Convolutive Sparse Decomposition in Inverse Problems: A Sensor Space Approach

Jean-Baptiste Malagnoux, Matthieu Kowalski

Session: General

Keywords: inverse problem, convolutive sparse coding, low rank decomposition

TL;DR: Better to perform a low rank decomposition in the sensor space than in the source one.

Abstract:

This paper presents a novel approach for solving inverse problems using convolutive sparse decomposition (CDL)
directly in the sensor domain. We demonstrate that performing CDL in the sensor domain before resolving the inverse problem
on the spatial dictionary can produce results comparable to those obtained by decomposing in the source domain. This
method enhances computational efficiency and improves spatial localization accuracy, especially in scenarios characterized by
sparse source activities. Theoretically, we establish a formal equivalence between CDL decomposition in the source and
sensor domains, laying the groundwork for a practical transition between these two approaches. Using synthetic data, we compare
two methodologies: (1) resolving the inverse problem first and then performing CDL decomposition, and (2) initially decom-
posing in the sensor domain, followed by solving the inverse problem on the spatial dictionary. Our results show that the
second approach offers improved reconstruction quality and reduced computational costs, making it a promising strategy for
addressing complex inverse problems in data-intensive fields.

Submission Number: 3