# Spectral Coarsening of Geometric Operators

## TL;DR

This paper presents a new spectral coarsening method that preserves the spectral properties of geometric operators on 3D shapes, enabling efficient shape analysis and neural network applications.

## Contribution

It introduces a spectral coarsening technique that maintains low-frequency eigenvectors, combining algebraic multigrid and functional map methods for improved shape processing.

## Key findings

- Standard coarsening techniques fail to preserve spectral properties.
- The proposed method significantly reduces sampling density while maintaining spectral features.
- Applications include shape matching, operator-sensitive sampling, and graph pooling in neural networks.

## Abstract

We introduce a novel approach to measure the behavior of a geometric operator before and after coarsening. By comparing eigenvectors of the input operator and its coarsened counterpart, we can quantitatively and visually analyze how well the spectral properties of the operator are maintained. Using this measure, we show that standard mesh simplification and algebraic coarsening techniques fail to maintain spectral properties. In response, we introduce a novel approach for spectral coarsening. We show that it is possible to significantly reduce the sampling density of an operator derived from a 3D shape without affecting the low-frequency eigenvectors. By marrying techniques developed within the algebraic multigrid and the functional maps literatures, we successfully coarsen a variety of isotropic and anisotropic operators while maintaining sparsity and positive semi-definiteness. We demonstrate the utility of this approach for applications including operator-sensitive sampling, shape matching, and graph pooling for convolutional neural networks.