SVarM: Linear Support Varifold Machines for Classification and Regression on Geometric Data

2025-06-01

AI Summary

Statistical modeling and generalization for geometric data—such as curves, graphs, and surfaces—in non-Euclidean shape spaces are hindered by parameterization invariance. Method: This paper introduces the first extension of linear support vector machines to the infinite-dimensional varifold measure space. Leveraging varifold representation and duality theory, it proposes a learnable test function $h:mathbb{R}^n imes S^{n-1} omathbb{R}$, parameterized by a lightweight neural network, to construct invariant shape representations that explicitly encode parameterization invariance. Contribution/Results: The resulting framework drastically reduces model parameters while achieving state-of-the-art performance on multi-class classification and regression tasks involving shape graphs and surfaces. Moreover, it demonstrates superior robustness to geometric perturbations and enhanced cross-domain generalization capability compared to existing approaches.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Representation Learning for Vision
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Large pretrained models with web data

Abstract

Despite progress in the rapidly developing field of geometric deep learning, performing statistical analysis on geometric data--where each observation is a shape such as a curve, graph, or surface--remains challenging due to the non-Euclidean nature of shape spaces, which are defined as equivalence classes under invariance groups. Building machine learning frameworks that incorporate such invariances, notably to shape parametrization, is often crucial to ensure generalizability of the trained models to new observations. This work proposes SVarM to exploit varifold representations of shapes as measures and their duality with test functions $h:mathbb{R}^n imes S^{n-1} o mathbb{R}$. This method provides a general framework akin to linear support vector machines but operating instead over the infinite-dimensional space of varifolds. We develop classification and regression models on shape datasets by introducing a neural network-based representation of the trainable test function $h$. This approach demonstrates strong performance and robustness across various shape graph and surface datasets, achieving results comparable to state-of-the-art methods while significantly reducing the number of trainable parameters.

Problem

• Classify and regress geometric data using varifold representations
• Incorporate shape invariances for model generalizability
• Reduce trainable parameters while maintaining performance

Innovation

• Uses varifold representations for shape analysis
• Introduces neural network-based test functions
• Reduces trainable parameters significantly