# A Unified Definition and Computation of Laplacian Spectral Distances

## TL;DR

This paper introduces a unified framework for defining and computing Laplacian spectral distances in both continuous and discrete settings, enabling efficient approximation and analysis of spectral kernels on graphs and manifolds.

## Contribution

It extends recent discrete results to the continuous case, providing a novel definition and efficient computation methods for Laplacian spectral kernels and distances.

## Key findings

- Discrete spectral kernels are approximated by sparse linear systems.
- The proposed methods are efficient and well-conditioned.
- The paper discusses the stability and optimality of spectral distances.

## Abstract

Laplacian spectral kernels and distances (e.g., biharmonic, heat diffusion, wave kernel distances) are easily defined through a filtering of the Laplacian eigenpairs. They play a central role in several applications, such as dimensionality reduction with spectral embeddings, diffusion geometry, image smoothing, geometric characterisations and embeddings of graphs. Extending the results recently derived in the discrete setting~\citep{PATANE-STAR2016,PATANE-CGF2017} to the continuous case, we propose a novel definition of the Laplacian spectral kernels and distances, whose approximation requires the solution of a set of inhomogeneous Laplace equations. Their discrete counterparts are equivalent to a set of sparse, symmetric, and well-conditioned linear systems, which are efficiently solved with iterative methods. Finally, we discuss the optimality of the Laplacian spectrum for the approximation of the spectral kernels, the relation between the spectral and Green kernels, and the stability of the spectral distances with respect to the evaluation of the Laplacian spectrum and to multiple Laplacian eigenvalues.