Entropy-Controlled Flow Matching

TL;DR

This paper introduces Entropy-Controlled Flow Matching (ECFM), a novel method that enforces entropy constraints in flow-based generative models to improve mode coverage and semantic preservation.

Contribution

ECFM formulates a convex optimization in Wasserstein space with entropy constraints, providing theoretical guarantees and a stochastic-control interpretation for flow matching.

Findings

ECFM recovers entropic OT geodesics in the pure transport regime.

Provides mode-coverage and density-floor guarantees with Lipschitz stability.

Constructs near-optimal collapse counterexamples for unconstrained flow matching.

Abstract

Modern vision generators transport a base distribution to data through time-indexed measures, implemented as deterministic flows (ODEs) or stochastic diffusions (SDEs). Despite strong empirical performance, standard flow-matching objectives do not directly control the information geometry of the trajectory, allowing low-entropy bottlenecks that can transiently deplete semantic modes. We propose Entropy-Controlled Flow Matching (ECFM): a constrained variational principle over continuity-equation paths enforcing a global entropy-rate budget d/dt H(mu_t) >= -lambda. ECFM is a convex optimization in Wasserstein space with a KKT/Pontryagin system, and admits a stochastic-control representation equivalent to a Schrodinger bridge with an explicit entropy multiplier. In the pure transport regime, ECFM recovers entropic OT geodesics and Gamma-converges to classical OT as lambda -> 0. We further obtain certificate-style mode-coverage and density-floor guarantees with Lipschitz stability, and construct near-optimal collapse counterexamples for unconstrained flow matching.