Coreset-Induced Conditional Velocity Flow Matching

2026-05-12

AI Summary

This work addresses the inefficiency in hierarchical flow matching when mapping from isotropic Gaussian noise to multimodal target velocity distributions. To overcome this limitation, the authors propose constructing a closed-form Gaussian mixture source distribution via entropy-regularized Sinkhorn coresets, replacing the conventional isotropic noise source. This approach eliminates the need for neural samplers and instead employs a lightweight correction flow to learn the residual mapping. Theoretical analysis demonstrates that the proposed source distribution yields a lower optimal transport cost compared to standard noise-based schemes. By integrating conditional velocity flow matching with this efficient source modeling strategy, the method achieves competitive few-step generation performance on MNIST, CIFAR-10, ImageNet-32, and CelebA-HQ benchmarks.

Abstract

We propose Coreset-Induced Conditional Velocity Flow Matching (CCVFM), a generative model that augments hierarchical rectified flow with a data-informed source distribution. Hierarchical flow matching models the full conditional velocity law in velocity space, but its inner flow is asked to transport isotropic Gaussian noise to a multimodal target velocity distribution from scratch. Our key observation is that this inner source can be replaced by a closed-form surrogate built from a coreset of the target. CCVFM first compresses the target into weighted atoms using an entropic Sinkhorn coreset and lifts them to a Gaussian mixture. The induced conditional velocity law is then a closed-form Gaussian mixture that can be sampled without a learned neural sampler. A lightweight correction flow, trained from this exact surrogate source, then refines the remaining surrogate-to-target residual rather than learning an entire noise-to-data map. We prove that the surrogate transport cost equals the target--surrogate Wasserstein gap under an explicit compression assumption, whereas the noise-source analogue has a dimension-scale lower bound. We further characterize the conditional second moment of the direct surrogate-source training target and show that its source-dependent excess is small when the surrogate conditional law is close to the true conditional velocity law in mean and covariance. Empirically, on MNIST, CIFAR-10, ImageNet-32, and CelebA-HQ, the proposed method reaches competitive few-step generation under matched architectures.

Problem

• flow matching
• conditional velocity
• generative modeling
• multimodal distribution
• Wasserstein distance

Innovation

• Coreset
• Conditional Velocity Flow Matching
• Rectified Flow
• Gaussian Mixture Surrogate
• Wasserstein Gap