7 papers
On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations
Charles Fanning, Mehmet Emin Aktas
Persistence diagrams are fundamental descriptors in topological data analysis. Many statistical and machine learning methods require persistence diagrams to be mapped into vector s…
Reproducing Kernel Hilbert Spaces for Virtual Persistence Diagrams
Charles Fanning, Mehmet Aktas
A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Existing functional and kernel representatio…
Square Metric Spaces
Charles Fanning, Mehmet Aktas
Product decompositions of metric spaces are built from coordinate maps, but these maps are not part of the resulting metric space. We recover this missing coordinate structure thro…
Higher-order Persistence Diagrams
Charles Fanning, Mehmet Aktas
Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relatio…
Random Walks on Virtual Persistence Diagrams
Charles Fanning, Mehmet Aktas
In the uniformly discrete case of virtual persistence diagram groups , we construct a translation-invariant heat semigroup. The kernels are supported on a countable subgrou…
Reproducing Kernel Hilbert Spaces on Banach Completions of Virtual Persistence Diagram Groups
Charles Fanning, Mehmet Aktas
Persistent homology maps a simplicial complex filtered by elements in to finite formal sums of elements of $\mathbb R_{\leq}^{2} = \{ (b,d) \in \mathbb R^2 \cup \{ \inf…