Rectified Schrödinger Bridge Matching for Few-Step Visual Navigation

Wuyang Luan; Junhui Li; Weiguang Zhao; Wenjian Zhang; Tieru Wu; Rui Ma

cs.ROApr 7, 2026

Rectified Schrödinger Bridge Matching for Few-Step Visual Navigation

Authors: Wuyang Luan, Junhui Li, Weiguang Zhao, Wenjian Zhang, Tieru Wu, Rui Ma

Organizations: School of Artificial Intelligence, Jilin University, Changchun 130012, China · College of Computer Science, Chongqing University, Chongqing 400044, China · Department of Computer Science, University of Liverpool, Liverpool L69 3BX, UK · Changchun GenY Technology Co., Ltd., Changchun, China

Abstract

Visual navigation is a core challenge in Embodied AI, requiring autonomous agents to translate high-dimensional sensory observations into continuous, long-horizon action trajectories. While generative policies based on diffusion models and Schrödinger Bridges (SB) effectively capture multimodal action distributions, they require dozens of integration steps due to high-variance stochastic transport, posing a critical barrier for real-time robotic control. We propose Rectified Schrödinger Bridge Matching (RSBM), a framework that exploits a shared velocity-field structure between standard Schrödinger Bridges ( $\varepsilon=1$ , maximum-entropy transport) and deterministic Optimal Transport ( $\varepsilon\to 0$ , as in Conditional Flow Matching), controlled by a single entropic regularization parameter $\varepsilon$ . We prove two key results: (1) the conditional velocity field's functional form is invariant across the entire $\varepsilon$ -spectrum (Velocity Structure Invariance), enabling a single network to serve all regularization strengths; and (2) reducing $\varepsilon$ linearly decreases the conditional velocity variance, enabling more stable coarse-step ODE integration. Anchored to a learned conditional prior that shortens transport distance, RSBM operates at an intermediate $\varepsilon$ that balances multimodal coverage and path straightness. Empirically, while standard bridges require $\geq 10$ steps to converge, RSBM achieves over 94% cosine similarity and 92% success rate in merely 3 integration steps -- without distillation or multi-stage training -- substantially narrowing the gap between high-fidelity generative policies and the low-latency demands of Embodied AI.

Figures & tables

Fig. 1: Denoising progression on two toy trajectories (star patrol and figure-8 loop). At $t\!=\!T$ all methods start as unstructured point clouds (Gaussian noise for DDPM/FM; learned prior for NaviBridger/RSBM). By $k\!=\!2$ (NFE $=$ 3), only RSBM closely matches the GT, while baselines require $k\!\geq\!10$ to converge. At $k\!=\!30$ (NFE $=$ 59), all methods have converged, confirming the quantitative trends in Table III .

Fig. 2: Overview of the RSBM framework. Left : A dual-stream EfficientNet-B0 vision encoder $f_{\phi}$ (§ III-A ) extracts observation and goal features, which are fused via positional encoding and self-attention into a context vector $\mathbf{c}\in\mathbb{R}^{256}$ . Center : A learned variational prior network $g_{\psi}$ (§ III-A ) produces a coarse action prior $\mathbf{a}_{T}$ . Right : A conditional U-Net 1D velocity network $\mathbf{v}_{\theta}$ (§ III-C ) with FiLM conditioning iteratively refines $\mathbf{a}_{T}$ into the output trajectory $\mathbf{a}_{0}$ via an $\varepsilon$ -Rectified Schrödinger Bridge (§ III-B – III-D ). The three trajectory snapshots illustrate the progressive refinement: from scattered prior waypoints (green), through partially converged intermediate states (orange), to the final high-fidelity trajectory (red) closely matching the ground truth (dashed gray).

