Topological Dictionary Learning

• URL: http://arxiv.org/abs/2503.11470v1
• Date: Fri, 14 Mar 2025 14:56:23 GMT
• Title: Topological Dictionary Learning
• Authors: Enrico Grimaldi, Claudio Battiloro, Paolo Di Lorenzo,
• Abstract summary: We introduce a novel dictionary learning algorithm for sparse signals defined over topological spaces, specifically, regular cell complexes.<n>The learning problem is cast to jointly infer the underlying cell complex and optimize the sparse signal representation.<n> Numerical results on both synthetic and real data show the effectiveness of the proposed procedure.
• Score: 13.291627429657416
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The aim of this paper is to introduce a novel dictionary learning algorithm for sparse representation of signals defined over combinatorial topological spaces, specifically, regular cell complexes. Leveraging Hodge theory, we embed topology into the dictionary structure via concatenated sub-dictionaries, each as a polynomial of Hodge Laplacians, yielding localized spectral topological filter frames. The learning problem is cast to jointly infer the underlying cell complex and optimize the dictionary coefficients and the sparse signal representation. We efficiently solve the problem via iterative alternating algorithms. Numerical results on both synthetic and real data show the effectiveness of the proposed procedure in jointly learning the sparse representations and the underlying relational structure of topological signals.

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