On the continuity of flows

• URL: http://arxiv.org/abs/2512.12821v1
• Date: Sun, 14 Dec 2025 20:00:39 GMT
• Title: On the continuity of flows
• Authors: Congzhou M Sha,
• Abstract summary: We show that the optimal velocity field under flow matching objectives may exhibit spatial discontinuities.<n>We argue that this discontinuity arises from the requirement that continuous flows must bifurcate to map a single mode to multiple modes.<n>Our analysis suggests that this phenomenon is not specific to $L2$ loss, but rather may be a consequence of topological mismatch between distributions.
• Score: 0.10152838128195464
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Flow matching has emerged as a powerful framework for generative modeling through continuous normalizing flows. We investigate a potential topological constraint: when the prior distribution and target distribution have mismatched topology (e.g., unimodal to multimodal), the optimal velocity field under standard flow matching objectives may exhibit spatial discontinuities. We suggest that this discontinuity arises from the requirement that continuous flows must bifurcate to map a single mode to multiple modes, forcing particles to make discrete routing decisions at intermediate times. Through theoretical analysis on bimodal Gaussian mixtures, we demonstrate that the optimal velocity field exhibits jump discontinuities along decision boundaries, with magnitude approaching infinity as time approaches the target distribution. Our analysis suggests that this phenomenon is not specific to $L^2$ loss, but rather may be a consequence of topological mismatch between distributions. We validate our theory empirically and discuss potential implications for flow matching on manifolds, connecting our findings to recent work on Riemannian flow matching and the challenge of learning discontinuous representations in neural networks.

Related papers

• Foundations of Diffusion Models in General State Spaces: A Self-Contained Introduction [54.95522167029998]
  This article is a self-contained primer on diffusion over general state spaces.<n>We develop the discrete-time view (forward noising via Markov kernels and learned reverse dynamics) alongside its continuous-time limits.<n>A common variational treatment yields the ELBO that underpins standard training losses.
  arXiv  Detail & Related papers  (2025-12-04T18:55:36Z)
• Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching [60.37045080890305]
  We provide the first analysis of the sample complexity for flow-matching based generative models.<n>We decompose the velocity field estimation error into neural-network approximation error, statistical error due to the finite sample size, and optimization error due to the finite number of optimization steps for estimating the velocity field.
  arXiv  Detail & Related papers  (2025-12-01T05:14:25Z)
• Error Analysis of Discrete Flow with Generator Matching [34.847768962128164]
  We develop a unified framework grounded in calculus theory to investigate the theoretical properties of discrete flow models.<n>Specifically, we derive the KL divergence of two path measures regarding two continuous-time Markov chains with different transition rates.<n>We provide a comprehensive analysis that encompasses the error arising from transition rate estimation and early stopping.
  arXiv  Detail & Related papers  (2025-09-26T05:41:45Z)
• Provable Maximum Entropy Manifold Exploration via Diffusion Models [58.89696361871563]
  Exploration is critical for solving real-world decision-making problems such as scientific discovery.<n>We introduce a novel framework that casts exploration as entropy over approximate data manifold implicitly defined by a pre-trained diffusion model.<n>We develop an algorithm based on mirror descent that solves the exploration problem as sequential fine-tuning of a pre-trained diffusion model.
  arXiv  Detail & Related papers  (2025-06-18T11:59:15Z)
• Theoretical Guarantees for High Order Trajectory Refinement in Generative Flows [40.884514919698596]
  Flow matching has emerged as a powerful framework for generative modeling.<n>We prove that higher-order flow matching preserves worst case optimality as a distribution estimator.
  arXiv  Detail & Related papers  (2025-03-12T05:07:07Z)
• On the Wasserstein Convergence and Straightness of Rectified Flow [54.580605276017096]
  Rectified Flow (RF) is a generative model that aims to learn straight flow trajectories from noise to data.<n>We provide a theoretical analysis of the Wasserstein distance between the sampling distribution of RF and the target distribution.<n>We present general conditions guaranteeing uniqueness and straightness of 1-RF, which is in line with previous empirical findings.
  arXiv  Detail & Related papers  (2024-10-19T02:36:11Z)
• Solving Prior Distribution Mismatch in Diffusion Models via Optimal Transport [24.90486913773359]
  In recent years, the knowledge surrounding diffusion models(DMs) has grown significantly, though several theoretical gaps remain. This paper explores the deeper relationship between optimal transport(OT) theory and DMs with discrete initial distribution. We prove that as the diffusion termination time increases, the probability flow exponentially converges to the gradient of the solution to the classical Monge-Ampere equation.
  arXiv  Detail & Related papers  (2024-10-17T10:54:55Z)
• Convergence of Score-Based Discrete Diffusion Models: A Discrete-Time Analysis [56.442307356162864]
  We study the theoretical aspects of score-based discrete diffusion models under the Continuous Time Markov Chain (CTMC) framework.<n>We introduce a discrete-time sampling algorithm in the general state space $[S]d$ that utilizes score estimators at predefined time points.<n>Our convergence analysis employs a Girsanov-based method and establishes key properties of the discrete score function.
  arXiv  Detail & Related papers  (2024-10-03T09:07:13Z)
• Gaussian Interpolation Flows [11.340847429991525]
  This work investigates the well-posedness of simulation-free continuous normalizing flows built on Gaussian denoising. We establish the Lipschitz regularity of the flow velocity field, the existence and uniqueness of the flow, and the continuity of the flow map. We also study the stability of these flows in source distributions and perturbations of the velocity field, using the quadratic Wasserstein distance as a metric.
  arXiv  Detail & Related papers  (2023-11-20T00:59:20Z)
• Building Normalizing Flows with Stochastic Interpolants [11.22149158986164]
  A simple generative quadratic model based on a continuous-time normalizing flow between any pair of base and target distributions is proposed. The velocity field of this flow is inferred from the probability current of a time-dependent distribution that interpolates between the base and the target in finite time.
  arXiv  Detail & Related papers  (2022-09-30T16:30:31Z)
• Convex Analysis of the Mean Field Langevin Dynamics [49.66486092259375]
  convergence rate analysis of the mean field Langevin dynamics is presented. $p_q$ associated with the dynamics allows us to develop a convergence theory parallel to classical results in convex optimization.
  arXiv  Detail & Related papers  (2022-01-25T17:13:56Z)
• Localisation in quasiperiodic chains: a theory based on convergence of local propagators [68.8204255655161]
  We present a theory of localisation in quasiperiodic chains with nearest-neighbour hoppings, based on the convergence of local propagators. Analysing the convergence of these continued fractions, localisation or its absence can be determined, yielding in turn the critical points and mobility edges. Results are exemplified by analysing the theory for three quasiperiodic models covering a range of behaviour.
  arXiv  Detail & Related papers  (2021-02-18T16:19:52Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.