			Custom Indoor					CitySim (Outdoor)
Method	$k$	NFE	MSE $\downarrow$	CosSim $\uparrow$	FDE $\downarrow$	Col.% $\downarrow$	Suc.% $\uparrow$	MSE $\downarrow$	CosSim $\uparrow$	FDE $\downarrow$	Col.% $\downarrow$	Suc.% $\uparrow$
ViNT [ 9 ] (1-shot)	1	1	6.50	0.720	2.85	1.58	28	8.20	0.650	4.50	0.41	38
NoMaD [ 19 ]	20	20	3.60	0.820	1.95	1.32	32	5.80	0.740	3.20	0.34	52
NoMaD [ 19 ]	3	6	8.40	0.610	3.75	2.25	18	10.50	0.540	5.20	0.55	22
DDPM [ 20 ]	50	50	3.80	0.820	2.05	0.98	64	5.50	0.750	3.10	0.35	50
DDPM [ 20 ]	3	6	14.80	0.320	6.10	3.60	6	16.20	0.280	7.80	0.72	4

TABLE I: Comprehensive comparison. Each method is shown at its default

k

and at

k\!=\!3

, revealing degradation under zero-shot step reduction. Gray rows :

k\!=\!3

variants. RSBM is shown at both

k\!=\!3

and

k\!=\!10

to demonstrate early saturation. Bold : best overall per column.

Fig. 3: Effect of $\varepsilon$ . $\varepsilon=0.5$ (red) provides stable low- $k$ performance.

Fig. 4: Quality–cost Pareto frontier. Each marker represents a method at a given sampling budget ( $k$ ). (a) CosSim vs. NFE; (b) Success Rate vs. NFE. RSBM at $k\!=\!3$ (NFE $\!=\!5$ ) lies on the favorable frontier region, providing strong quality at substantially lower evaluations.

		Action MSE $\downarrow$						CosSim $\uparrow$
Method	$k$	HuRoN	Recon	SACSoN	SCAND	GoStan.	Avg	HuRoN	Recon	SACSoN	SCAND	GoStan.	Avg
DDPM [ 20 ]	10	1.38	1.19	1.74	0.72	4.12	1.83	0.880	0.900	0.860	0.920	0.780	0.868
DDPM [ 20 ]	3	8.50	6.20	9.80	4.60	15.30	8.88	0.350	0.420	0.280	0.480	0.210	0.348
FM [ 39 ]	10	1.10	0.97	1.55	0.65	3.65	1.58	0.910	0.920	0.885	0.935	0.820	0.894
FM [ 39 ]	3	3.20	2.60	4.10	1.80	7.50	3.84	0.720	0.740	0.670	0.790	0.560	0.696
NaviBridger [ 31 ]	10	0.27	0.88	1.43	0.59	3.22	1.28	0.955	0.935	0.920	0.960	0.875	0.929

TABLE II: Per-dataset generalization. Action MSE

\downarrow

and CosSim

\uparrow

across five diverse real-world datasets. RSBM(

k\!=\!3

) consistently matches or exceeds NaviBridger(

k\!=\!10

); NaviBridger(

k\!=\!3

) degrades significantly across all domains. Bold : best per column among

k\!=\!3

methods.

	$k\!=\!3$ (NFE=5)			$k\!=\!10$ (NFE=19)
Target	MSE $\downarrow$	CosSim $\uparrow$	FDE $\downarrow$	MSE $\downarrow$	CosSim $\uparrow$	FDE $\downarrow$
$\epsilon$ -pred	3.50	0.895	1.42	1.95	0.940	0.88
$x_{0}$ -pred	2.95	0.920	1.15	1.80	0.946	0.84
$v$ -pred (Ours)	1.90	0.945	0.80	1.72	0.949	0.78

TABLE III: Prediction target ablation (RSBM,

\varepsilon\!=\!0.5

, Custom Indoor). Bold : best per column.

Fig. 5: Qualitative trajectory comparison across eight challenging scenarios ( $2\!\times\!4$ grid, $k\!=\!3$ , NFE $\!=\!5$ ). Top row : four indoor/structured environments. Bottom row : four large-scale environments. Baselines collide early (×); faint dotted lines show invalid ghost continuations. RSBM (green) remains collision-free and closely tracks the ground truth (dashed gray).

