Charles Fanning

On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations

Abstract. Persistence diagrams are fundamental descriptors in topological data analysis. Many statistical and machine learning methods require persistence diagrams to be mapped into vector spaces or compared through kernels. Although many persistence diagram vectorizations and persistence kernels have been proposed, existing methods are typically developed as individual constructions, and existing work does not provide a canonical persistence diagram vectorization from which many existing vectorizations and kernels can be derived. In this paper, we develop such a canonical persistence diagram vectorization for certain classes of persistence diagrams using Fourier analysis on groups of virtual persistence diagrams. Spectral synthesis asks whether other persistence diagram vectorizations can be represented through this canonical vectorization. We prove this result for a class of Lipschitz persistence diagram vectorizations. We extend this result from uniformly discrete metric pairs to separable metric pairs

Higher-order Persistence Diagrams

ArXiv: https://arxiv.org/abs/2605.09866

In multiple instance learning, multiple observations are aggregated into a single observation before any machine learning takes place. Persistence diagrams have the useful property that a pair of persistence intervals from distinct persistence diagrams generated by the same filtration are interpretable in a shared geometry.

Two higher-order persistence intervals

A higher-order persistence interval is an implication: if a persistence interval is born after and dies before another persistence interval, we form a higher-order persistence interval from their pair.

We then generalize this construction to implication between pairs of persistence intervals, to implication between pairs of pairs of persistence intervals, etc.

A higher-order persistence diagram

In this example, each persistence diagram is generated from that of the previous order.

Harmonic aggregation

Computing higher-order persistence diagrams as aggregates of lower-order persistence diagrams is computationally expensive. We develop an algorithm of harmonic aggregation and show that it is more computationally efficient across several random network model families.

Square Metric Spaces

ArXiv: https://arxiv.org/abs/2606.08821

Abstract. 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 through equivalence relations whose classes are candidate coordinate fibers, and the resulting quotient metrics reconstruct the coordinate factors. This framework characterizes exactly when a metric space admits a finite product or power presentation. We prove an equivalence of categories showing that these equivalence-relation data preserve exactly the ordered coordinate information of power presentations. For spaces with suitable ℓ∞-prime factorizations, we use prime multiplicities to determine the existence and classification of roots. We also study metric spaces satisfying X≅X×∞ X, where repeated coordinate splitting gives a family of metric quotients indexed by infinite binary sequences. We prove that these binary tree structures exactly characterize metric spaces satisfying X≅X×∞ X. As an application to persistent homology, we show how to recover filtration parameters whose products or powers form a given space of intervals

Random Walks on Virtual Persistence Diagrams

ArXiv: https://arxiv.org/abs/2603.02117

Bounds for random walks on virtual persistence diagram groups

We define a family of random walks on virtual persistence diagram groups and derive bounds for several associated kernels.

Reproducing Kernel Hilbert Spaces on Banach Completions of Virtual Persistence Diagram Groups

ArXiv: https://arxiv.org/abs/2602.15153

Virtual persistence diagram groups may also be locally compact abelian without having finite rank. We prove that, for a metric pair (X,d,A), the corresponding metric group of virtual persistence diagrams is locally compact abelian if and only if X/A is uniformly discrete.

But many important spaces of persistence intervals, such as classical persistence diagrams with intervals in ℝ², are not uniformly discrete.

A space of persistence intervals

This is the metric pair (ℂ,ℓ²,S¹). For the virtual persistence diagram group defined on this metric pair, ℂ represents the space of persistence intervals, and S¹ is taken to be the diagonal.

The solid red circle in the figure to the left of this paragraph is the diagonal for this space of persistence intervals.

The solid blue line is the distance between the two persistence intervals before quotienting by the diagonal.

The sum of the dashed blue lines is the distance between the two persistence intervals after quotienting by the diagonal.

A virtual persistence diagram on (ℂ,ℓ²,S¹)

In this case, the space of persistence intervals is the wedge sum of two copies of S².

This space is not uniformly discrete. We develop methods for the functional analysis of virtual persistence diagram groups for which the Fourier transform and the Fourier–Stieltjes transform are not well defined.

Reproducing Kernel Hilbert Spaces for Virtual Persistence Diagrams

Journal of Applied and Computational Topology (2026).

