Observation Quotients and Learning-as-Lifting
Takahashi, K · Zenodo (CERN European Organization for Nuclear Research) · 2025
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)(\mathsf X,d_{\mathsf X})(X,dX) and a Borel map O:X → ZO:\mathsf X\!\to\!\mathsf ZO:X→Z, we endow the image with the induced quotient metric dZd_{\mathsf Z}dZ (or a declared surrogate metric d^Z\hat d_{\mathsf Z}d^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: λ\lambdaλ-EVI gradient flows of FFF on (P2(X),W2,X)(\mathcal P_2(\mathsf X),W_{2,\mathsf X})(P2(X),W2,X) push forward to relaxed/exact λ\lambdaλ-EVI flows of GGG on (P2(Z),W2,Z)(\mathcal P_2(\mathsf Z),W_{2,\mathsf 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)(\epsilon,\delta)(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])(\epsilon,\delta,\widehat{\mathbb E}[r_{\rm sel}])(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.