Flow Matching under Noisy Latent Structure: Beyond Exact Low-Dimensional Support
Abstract
Flow Matching (FM) learns a velocity field whose ODE transports a simple source distribution to a target law. Existing finite-sample theory largely treats ambient-space regularity or data supported exactly on low-dimensional sets. We study linear FM under a noisy latent-generator model, where a low-dimensional Hölder map is perturbed by nondegenerate ambient Gaussian noise, so the target law is full-dimensional despite its latent structure. We construct a spatially regular ReLU velocity class and establish non-asymptotic high-probability approximation and estimation bounds whose leading sample-size exponent is governed by the latent dimension rather than the ambient dimension, with ambient and noise dependence kept explicit. Fixed positive target noise keeps the interpolation nondegenerate over the full time interval. The same spatial regularity propagates the learned velocity error through the transport ODE, yielding a corresponding Wasserstein convergence guarantee. These results show that exact low-dimensional support is not necessary for Flow Matching to retain latent-dimensional statistical behavior.
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BibTeX
@misc{hao2026flow,
title = {{Flow Matching under Noisy Latent Structure: Beyond Exact Low-Dimensional Support}},
author = {Lifeng Hao and Shaolin Ji},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38918},
eprint = {2609.38918},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38918},
}
RIS
TY - UNPB
TI - Flow Matching under Noisy Latent Structure: Beyond Exact Low-Dimensional Support
AU - Lifeng Hao
AU - Shaolin Ji
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38918
UR - https://arxiv.org/abs/2609.38918
AB - Flow Matching (FM) learns a velocity field whose ODE transports a simple source distribution to a target law. Existing finite-sample theory largely treats ambient-space regularity or data supported exactly on low-dimensional sets. We study linear FM under a noisy latent-generator model, where a low-dimensional Hölder map is perturbed by nondegenerate ambient Gaussian noise, so the target law is full-dimensional despite its latent structure. We construct a spatially regular ReLU velocity class and establish non-asymptotic high-probability approximation and estimation bounds whose leading sample-size exponent is governed by the latent dimension rather than the ambient dimension, with ambient and noise dependence kept explicit. Fixed positive target noise keeps the interpolation nondegenerate over the full time interval. The same spatial regularity propagates the learned velocity error through the transport ODE, yielding a corresponding Wasserstein convergence guarantee. These results show that exact low-dimensional support is not necessary for Flow Matching to retain latent-dimensional statistical behavior.
ER -