Sampling and Recovery of Signals on a Simplicial Complex using Neighbourhood Aggregation

TL;DR

This paper introduces a novel sampling and recovery method for multi-order bandlimited signals on simplicial complexes, leveraging Hodge Laplacian-based aggregation and Helmholtz decomposition, with theoretical guarantees and numerical validation.

Contribution

It proposes a new aggregation sampling scheme and least squares recovery method for simplicial signals, with theoretical conditions for faithful reconstruction.

Findings

Effective recovery of multi-order signals demonstrated through experiments

Theoretical bounds on sampling requirements established

Method outperforms existing approaches in accuracy and efficiency

Abstract

In this work, we focus on sampling and recovery of signals over simplicial complexes. In particular, we subsample a simplicial signal of a certain order and focus on recovering multi-order bandlimited simplicial signals of one order higher and one order lower. To do so, we assume that the simplicial signal admits the Helmholtz decomposition that relates simplicial signals of these different orders. Next, we propose an aggregation sampling scheme for simplicial signals based on the Hodge Laplacian matrix and a simple least squares estimator for recovery. We also provide theoretical conditions on the number of aggregations and size of the sampling set required for faithful reconstruction as a function of the bandwidth of simplicial signals to be recovered. Numerical experiments are provided to show the effectiveness of the proposed method.