Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning Dirac Spectral Transforms for Topological Signals

Leonardo Di Nino, Tiziana Cattai, Sergio Barbarossa, Ginestra Bianconi, Paolo Di Lorenzo

TL;DR

This paper proposes a parameterized nonredundant transform whose eigenmodes incorporate a mode-specific mass parameter that captures the interplay between node and edge modes and shows 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.

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.

Learning Dirac Spectral Transforms for Topological Signals

TL;DR

This paper proposes a parameterized nonredundant transform whose eigenmodes incorporate a mode-specific mass parameter that captures the interplay between node and edge modes and shows 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.

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.
Paper Structure (9 sections, 1 theorem, 23 equations, 2 figures)

This paper contains 9 sections, 1 theorem, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Representation of Topological Signals via Dirac-Laplacian Frames
  4. Learning Dirac Spectral Transforms
  5. Dirac equation of networks
  6. Spectral Transform based on mixtures of Dirac equations
  7. Dirac-driven transform learning from signals
  8. Numerical results
  9. Conclusions

Key Result

Proposition 1

Let $\mathbf{\Phi}$ and $\mathbf{\Theta}$ denote the orthonormal eigenbases of the Dirac operator and of the super-Laplacian operator, respectively, and define the dictionary Then $\mathbf{F}$ is a tight frame with frame bound $A=2$, i.e., $\mathbf{F}\mathbf{F}^\top = 2 \mathbf{I}_{V+E}.$ Consequently, every spinor $\mathbf{s}\in\mathbb{R}^{V+E}$ satisfies the Parseval-type reconstruction formula

Figures (2)

  • Figure 1: Synthetic results for sparsity-reconstruction trade-off with different classes of signals on different basis
  • Figure 2: NMSE vs. SNR, for different algorithms.

Theorems & Definitions (1)

  • Proposition 1: Dirac–Laplacian frame