Local Manifold Identification with Latent Linear Models and OT Flows

Identifying manifold structure reveals hidden signals from datasets, removing noise. In this paper, we study the problem of identifying local manifold structure from data. Given a query point $y_0$ from a dataset, our goal is to recover the geometry of the data manifold in a neighborhood of $y_0$ using a \emph{pretrained} optimal transport flow from the reference to the data distribution. First, we prove that the Brenier optimal transport map preserves manifold structure: the preimage of an $m$-dimensional data manifold is itself an $m$-dimensional manifold in the reference space. Second, motivated by this result, we propose a latent variable model that maps a linear model through the transport flow. We prove that the linear approximation error is significantly reduced by the optimal transport map, leading to a tight fit of the non-linear data manifold. Third, noting the intractability of the resulting likelihood, we deploy denoising Fisher score estimation --- a recent development from simulation-based inference that learns the Fisher score over parameter--observation pairs --- to perform likelihood-based inference effectively. Experiments on both synthetic and real-world datasets demonstrate the effectiveness of the proposed method.