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

2025/06/01 by Emmanuel Hartman, Hartman, Emmanuel, Nicolas Charon +1
Computer Science · Engineering · Mathematics · #3D Shape Modeling and Analysis #46N10 #49Q15 #53C42 #Computer Vision and Pattern Recognition (cs.CV) #Differential Geometry (math.DG) #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #I.4.0 #I.5.1 #Machine Learning (cs.LG) #Morphological variations and asymmetry #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2506.01189

openalex publication_date 2025/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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:ℝⁿ × S^n-1 → 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.

Citations

• Universal approximation theorem for neural networks with inputs from a topological vector space
• Neural varifolds: an aggregate representation for quantifying the geometry of point clouds
• Sparse Nystrom Approximation of Currents and Varifolds
• Lightweight Deep Learning for Resource-Constrained Environments: A Survey
• Basis restricted elastic shape analysis on the space of unregistered surfaces
• VariGrad: A Novel Feature Vector Architecture for Geometric Deep Learning on Unregistered Data
• Toward Mesh-Invariant 3D Generative Deep Learning with Geometric Measures
• 3D Shape Sequence of Human Comparison and Classification using Current and Varifolds
• Geometric Deep Learning and Equivariant Neural Networks
• Weak and approximate curvatures of a measure: a varifold perspective
• Metrics, quantization and registration in varifold spaces
• Lipschitz regularity of deep neural networks: analysis and efficient estimation
• A relaxed approach for curve matching with elastic metrics
• Dynamic Graph CNN for Learning on Point Clouds
• Dynamic Graph CNN for Learning on Point Clouds
• The Expressive Power of Neural Networks: A View from the Width
• PointNet: Deep Learning on Point Sets for 3D Classification and\n Segmentation
• Geometric Deep Learning: Going beyond Euclidean data
• Multilayer feedforward networks are universal approximators

Related

• VariGrad: A Novel Feature Vector Architecture for Geometric Deep Learning on Unregistered Data
• ABC: A Big CAD Model Dataset For Geometric Deep Learning
• Neural varifolds: an aggregate representation for quantifying the geometry of point clouds
• SAL: Sign Agnostic Learning of Shapes from Raw Data
• Learning from landmarks, curves, surfaces, and shapes in Geomstats