Observation Quotients and Learning-as-Lifting

This paper develops a model-agnostic theory of learning and inference from compressed observations by treating observation as a metric quotient. Starting from a Polish geodesic latent space (X,dX)( X,dX)(X,dX) and a Borel map O:X ⁣→ ⁣ZO: X\!→\! ZO:X→Z, we endow the image with the induced quotient metric dZdZdZ (or a declared surrogate metric d^Z dZd^Z with Lipschitz constant LLL). On probability measures we study the inf-projection GGG of an energy FFF along OOO and prove an image–EVI result: λλλ-EVI gradient flows of FFF on (P2(X),W2,X)( P_2( X),W2, X)(P2(X),W2,X) push forward to relaxed/exact λλλ-EVI flows of GGG on (P2(Z),W2,Z)( P_2( Z),W2, Z)(P2(Z),W2,Z). We quantify the loss from compression via a selector-based fiber radius, derive a Base–Fiber transport bound that separates image motion from intra-fiber dispersion, and establish inexact minimizing-movement stability that preserves EVI under optimization and feasibility errors. Main contributions (searchable): Image–EVI theorem for pushforward of Wasserstein-2 gradient flows under observation/quotient maps; exactness under attainment of the inf-projection. Fiber quantification: selector radius bounds (Fréchet variance / i.i.d. intra-fiber distances) controlling risk degradation in representation learning. Base–Fiber decomposition: W2W_2W2 upper bounds splitting image transport and intra-fiber variance with measurable selections on Radon spaces. Inexact JKO stability: EVI-preserving minimizing movements with explicit (ϵ,δ)(ε,δ)(ϵ,δ) error summability conditions. Auditable pipeline (QJKO-AL): quotient-first solver with Sinkhorn, sliced Wasserstein (SQT), and Lipschitz coarsening (Q-Tree); logs (ϵ,δ,E^[rsel])(ε,δ,{ E}[rₛₑₗ])(ϵ,δ,E[rsel]). Scope and relevance: The framework links optimal transport (OT), Wasserstein gradient flows, and representation learning (UMAP/t-SNE, geometric deep learning, Neural ODEs/continuous normalizing flows). It covers unbalanced OT / entropy–transport, submetry / Riemannian submersion cases, and provides a validation specification (geometry, energies, audit schema) to enable reproducible empirical studies of quotient-first learning. Keywords: optimal transport; Wasserstein-2; gradient flows; EVI; minimizing movement; JKO; quotient metric; observation map; pushforward; inf-projection; entropy–transport; sliced Wasserstein; Sinkhorn; representation learning; dimension reduction; Lipschitz coarsening; auditable algorithms; Base–Fiber bound; selector radius; Radon/Polish spaces.