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

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 (\ie, radial and angular sub-manifolds). Under this new geometric view, we identify three overlooked limitations in them: \textit{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. \textit{2) Radial dynamics neglect}: Feature normalization discards modality confidence, failing to distinguish out-of-distribution and in-distribution data, and abandoning crucial radial dynamics. \textit{3) Context-agnostic unconditional flow}: Dataset-specific information loss during pre-trained cross-modal feature extraction remains unrecovered. To resolve these issues, we propose \textbf{warped product flow matching (WP-FM)}, a unified Riemannian framework that reformulates alignment on a warped product manifold. Within this framework, we derive \textbf{direct product flow matching (DP-FM)} by introducing a constant-warping metric, which yields a decoupled cylindrical manifold (\ie, 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.