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

TL;DR

SVarM introduces a novel support varifold machine framework for classification and regression on geometric data, leveraging varifold representations and neural networks to handle non-Euclidean shapes efficiently.

Contribution

It proposes a new linear support varifold machine approach that effectively models shapes using varifold measures and neural networks, improving robustness and reducing parameters.

Findings

Achieves competitive accuracy on shape datasets

Demonstrates robustness across diverse geometric data

Reduces number of trainable parameters compared to state-of-the-art

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 \textit{SVarM} to exploit varifold representations of shapes as measures and their duality with test functions $h:\mathbb{R}^n \times S^{n-1} \rightarrow \mathbb{R}$. 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.