An inverse problem for the collapsing sum

TL;DR

This paper explores the inverse problem of the collapsing sum operator, framing it as a matrix completion challenge and providing conditions for reconstructing preimages using bipartite graph techniques.

Contribution

It introduces a novel discrete tomographical perspective on the collapsing sum and establishes a necessary and sufficient condition for matrix preimage extension.

Findings

Derived a condition for matrix preimage extension

Connected collapsing sum to matrix completion problems

Applied bipartite graphs to characterize recoverability

Abstract

Gaussian filters have applications in a variety of areas in computer science, from computer vision to speech recognition. The collapsing sum is a matrix operator that was recently introduced to study Gaussian filters combinatorially. In this paper, we view the collapsing sum from a discrete tomographical perspective and examine the recoverability of its preimages as a matrix completion problem. Using bipartite graphs, we derive a necessary and sufficient condition for a partial matrix to be extended to a preimage of a given matrix.