REINS: Learning Inertia-Induced Geometry for Physics-Consistent Motion Representation in Clinical Gait Phenotyping

Parkinson's disease and other gait-related neurological disorders manifest in subtle but biomechanically meaningful changes in how patients move, making computational gait phenotyping a valuable tool for clinical assessment. Yet existing latent representations of motion are typically learned from data alone and miss the biomechanical structure that gives clinical motion its diagnostic meaning, leaving them vulnerable when training cohorts are small or acquisition sites differ. We propose REINS (Riemannian Embedding for INertia-induced motion Spaces), an Euler-Lagrange-induced motion representation that uses the generalized inertia matrix of articulated systems to define a biomechanically grounded geometry on configuration space. Because kinetic energy is itself a quadratic form in velocity weighted by the inertia matrix, this metric is a mechanically determined Riemannian structure, and we learn a latent embedding of the resulting structured space. By defining the representation space itself through rigid-body dynamics rather than enforcing physics as a post-hoc penalty, REINS aligns the geometry of motion representation with the same physical laws clinicians rely on to interpret gait. We apply the framework on Parkinson's disease gait analysis, where deviation from healthy motion is measured as a geodesic distance under the inertia-induced metric, and report results on healthy-control vs.\ PD discrimination and UPDRS-gait severity correlation.