Method	$k$	MSE $\downarrow$	CosSim $\uparrow$	Suc.% $\uparrow$
ViNT (1-shot)	1	$6.50\pm 0.15$	$0.720\pm 0.005$	$28\pm 2$
NoMaD	20	$3.60\pm 0.12$	$0.820\pm 0.004$	$32\pm 3$
DDPM	50	$3.80\pm 0.14$	$0.820\pm 0.005$	$64\pm 3$
FM	10	$2.80\pm 0.10$	$0.910\pm 0.003$	$82\pm 2$
NaviBridger ( $k$ =10)	10	$1.82\pm 0.08$	$0.942\pm 0.003$	$88\pm 2$
NaviBridger ( $k$ =3)	3	$12.00\pm 0.45$	$0.710\pm 0.015$	$35\pm 4$

TABLE IV: Mean

\pm

std over 3 random seeds (Custom Indoor). Bold : best per column.

Action MSE $\downarrow$
Method	$k$	HuRoN	Recon	SACSoN	SCAND	GoStan.	Avg
DDPM	3	8.50	6.20	9.80	4.60	15.30	8.88
FM	3	3.20	2.60	4.10	1.80	7.50	3.84
NaviBridger	10	0.27	0.88	1.43	0.59	3.22	1.28
NaviBridger	3	2.80	3.45	5.20	2.15	8.50	4.42
RSBM	3	0.25	0.82	1.35	0.48	3.05	1.19

TABLE V: Per-dataset comparison (

k\!=\!3

unless noted). Bold : best.

Solver	$k$	NFE	MSE $\downarrow$	CosSim $\uparrow$	FDE $\downarrow$	Suc.% $\uparrow$
Euler (1st)	3	3	2.45	0.928	1.15	82
Euler	5	5	2.05	0.938	0.95	86
Euler	10	10	1.80	0.947	0.82	90
Heun (2nd)	3	5	1.90	0.945	0.80	92
Heun	5	9	1.78	0.948	0.76	93
Heun	10	19	1.72	0.949	0.74	93

TABLE VI: Solver ablation (RSBM,

\varepsilon\!=\!0.5

, Custom Indoor). Bold : best per column.

Configuration	MSE $\downarrow$	CosSim $\uparrow$	FDE $\downarrow$	Suc.% $\uparrow$
Prior $g_{\psi}$ only (no bridge)	5.80	0.780	2.60	45
Gauss. + RSBM ( $\varepsilon\!=\!0.5$ )	4.20	0.860	1.95	62
Gauss. + Std. SB ( $\varepsilon\!=\!1.0$ )	12.00	0.710	4.20	35
Prior + Std. SB ( $\varepsilon\!=\!1.0$ )	5.50	0.810	2.35	52
Prior + RSBM ( $\varepsilon\!=\!0.5$ )	1.90	0.945	0.80	92

TABLE VII: Prior initialization ablation (Custom Indoor,

k\!=\!3

Schrödinger Bridges

Schrödinger bridges are stochastic transports that match prescribed distributions while staying close to a reference process.

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Diffusion Policy Robot Motion Generation Vision-Language-Action Models Language-Conditioned Robot Control Robot Policy Generalization

Open paper AskPDF arXiv

Bridging high-level semantic understanding with low-level physical control remains a persistent challenge in embodied intelligence, stemming from the fundamental spatiotemporal scale mismatch between cognition and action. Existing generative VLA policies typically adopt a "Generation-from-Noise" paradigm, which disregards this disparity, leading to representation inefficiency and weak condition alignment during optimization. In this work, we propose ResVLA, an architecture that shifts the paradigm to "Refinement-from-Intent." Recognizing that robotic motion naturally decomposes into global intent and local dynamics, ResVLA utilizes spectral analysis to decouple control into a deterministic low-frequency anchor and a stochastic high-frequency residual. By anchoring the generative process on the predicted intent, our model focuses strictly on refining local dynamics via a residual diffusion bridge. Extensive simulation experiments show that ResVLA achieves competitive performance, strong robustness to language and robot embodiment perturbations, and faster convergence than standard generative baselines. ResVLA also demonstrates strong performance in real-world robot experiments.

Abstract

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