用变流形表示形状，实现高效几何数据分类与回归
SVarM: Linear Support Varifold Machines for Classification and Regression on Geometric Data
- 将形状视为测度，通过测试函数构建线性模型
- 在多个形状数据集上表现接近顶尖方法，参数量大幅减少
- 适合处理曲线、图、曲面等几何数据的机器学习任务
尽管几何深度学习进展迅速，但对几何数据（如曲线、图、曲面）进行统计分析仍具挑战，因其形状空间具有非欧几里得特性，且定义为在不变群下的等价类。构建能融入此类不变性的机器学习框架，特别是对形状参数化的不变性，对确保模型泛化能力至关重要。本文提出SVarM，利用形状的变流形表示作为测度，并基于其与测试函数 $h:\mathbb{R}^n \times S^{n-1} \rightarrow \mathbb{R}$ 的对偶性，构建一个类似线性支持向量机的通用框架，但作用于无限维的变流形空间。通过引入神经网络表示可训练的测试函数 $h$，在形状数据集上实现了分类与回归模型。该方法在多种图和表面数据集上表现出色，性能媲美当前最优方法，同时显著降低可训练参数数量。
原文摘要 · Abstract (English)
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.
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