest as tractable congruence transformations. By leveraging geometry-aware Riemannian mappings, we project these descriptors into a linearized Euclidean embedding, effectively decoupling signal structure from noise. Our approach introduces a training-free framework built upon fixed, pre-trained backbones, achieving strong zero-shot generalization without parameter updates. Extensive experiments confirm that our method achieves highly competitive performance against state-of-the-art baselines, particularly excelling in challenging zero-shot scenarios.