DeepShapeMatchingKit: Accelerated Functional Map Solver and Shape Matching Pipelines Revisited

Yizheng Xie; Lennart Bastian; Congyue Deng; Thomas W. Mitchel; Maolin Gao; Daniel Cremers

cs.CVApr 11, 2026

DeepShapeMatchingKit: Accelerated Functional Map Solver and Shape Matching Pipelines Revisited

Authors: Yizheng Xie, Lennart Bastian, Congyue Deng, Thomas W. Mitchel, Maolin Gao, Daniel Cremers

Organizations: Technical University of Munich · Simon Fraser University · Munich Center for Machine Learning · Imperial College London ·

Abstract

Deep functional maps, leveraging learned feature extractors and spectral correspondence solvers, are fundamental to non-rigid 3D shape matching. Based on an analysis of open-source implementations, we find that standard functional map implementations solve k independent linear systems serially, which is a computational bottleneck at higher spectral resolution. We thus propose a vectorized reformulation that solves all systems in a single kernel call, achieving up to a 33x speedup while preserving the exact solution. Furthermore, we identify and document a previously unnoticed implementation divergence in the spatial gradient features of the mainstay DiffusionNet: two variants that parameterize distinct families of tangent-plane transformations, and present experiments analyzing their respective behaviors across diverse benchmarks. We additionally revisit overlap prediction evaluation for partial-to-partial matching and show that balanced accuracy provides a useful complementary metric under varying overlap ratios. To share these advancements with the wider community, we present an open-source codebase, DeepShapeMatchingKit, that incorporates these improvements and standardizes training, evaluation, and data pipelines for common deep shape matching methods. The codebase is available at: https://github.com/xieyizheng/DeepShapeMatchingKit

Figures & tables

Panel of Figure 1 — Figure 1: Runtime vs. spectral resolution. Comparison between existing loop-based functional map solvers [ 15 , 3 , 10 , 5 , 26 , 42 ] and our batched implementation. Our batched solver achieves a $33\times$ speedup while preserving exact solutions.

Frameworks	Fast Solver	Diff. (A)	Diff. (B)	F2F	P2F	P2P	IoU	Bal. Acc
PyFM [ 31 ]	✗	✗	✗	✓	✓	✗	✗	✗
GeomFUM [ 40 ]	✗	✗	✓	✓	✓	✗	✗	✗
ULRSSM [ 10 ]	✗	✗	✓	✓	✓	✗	✗	✗
Hybrid FMaps [ 5 ]	✗	✗	✓	✓	✗	✗	✗	✗
AttentiveFmaps [ 26 ]	✗	✓	✗	✓	✗	✗	✗	✗
DPFM [ 3 ]	✗	✓	✗	✓	✓	✓	✓	✗

Table 1: We present DeepShapeMatchingKit , an open-source code base for accelerated FM computation, supporting both DiffusionNet variants, full-to-full (F2F), partial-to-full (P2F), and partial-to-partial (P2P) matching, and complementary overlap prediction metrics.

Figure 2: Overview of common deep functional map pipelines. A shared pipeline underlies most deep shape matching methods: a feature backbone, e.g., DiffusionNet, extracts per-vertex descriptors, a functional map solver, and method-specific components produce the final matching. We revisit three parts of this pipeline: a batched functional map solver, an analysis of DiffusionNet variants, and balanced accuracy as a complementary overlap prediction metric.

Panel of Figure 4 — Figure 4: Illustration of two linear transform families in $\mathbb{R}^{2}$ . Left: Arbitrary rotation and isotropic scaling, used by [ 39 , 3 , 42 , 26 , 14 , 44 , 2 ] ; Right: fixed rotation at 45 degree angle and arbitrary anisotropic scaling used by [ 10 , 5 , 40 , 9 , 7 , 8 , 17 , 37 ] .

Figure 6: Metric behavior vs. overlap ratio. X–axis: fraction of truly overlapping vertices

r\in[0,1]

; Y–axis: metric value. We simulate three degenerate predictors: Zeros (predict no overlap), Ones (predict all overlap), and Random (each vertex predict overlap with prob.

0.5

). Most metrics are biased by the class prior

r

(e.g., Accuracy, Precision, IoU, F1), whereas Balanced Accuracy remains constant at

0.5

for all three baselines, reflecting predictive performance under class imbalance.

Implementation	ULRSSM	Hybrid ULRSSM	AttentiveFmaps	AttentiveFmaps-Fast	DPFM	EchoMatch
Original	429 ms	555 ms	2050 ms	578 ms	115 ms	215 ms
Ours	82 ms	363 ms	448 ms	416 ms	95 ms	195 ms
Speedup ( $\times$ )	5.23	1.53	4.57	1.39	1.21	1.10
Memory ( $+$ )	126 MB (8.5%)	30 MB (1.0%)	142 MB (5.6%)	76 MB (3.2%)	2 MB (0.1%)	2 MB (0.02%)

Table 2: Per-iteration runtime and memory comparison between the original implementations of multiple shape matching methods and using our batched solver on the same hardware. Additional GPU memory usage of our implementation is also reported. Original GPU memory consumption of the full learning pipelines ranges from approximately 1.5 to 8.4 GB across methods.

Panel of Figure 7 — Figure 7: Qualitative comparison of DiffusionNet variants on DT4D-H and TOPKIDS using ULRSSM. Formulation A and B exhibit complementary strengths at non-isometry vs topological noise challenges.

