Direct Product Flow Matching: Decoupling Radial and Angular Dynamics for Few-Shot Adaptation

• URL: http://arxiv.org/abs/2605.05054v1
• Date: Wed, 06 May 2026 15:51:26 GMT
• Title: Direct Product Flow Matching: Decoupling Radial and Angular Dynamics for Few-Shot Adaptation
• Abstract summary: Recent flow matching (FM) methods improve the few-shot adaptation of vision-language models.<n>We argue that existing FM methods are inherently constrained by incompatible geometric priors on pre-trained cross-modal features.<n>We propose warped product flow matching (WP-FM) to reformulate alignment on a warped product manifold.
• Score: 28.087854592514162
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recent flow matching (FM) methods improve the few-shot adaptation of vision-language models, by modeling cross-modal alignment as a continuous multi-step flow. In this paper, we argue that existing FM methods are inherently constrained by incompatible geometric priors on pre-trained cross-modal features, resulting in suboptimal adaptation performance. We first analyze these methods from a polar decomposition perspective (i.e., radial and angular sub-manifolds). Under this new geometric view, we identify three overlooked limitations in them: 1) Angular dynamics distortion: The radial-angular coupling induces non-uniform speed on the angular sub-manifold, leading to regression training difficulty and extra truncation errors. 2) Radial dynamics neglect: Feature normalization discards modality confidence, failing to distinguish out-of-distribution and in-distribution data, and abandoning crucial radial dynamics. 3) Context-agnostic unconditional flow: Dataset-specific information loss during pre-trained cross-modal feature extraction remains unrecovered. To resolve these issues, we propose warped product flow matching (WP-FM), a unified Riemannian framework that reformulates alignment on a warped product manifold. Within this framework, we derive direct product flow matching (DP-FM) by introducing a constant-warping metric, which yields a decoupled cylindrical manifold (i.e., direct product manifold). DP-FM enables independent radial evolution and constant-speed angular geodesic transport, effectively eliminating angular dynamics distortion while preserving radial consistency. Meanwhile, we incorporate classifier-free guidance by conditioning the flow on the pre-trained VLMs' hidden states to inject missing dataset-specific information. Extensive results across 11 benchmarks have demonstrated that DP-FM achieves a new state-of-the-art for multi-step few-shot adaptation.

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