Direct Product Flow Matching: Decoupling Radial and Angular Dynamics for Few-Shot Adaptation
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| Main Authors: | , , , , , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866914535134199808 |
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| author | Chen, Hongxu Wang, Yanghao Zhu, Bowei Li, Hongxiang Wang, Zhen Jiang, Ziqi Li, Lin Liu, Rui Chen, Long |
| author_facet | Chen, Hongxu Wang, Yanghao Zhu, Bowei Li, Hongxiang Wang, Zhen Jiang, Ziqi Li, Lin Liu, Rui Chen, Long |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_05054 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Direct Product Flow Matching: Decoupling Radial and Angular Dynamics for Few-Shot Adaptation Chen, Hongxu Wang, Yanghao Zhu, Bowei Li, Hongxiang Wang, Zhen Jiang, Ziqi Li, Lin Liu, Rui Chen, Long Computer Vision and Pattern Recognition Artificial Intelligence Machine Learning 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. |
| title | Direct Product Flow Matching: Decoupling Radial and Angular Dynamics for Few-Shot Adaptation |
| topic | Computer Vision and Pattern Recognition Artificial Intelligence Machine Learning |
| url | https://arxiv.org/abs/2605.05054 |