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Same Flow, Different Paths: Variance Reduction in Flow Matching

Research Multimodal & Generative

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TL;DR - This paper shows that flow-matching paths inducing the same objective can produce different stochastic-gradient variance and SGD convergence rates. It develops a constrained path-optimization method for reducing variance without changing the underlying flow-matching problem.

  • Derives near-tight SGD iteration-complexity bounds for a linear velocity model with one-dimensional Gaussian data and identifies an optimal linear path.
  • Generalizes path selection as a variance-minimization problem constrained to preserve the marginal distributions and velocity field.
  • Shows that unconstrained variance reduction can paradoxically slow convergence, making flow-preserving constraints essential.
  • Reformulates otherwise intractable constraints into sample-estimable ones and supports the theory on synthetic and real datasets.

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Same Flow, Different Paths: Variance Reduction in Flow Matching

arXiv cs.LG Alexander Tyurin 2026-09-15 arXiv:2609.17287
Public signals Semantic Scholar citations 0 · Semantic Scholar influential citations 0
Providers: Hugging Face · N/A OpenAlex · N/A Publisher · N/A Semantic Scholar · Citations 0 · Influential citations 0 X · N/A Fetched 2026-09-26 14:15:46.164108 UTC

TL;DR - This paper shows that flow-matching paths inducing the same objective can produce different stochastic-gradient variance and SGD convergence rates. It develops a constrained path-optimization method for reducing variance without changing the underlying flow-matching problem.

  • Derives near-tight SGD iteration-complexity bounds for a linear velocity model with one-dimensional Gaussian data and identifies an optimal linear path.
  • Generalizes path selection as a variance-minimization problem constrained to preserve the marginal distributions and velocity field.
  • Shows that unconstrained variance reduction can paradoxically slow convergence, making flow-preserving constraints essential.
  • Reformulates otherwise intractable constraints into sample-estimable ones and supports the theory on synthetic and real datasets.
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