Dirac signal processing of higher-order topological signals

Abstract


Higher-order networks can sustain topological signals which are variables associated not only to the nodes, but also to the links, to the triangles and in general to the higher dimensional simplices of simplicial complexes. These topological signals can describe a large variety of real systems including currents in the ocean, synaptic currents between neurons and biological transportation networks.
In real scenarios topological signal data might be noisy and an important task is to process these signals {by} improving their signal to noise ratio. So far topological signals are typically processed independently of each other. For instance, node signals are processed independently of link signals, and algorithms that can enforce a consistent processing of topological signals across different dimensions 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 triangles of simplicial complexes in a consistent way. 
The proposed Dirac signal processing algorithm is formulated in terms of the discrete Dirac operator which can be interpreted as ``square root" of a higher-order Hodge Laplacian. We discuss in detail the properties of the Dirac operator including its spectrum and the chirality of its eigenvectors and we adopt this operator to formulate Dirac signal processing that can filter noisy signals defined 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.

Complex quantum networks: a topical review

“…The topological Dirac operator can also be used to characterize the dynamics of coupled (classical) topological signals [81]. In particular the Dirac operator allows to define a new class of dynamical processes on network and simplicial complexes, revealing new physical phenomena as demonstrated by its application to Dirac synchronization, Dirac Turing patterns and Dirac signal processing [338][339][340][341].…”

“…Another inference algorithm using the Dirac operator is Dirac signal processing [341] that allows to jointly process signals defined on simplices of different dimensions.…”

Higher-order connection Laplacians for directed simplicial complexes

“…The investigation of the dynamical state of simplicial complexes has instead revealed that this is only a special case and that in general each simplex (higher-order interaction) can be associated with a dynamical variable leading to the notion of topological signals. This change of paradigm has lead to novel understanding of topological synchronization [30][31][32][33][34] and higher-order diffusion dynamics [35][36][37][38] and to novel signal processing [39][40][41] and topological neural network algorithms [42,43]. In particular, higher-order diffusion dynamics is among the most basic topological dynamical processes, describing diffusion from n-dimensional simplices to n-dimensional simplices going either one dimension up or one dimension down.…”

The mass of simple and higher-order networks

“…Recently, growing attention is addressed to the discrete topological Dirac operator [1,2], originally introduced in non-commutative geometry [3][4][5][6][7], and then used in quantum graphs [8][9][10][11][12][13][14][15] and in network theory [16][17][18][19][20][21]. On a network, the discrete topological Dirac operator is defined over topological spinors, which are the direct sum of zero-cochain and one-cochains.…”