Learning Dirac Spectral Transforms for Topological Signals

TL;DR

This paper introduces a novel spectral transform based on the Dirac operator for topological signals, demonstrating improved performance in capturing cross-domain interactions and optimizing the distortion-sparsity tradeoff.

Contribution

It proposes a parameterized Dirac spectral transform with learnable mode-specific parameters, outperforming traditional methods in topological signal processing.

Findings

Overcomplete basis improves performance over single approaches.

Learning mode-specific parameters enhances distortion-sparsity tradeoff.

Dirac-based transform captures cross-domain interactions effectively.

Abstract

The Dirac operator provides a unified framework for processing signals defined over different order topological domains, such as node and edge signals. Its eigenmodes define a spectral representation that inherently captures cross-domain interactions, in contrast to conventional Hodge-Laplacian eigenmodes that operate within a single topological dimension. In this paper, we compare the two alternatives in terms of the distortion/sparsity trade-off and we show how an overcomplete basis built concatenating the two dictionaries can provide better performance with respect to each approach. Then, we propose a parameterized nonredundant transform whose eigenmodes incorporate a mode-specific mass parameter that captures the interplay between node and edge modes. Interestingly, we show that learning the mass parameters from data makes the proposed transform able to achieve the best distortion-sparsity tradeoff with respect to both complete and overcomplete bases.