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

2026-03-24

AI Summary

Existing methods for non-rigid 3D shape correspondence often struggle to balance accuracy and efficiency due to their neglect of spectral basis optimization and reliance on computationally expensive solvers. This work proposes the Advanced Functional Maps framework, which, for the first time, enables end-to-end unsupervised learning of spectral basis optimization and reveals its equivalence to spectral convolution. By introducing a learnable suppression function that jointly optimizes feature representations and the spectral basis, and integrating a heat diffusion module with an unsupervised loss, the method constructs a lightweight architecture that avoids complex solvers or auxiliary losses. The approach significantly outperforms existing techniques under challenging conditions such as non-isometric deformations and topological noise, while maintaining high computational efficiency.

Technology Category

• Machine Learning: Unsupervised & Self-Supervised Learning
• Search and Optimization: Non-convex Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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.

Problem

• shape matching
• spectral basis
• functional maps
• non-rigid 3D shapes
• computational efficiency

Innovation

• spectral basis learning
• functional maps
• unsupervised shape matching
• learnable basis functions
• spectral convolution