Local Manifold Identification with Latent Linear Models and OT Flows

The manifold hypothesis states that high-dimensional data often concentrate near low-dimensional structures. Identifying these local manifold structures allows us to separate signals from ambient 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.