Dirac signal processing of higher-order topological signals

Abstract

We consider topological signals corresponding to variables supported on nodes, links and triangles of higher-order networks and simplicial complexes. So far such signals are typically processed independently of each other, and algorithms that can enforce a consistent processing of topological signals across different levels are largely lacking. Here we propose Dirac signal processing, an adaptive, unsupervised signal processing algorithm that learns to jointly filter topological signals supported on nodes, links and (filled) triangles of simplicial complexes in a consistent way. The proposed Dirac signal processing algorithm is rooted in algebraic topology and formulated in terms of the discrete Dirac operator which can be interpreted as "square root" of a higher-order (Hodge) Laplacian matrix acting on nodes, links and triangles of simplicial complexes. We test our algorithms on noisy synthetic data and noisy data of drifters in the ocean and find that the algorithm can learn to efficiently reconstruct the true signals outperforming algorithms based exclusively on the Hodge Laplacian.

Persistent Dirac for molecular representation

“…By setting G n as identity matrices, we obtain the Dirac matrices whose eigenvalues has been shown to be always real [50]. In [50], the special case is discussed and has been applied to signal processing by proposing the use of topological spinors obtained from eigenspectrum of Dirac matrices. Essentially, an n-dimensional topological spinor s can be written as…”

“…Note that D This means that the weighted Dirac matrix D 1 admits the following Dirac decomposition [50]:…”

“…Inspired by the success of Hodge Laplacian matrix in molecular sciences, here we propose persistent Dirac based molecular representation and fingerprint. The discrete Dirac operator [49][50][51][52][53][54][55] is a first-order differential operator which can be interpreted as the square root of Hodge Laplace operator. This operator has been developed on graphs and simplicial complexes and used in TDA and for investigating dynamics of topological signals [50,[56][57][58].…”

“…The discrete Dirac operator [49][50][51][52][53][54][55] is a first-order differential operator which can be interpreted as the square root of Hodge Laplace operator. This operator has been developed on graphs and simplicial complexes and used in TDA and for investigating dynamics of topological signals [50,[56][57][58]. Moreover, the persistent Dirac model can be used in the quantum algorithm of persistent homology [52,53,59].…”

“…The eigenvectors of the weighted Dirac matrix are formed by the direct sum of the eigenvectors of the weighted Hodge Laplacians. The normalized weighted Dirac matrix has positive, zero and negative eigenvalues λ n that have absolute value smaller or equal to one [50] |λ n | ≤ 1.…”