Lipschitz persistence diagram vectorizations whose scalar coordinates are additive plus Fourier–Stieltjes transforms generate varieties that can be densely approximated by exponential monomials.
Higher-order Persistence Diagrams
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abstract
Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relations among persistence intervals that are essential for faithful structural comparison and interpretability. We introduce higher-order persistence diagrams, a recursive construction in which containment relations among persistence intervals define higher-order persistence intervals. This construction performs comparison and aggregation directly on persistence diagrams and preserves interval-level structure. We use harmonic analysis to reduce frequency-space evaluations of aggregated diagrams to zeta transforms. This reduction avoids explicit construction of higher-order diagrams and replaces quadratic pair enumeration with nearly linear-time evaluation. Experiments on random network models show substantial speedups over explicit aggregation. Anonymized code is available at https://anonymous.4open.science/r/higher-order-persistence-8201.
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On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations
Lipschitz persistence diagram vectorizations whose scalar coordinates are additive plus Fourier–Stieltjes transforms generate varieties that can be densely approximated by exponential monomials.