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Entropy-Controlled Flow Matching

Chika Maduabuchi

TL;DR

Entropy-Controlled Flow Matching (ECFM) is proposed: a constrained variational principle over continuity-equation paths enforcing a global entropy-rate budget d/dt H(mu_t)>= -lambda, and admits a stochastic-control representation equivalent to a Schrodinger bridge with an explicit entropy multiplier.

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.

Entropy-Controlled Flow Matching

TL;DR

Entropy-Controlled Flow Matching (ECFM) is proposed: a constrained variational principle over continuity-equation paths enforcing a global entropy-rate budget d/dt H(mu_t)>= -lambda, and admits a stochastic-control representation equivalent to a Schrodinger bridge with an explicit entropy multiplier.

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.
Paper Structure (142 sections, 253 theorems, 923 equations, 1 figure, 1 algorithm)

This paper contains 142 sections, 253 theorems, 923 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Optimal transport as geometry, Schrödinger bridges as regularization.
  3. Our idea: entropy-controlled flow matching.
  4. Contributions.
  5. Relevance to vision generators.
  6. Practical enforcement and certification.
  7. Relation to Schrödinger bridges and entropic OT (what is new).
  8. Roadmap.
  9. Theory
  10. Setup and notation
  11. Ambient space.
  12. Time-indexed measures and velocities.
  13. Continuity equation.
  14. Entropy and information functionals.
  15. Admissible endpoints and paths.
  16. ...and 127 more sections

Key Result

Theorem 1.2.1

Assume $(\mu^\lambda,v^\lambda)$ solves eq:ECFM_primal and $\mu_t^\lambda=\rho_t^\lambda dx$ satisfies the regularity conditions in Apps. app:A--app:C. Then there exist $\phi^\lambda$ and $\eta^\lambda(t)\ge0$ such that:

Figures (1)

  • Figure : Primal--dual ECFM training with entropy-rate constraints

Theorems & Definitions (624)

  • Theorem 1.2.1: KKT optimality system (core conditions)
  • Theorem 1.2.2: Existence of an ECFM minimizer
  • Theorem 1.2.3: Uniqueness in strict convex regimes
  • proof : Proof sketch
  • Theorem 1.2.4: Equivalence: ECFM as an entropy-controlled Schrödinger bridge
  • proof : Proof sketch
  • Theorem 1.2.5: Entropy-controlled geodesics are entropic OT geodesics
  • proof : Proof sketch
  • Theorem 1.2.6: $\Gamma$-convergence to classical OT
  • proof : Proof sketch
  • ...and 614 more