Pseudo-inverses of difference matrices and their application to sparse signal approximation

TL;DR

This paper derives explicit formulas for Moore-Penrose inverses of symmetric difference matrices and applies them to develop a new sparse signal approximation method using PDE-based regularization and OMP for data selection.

Contribution

It introduces explicit expressions for generalized inverses of difference matrices and a novel sparse approximation technique combining PDE regularization with orthogonal patching pursuit.

Findings

Explicit formulas for Moore-Penrose inverses of symmetric difference matrices

A new PDE-based regularization approach for scattered data interpolation

Effective sparse signal approximation using the derived inverses and OMP

Abstract

We derive new explicit expressions for the components of Moore-Penrose inverses of symmetric difference matrices. These generalized inverses are applied in a new regularization approach for scattered data interpolation based on partial differential equations. The columns of the Moore-Penrose inverse then serve as elements of a dictionary that allow a sparse signal approximation. In order to find a set of suitable data points for signal representation we apply the orthogonal patching pursuit (OMP) method.