Path-Decoupled Hyperbolic Flow Matching for Few-Shot Adaptation

Lin Li; Ziqi Jiang; Gefan Ye; Zhenqi He; Jiahui Li; Jun Xiao; Kwang-Ting Cheng; Long Chen

cs.CVFeb 24, 2026

Path-Decoupled Hyperbolic Flow Matching for Few-Shot Adaptation

Authors: Lin Li, Ziqi Jiang, Gefan Ye, Zhenqi He, Jiahui Li, Jun Xiao, Kwang-Ting Cheng, Long Chen

Organizations: The Hong Kong University of Science and Technology (HKUST) · Zhejiang University

Abstract

Recent advances in cross-modal few-shot adaptation treat visual-semantic alignment as a continuous feature transport problem via Flow Matching (FM). However, we argue that Euclidean-based FM overlooks fundamental limitations of flat geometry, where polynomial volume growth fails to accommodate diverse feature distributions, leading to severe path entanglement. To this end, we propose path-decoupled Hyperbolic Flow Matching (HFM), leveraging the Lorentz manifold's exponential expansion for trajectory decoupling. HFM structures the transport via two key designs: 1) Centripetal hyperbolic alignment: It constructs a centripetal hierarchy by anchoring textual roots, which pushes visual leaves to the boundary to initialize orderly flows. 2) Path-decoupled objective: It acts as a ``semantic guardrail'' rigidly confining trajectories within isolated class-specific geodesic corridors via step-wise supervision. Furthermore, we devise an adaptive diameter-based stopping to prevent over-transportation into the crowded origin based on the intrinsic semantic scale. Extensive ablations on 11 benchmarks have shown that HFM establishes a new state-of-the-art, consistently outperforming its Euclidean counterparts. Our codes and models will be released.

Figures & tables

Panel of Figure 1 — Figure 1: Illustration of Path Entanglement. (a) Euclidean Flow Matching suffers from severe trajectory collisions ( e.g . , “cat” intersecting “tiger” and “dog” merging with “cat”) due to the limited polynomial capacity of flat geometry. (b) Hyperbolic Flow Matching uses exponential volume expansion to achieve path decoupling.

Panel of Figure 2 — Figure 2: The overview of HFM. (a) Constructing Centripetal Hyperbolic Space : Establish a centripetal cross-modal hierarchy, optimizing textual roots near the origin and visual leaves toward the boundary. (b) Learning Path-Decoupled Flows : Tangent velocity fields $\mathcal{F}_{\theta}$ are optimized to guide features along isolated geodesic corridors. The step-wise consistency enforces trajectory decoupling in hyperbolic space. (c) Inference with Diameter-based Stopping : The flow terminates at $t^{*}$ once the distance to text prototypes drops below a dynamic threshold scaled by semantic diameter $d_{\text{txt}}$ to alleviate over-transportation into the crowded origin.

		Difficult						Easy
Shots	Method	Aircraft	SAT	DTD	SUN	UCF	Avg	Cars	Net	Flowers	Food	Pets	Caltech	Avg
0	CLIP 2021	24.8	47.8	43.8	62.5	66.7	47.6	65.5	66.7	67.4	85.3	89.1	92.9	77.7
1	CoOp 2022b	20.8	56.4	50.1	67.0	71.2	53.1	67.5	65.7	78.3	84.3	90.2	92.5	79.8
	CoCoOp 2022a	28.1	55.4	52.6	68.7	70.4	55.0	67.6	69.4	73.4	84.9	91.9	94.1	80.2
	TIP-Adapter 2022	28.8	67.8	51.6	67.2	73.4	57.8	67.1	69.4	83.8	85.8	90.6	94.0	81.8
	CLIP-Adapter 2024	25.2	49.3	44.2	65.4	66.9	50.2	65.7	67.9	71.3	86.1	89.0	92.0	78.7

Table 1: Quantitative results. Performance comparison (§ 4.1 ) of the state-of-the-art approaches. Based on CLIP-LoRA, we add HFM to further improve the performance. Top-1 accuracy averaged over 3 random seeds is reported. The highest value of each dataset is bolded.

	Components			Difficult Datasets						Easy Datasets
Shots	CHA	PO	DS	FGVC	EuroSAT	DTD	SUN	UCF	Avg	Cars	Net	Flowers	Food	Pets	Caltech	Avg
4	✗	✗	✗	38.8	83.5	64.0	72.8	81.1	68.0	77.4	71.4	92.9	82.6	90.6	95.0	85.0
	✓	✗	✗	38.9	88.4	65.9	74.0	81.8	69.8	77.6	71.4	94.4	86.7	92.6	95.7	86.4
	✓	✓	✗	43.5	88.6	68.0	75.4	83.6	71.8	80.5	71.9	96.4	86.8	92.7	95.8	87.3
	✓	✓	✓	43.5	90.3	68.6	75.5	83.6	72.3	80.1	72.1	96.2	86.9	93.2	96.2	87.4
16	✗	✗	✗	54.7	90.7	73.0	76.0	86.2	76.1	86.0	73.4	97.9	84.2	91.6	96.1	88.2

Table 2: Ablation Study. Analysis of component effectiveness (§ 4.2 ) under 4/16-shot settings. CHA : C entripetal H yperbolic A lignment to construct the centripetal hierarchy, PO : flow matching with the P ath-decoupled O bjective, DS : inference with D iameter-based S topping.