Geodesic Error ( $\times$ 100)	SMAL	DT4D-H		TOPKIDS
Geodesic Error ( $\times$ 100)		intra-class	inter-class
ULRSSM (B)	3.9	0.9	4.1	9.2
ULRSSM (A)	3.9	1.0	7.9	5.7
Hybrid ULRSSM (B)	3.3	1.0	3.5	4.6
Hybrid ULRSSM (A)	3.3	0.9	3.9	4.6

Table 3: Evaluation on non-isometric and topologically noisy datasets. Average geodesic error (

\times 100

) across standard benchmarks.

Train	FAUST			SCAPE			FAUST + SCAPE
Test	FAUST	SCAPE	SHREC’19	FAUST	SCAPE	SHREC’19	FAUST	SCAPE	SHREC’19
ULRSSM (B)	1.6	2.2	6.7	1.6	1.9	5.7	1.6	2.1	4.6
ULRSSM (A)	1.6	2.2	5.4	1.6	1.9	5.1	1.6	2.2	4.5
Hybrid ULRSSM (B)	1.4	2.1	5.5	1.4	1.8	5.4	1.5	2.0	3.6
Hybrid ULRSSM (A)	1.4	1.9	4.0	1.4	1.8	5.1	1.4	1.9	3.7

Table 4: Cross-dataset near-isometric evaluation on FAUST, SCAPE, and SHREC’19 datasets. Average geodesic error (

\times 100

) across standard benchmarks.

Train	CUTS		HOLES
Test	CUTS	HOLES	CUTS	HOLES
ULRSSM (B)	3.2	13.5	5.6	8.2
ULRSSM (A)	2.6	13.0	5.8	8.7

Table 5: Evaluation on partial-shape benchmark (SHREC’16). Geodesic error (

\times 100

) on partial-to-partial matching tasks.

Methods		CP2P24	PSMAL	BeCoS
EchoMatch (B)	XYZ	90.62 / 79.06	85.25 / 72.19	55.48 / 25.49
EchoMatch (B)	DINOv2	91.94 / 81.91	91.72 / 83.30	67.90 / 64.54
EchoMatch (A)	XYZ	91.03 / 80.10	85.99 / 72.71	60.13 / 52.40
EchoMatch (A)	DINOv2	93.53 / 84.72	92.75 / 84.75	67.05 / 64.68

Table 6: Balanced Accuracy / Mean IoU (

{\times 100}

) on partial-to-partial datasets. Each entry reports balanced accuracy / mean IoU , averaged across all test pairs.

Panel of Figure 8 — Figure 8: PCK curves with Balanced Accuracy / Mean IoU ( $\uparrow$ ) (legend values) for DPFM and EchoMatch on BeCoS with XYZ features. Despite lower IoU, DPFM achieves better geodesic accuracy in the low-error regime (0–0.25), a distinction reflected by balanced accuracy.

Methods		CP2P24	PSMAL	BeCoS
DPFM	XYZ	80.10 / 63.86	81.84 / 67.04	62.35 / 48.18
DPFM	DINOv2	89.38 / 74.15	87.31 / 73.67	65.10 / 51.02
EchoMatch	XYZ	91.03 / 80.10	85.99 / 72.71	60.13 / 52.40
EchoMatch	DINOv2	93.53 / 84.72	92.75 / 84.75	67.05 / 64.68

Table 7: Balanced Accuracy / Mean IoU (

{\times 100}

) on partial-to-partial datasets. Each entry reports balanced accuracy / mean IoU , averaged across all test pairs. EchoMatch leads on IoU across all settings; however, on BeCoS with XYZ input, DPFM achieves higher balanced accuracy, reflecting its stronger performance in the low-error regime.

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Mar 24, 2026J/K move · Enter open · S save

From Feature Learning to Spectral Basis Learning: A Unifying and Flexible Framework for Efficient and Robust Shape Matching

Feifan Luo, Hongyang Chen

College of Computer Science and Technology, Zhejiang University, China · Research Center for Computational Earth and Space Science, Zhejiang Lab, China

Functional Maps Non-Rigid Shape Correspondence Unsupervised Representation Learning

Open paper AskPDF arXiv

Shape matching is a fundamental task in computer graphics and vision, with deep functional maps becoming a prominent paradigm. However, existing methods primarily focus on learning informative feature representations by constraining pointwise and functional maps, while neglecting the optimization of the spectral basis-a critical component of the functional map pipeline. This oversight often leads to suboptimal matching results. Furthermore, many current approaches rely on conventional, time-consuming functional map solvers, incurring significant computational overhead. To bridge these gaps, we introduce Advanced Functional Maps, a framework that generalizes standard functional maps by replacing fixed basis functions with learnable ones, supported by rigorous theoretical guarantees. Specifically, the spectral basis is optimized through a set of learned inhibition functions. Building on this, we propose the first unsupervised spectral basis learning method for robust non-rigid 3D shape matching, enabling the joint, end-to-end optimization of feature extraction and basis functions. Our approach incorporates a novel heat diffusion module and an unsupervised loss function, alongside a streamlined architecture that bypasses expensive solvers and auxiliary losses. Extensive experiments demonstrate that our method significantly outperforms state-of-the-art feature-learning approaches, particularly in challenging non-isometric and topological noise scenarios, while maintaining high efficiency. Finally, we reveal that optimizing basis functions is equivalent to spectral convolution, where inhibition functions act as filters. This insight enables enhanced representations inspired by spectral graph networks, opening new avenues for future research. Our code is available at https://github.com/LuoFeifan77/Unsupervised-Spectral-Basis-Learning.

Abstract

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