Entropy-Controlled Flow Matching
arXiv:2602.22265v1 Announce Type: new 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-f
arXiv:2602.22265v1 Announce Type: new 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.
Executive Summary
This article proposes Entropy-Controlled Flow Matching (ECFM), a novel approach to vision generators that addresses a critical limitation in standard flow-matching objectives. ECFM introduces a global entropy-rate budget to control the information geometry of the trajectory, preventing low-entropy bottlenecks that can deplete semantic modes. The method is based on a constrained variational principle over continuity-equation paths, which is formulated as a convex optimization in Wasserstein space. ECFM is shown to recover entropic OT geodesics and Gamma-converge to classical OT as the entropy multiplier approaches zero. The article also provides guarantees on mode-coverage and density-floor, as well as a near-optimal collapse counterexample for unconstrained flow matching. ECFM has significant implications for the development of more efficient and effective vision generators.
Key Points
- ▸ Entropy-Controlled Flow Matching (ECFM) addresses the limitation of standard flow-matching objectives in controlling the information geometry of the trajectory.
- ▸ ECFM introduces a global entropy-rate budget to prevent low-entropy bottlenecks that can deplete semantic modes.
- ▸ The method is formulated as a convex optimization in Wasserstein space and is shown to recover entropic OT geodesics and Gamma-converge to classical OT as the entropy multiplier approaches zero.
Merits
Strength
ECFM provides a more robust and effective approach to vision generators by controlling the information geometry of the trajectory, which can prevent low-entropy bottlenecks that can deplete semantic modes.
Convex optimization
The formulation of ECFM as a convex optimization in Wasserstein space ensures that the method is computationally efficient and scalable.
Guarantees on mode-coverage and density-floor
The article provides guarantees on mode-coverage and density-floor, which is a significant step towards ensuring the quality and robustness of vision generators.
Demerits
Limitation
The method requires the specification of an entropy multiplier, which can be challenging to set in practice.
Computational complexity
The convex optimization formulation of ECFM can be computationally intensive, especially for large-scale problems.
Interpretability
The article does not provide a clear interpretation of the entropy multiplier and its relationship to the semantic modes of the vision generator.
Expert Commentary
The article proposes a novel approach to vision generators that addresses a critical limitation in standard flow-matching objectives. The method is based on a constrained variational principle over continuity-equation paths, which is formulated as a convex optimization in Wasserstein space. ECFM is shown to recover entropic OT geodesics and Gamma-converge to classical OT as the entropy multiplier approaches zero. The article also provides guarantees on mode-coverage and density-floor, as well as a near-optimal collapse counterexample for unconstrained flow matching. While the method requires the specification of an entropy multiplier, which can be challenging to set in practice, the article provides a clear formulation of the method and its relationship to classical OT. The implications of ECFM are significant, and the method has the potential to revolutionize the development of vision generators.
Recommendations
- ✓ Recommendation 1: Further research is needed to investigate the computational complexity of ECFM and to develop more efficient algorithms for solving the convex optimization problem.
- ✓ Recommendation 2: The article should be expanded to provide a more detailed interpretation of the entropy multiplier and its relationship to the semantic modes of the vision generator.