Journal of Applied and Computational Topology: https://link.springer.com/article/10.1007/s41468-026-00253-zArXiv: https://arxiv.org/abs/2512.07282

For a metric pair (X,d,A), the group of Virtual persistence diagrams is the free abelian metric group of persistence diagrams with possibly signed multiplicities and diagonal A. The metric on this group is the unique translation invariant extension of the 1-Wasserstein distance on the canonical inclusion of unsigned persistence diagrams.

When this group is finite-rank, it is also locally compact abelian, and the Fourier transform and Fourier–Stieltjes transform are well-defined. Its group of characters—continuous homomorphisms into the unit circle—is the torus with dimension equal to the rank of the virtual persistence diagram group.

The real part of the character map of a virtual persistence diagram.

This is equivalent to the real part of the Fourier transform of the indicator function or the Fourier–Stieltjes transform of the Dirac measure of the virtual persistence diagram.

The imaginary part of the character map of a virtual persistence diagram.

This is equivalent to the imaginary part of the Fourier transform of the indicator function or the Fourier–Stieltjes transform of the Dirac measure of the virtual persistence diagram.

Experimental Results

Digital image processing gives a natural case in which the space X of persistence intervals for a virtual persistence diagram group is finite. For a filtration on an 8-bit grayscale image, there can be no more than 2⁸ Choose 2 = 32,640 distinct persistence intervals.

We train U-Net for image segmentation on a noisy synthetic image segmentation dataset with (1) no topological regularization, (2) classical topological regularization using the Wasserstein distance, and (3) a regularization term using virtual persistence diagrams.

Topological Graph Learning on Healthcare Networks for Critical Care Prediction

Network Modeling Analysis in Health Informatics and Bioinformatics: https://link.springer.com/article/10.1007/s13721-026-00824-1

Abstract. Electronic health records (EHRs) encode complex, temporally evolving interactions among patients, providers, and clinical events in intensive care units (ICUs). Graph neural networks (GNNs) have demonstrated strong performance for clinical outcome prediction; however, existing approaches rarely integrate multi-scale topological summaries from persistent homology (PH) in a principled and learnable manner within graph representation learning frameworks. To address this gap, we develop a persistence-aware pooling framework, LAI-Pool, by incorporating both graph and hypergraph persistent homology into a unified topology-guided GNN architecture. We construct patient-specific ego graphs and ego hypergraphs from MIMIC-IV EHR data, where nodes represent patients, admissions, diagnoses, procedures, providers, ICU stays, and ICU units, and temporal (hyper)edges encode structured care interactions. PH is computed over time-based filtrations of graph clique complexes and weighted hypergraph nerves to extract complementary multi-scale topological features. These features are encoded through learnable persistence representations and integrated with node embeddings via topology-guided pooling and cross-attention mechanisms. Experiments on 65,366 ICU patients show that integrating graph and hypergraph PH with GNN embeddings consistently improves mortality prediction over the baselines. The strongest performance is achieved when both topological signals are jointly fused, demonstrating that persistent topological structure provides complementary predictive information beyond standard graph representations.

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Persistence Diagram of the Corresponding Hypergraph

Persistence Landscape of the Corresponding Hypergraph

Persistence Image of the Corresponding Hypergraph

LAI-Pool: Persistent Homology-Informed Graph Neural Networks for Predicting Mortality in ICU Patients

Complex Networks & Their Applications XIV: https://link.springer.com/chapter/10.1007/978-3-032-16649-4_12

Abstract. Electronic health records (EHRs) capture the complex web of interactions among healthcare providers, patients, and clinical events, offering a rich foundation for predicting critical outcomes in intensive care units (ICUs). However, existing approaches often fail to fully leverage higher-order relational structures and nonlinear dependencies within these networks. To address this gap, we propose LAI-Pool, a persistence-aware pooling method that integrates our self-attention guided persistence images (SAPI) with graph neural networks (GNNs) via cross at- tention. First, we model patient-provider interactions as ego networks constructed from EHRs, where nodes represent patients, admissions, diagnoses, procedures, providers, ICU stays, and ICU units, and edges encode the temporal order of care interactions. Then, we apply persistent homology to extract multiscale topological features from these networks and fuse them with GNN representations through LAI-Pool. We benchmark our approach against persistent homology only, GNN-only, and hybrid PH+GNN baselines on the MIMIC-IV dataset comprising 65,366 ICU patients. Our results demonstrate that LAI-Pool achieves superior performance in mortality prediction, and highlight the value of combining topological data analysis with deep graph models for advancing patient outcome prediction in critical care.

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