	Difficult						Easy
Method	Aircraft	SAT	DTD	SUN	UCF	Avg	Cars	Net	Flowers	Food	Pets	Caltech	Avg
CoOp 2022b	43.3	86.0	70.0	74.9	83.1	71.4	83.1	71.4	97.2	84.4	91.1	95.5	87.1
+FMA 2025	47.6	88.1	73.1	75.9	84.4	73.8 $+$ 2.4	85.4	72.5	98.2	85.0	91.4	95.7	88.0 $+$ 0.9
+ HFM (ours)	49.2	88.9	75.1	76.4	84.7	74.9 $+$ 3.5	84.9	72.2	98.2	86.6	93.8	96.3	88.7 $+$ 1.6
CoCoOp 2022a	33.8	75.5	65.8	72.8	76.0	64.8	72.4	71.1	87.1	87.4	93.2	95.2	84.4
+FMA 2025	36.9	86.9	71.9	73.4	80.3	69.9 $+$ 5.1	73.5	71.9	94.5	87.8	93.4	95.6	86.1 $+$ 1.7

Table 3: Ablation Study. Analysis of model-agnostic capability (§ 4.2 ) across various PEFT approaches under the 16-shot setting.

			Difficult						Easy
Shots	Backbone	Method	Aircraft	SAT	DTD	SUN	UCF	Avg	Cars	Net	Flowers	Food	Pets	Caltech	Avg
4	ViT-B32	CLIP-LoRA 2024	27.7	85.6	60.3	70.3	76.5	64.1	68.3	66.5	90.1	75.6	86.3	94.3	80.2
	ViT-B32	+ HFM (ours)	33.9	90.1	64.5	72.7	80.1	68.3 $+$ 4.2	71.6	66.7	93.1	80.7	90.6	95.4	83.0 $+$ 2.8
	ViT-B16	CLIP-LoRA 2024	38.8	83.5	64.0	72.8	81.1	68.0	77.4	71.4	92.9	82.6	90.6	95.0	85.0
	ViT-B16	+ HFM (ours)	43.5	90.3	68.6	75.5	83.6	72.3 $+$ 4.3	80.1	72.1	96.2	86.9	93.2	96.2	87.4 $+$ 2.4
	ViT-L14	CLIP-LoRA 2024	48.9	86.4	70.4	76.7	86.4	73.8	85.2	77.9	97.4	89.6	93.9	97.2	90.2

Table 4: Ablation study. Performance comparison (§ 4.2 ) between CLIP-LoRA and our HFM on different backbones (ViT-B32, ViT-B16, and ViT-L14) with 4 and 16 shots. Top-1 accuracy is reported. The highest value of each setting is bolded.

Figure 3: Qualitative Results. Visualization of transport trajectories (§ 4.3 ). Top: Euclidean flows suffer from severe path entanglement , exhibiting chaotic and intersecting paths due to spatial crowding. Bottom: HFM achieves path decoupling via a centripetal hierarchy . Visual features move centripetally from the boundary to central text roots along isolated geodesic corridors.

VLM Adaptation

Vision-language model adaptation tailors pretrained image-text and multimodal models to new tasks and domains while retaining transferable capabilities.

See all 545 papers

Direct Product Flow Matching: Decoupling Radial and Angular Dynamics for Few-Shot Adaptation
May 6, 2026Hongxu Chen, Yanghao Wang, Bowei Zhu +6Cross-Modal Learning Flow Matching
HKUST · USTC · Huawei Research
MC-RFM: Geometry-Aware Few-Shot Adaptation via Mixed-Curvature Riemannian Flow Matching
May 8, 2026Salim Khazem, Ibrahim Mohamed Serouis, Zakaria EzzahedFlow Matching Vision Foundation Model Adaptation
Talan Research Center, Paris, France
Smoother Flow Matching via Contrastive Trajectory Repulsion
Oct 1, 2026Ziqi Jiang, Zhenqi He, Long ChenFlow Matching Contrastive Flow Matching
Department of CSE, The Hong Kong University of Science and Technology

May 6, 2026J/K move · Enter open · S save

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

Hongxu Chen, Yanghao Wang, Bowei Zhu, Hongxiang Li+5

HKUST · USTC · Huawei Research

Cross-Modal Learning Flow Matching VLM Adaptation Cross-Modal Alignment

Open paper AskPDF arXiv

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.

Abstract

VLM Adaptation

Explore similar work

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

MC-RFM: Geometry-Aware Few-Shot Adaptation via Mixed-Curvature Riemannian Flow Matching

Smoother Flow Matching via Contrastive Trajectory Repulsion