arXiv subjects

Charles Fanning

Publications and source records attributed to Charles Fanning.

9 recordsLinked to original sources

arXiv2026

The K-theory of uniform Roe algebras for coarse structures generated by finite-rank free abelian subgroups

For a uniformly locally finite coarse space $X$, the uniform Roe algebra $C_u^*(X)$ is the operator norm closure of the controlled operators on $\ell^2(X)$. The $K$-theory of uniform Roe algebras is known in asymptotic dimension zero, but it is not fully understood in higher dimensions. We compute $K_0(C_u^*(G,\mathcal E))$ and $K_1(C_u^*(G,\mathcal E))$ for every countable discrete abelian group $G$ and every finite-rank free abelian subgroup $H\leq G$, where $\mathcal E$ is the coarse structure generated by $H$. We use the Proietti--Yamashita spectral sequence to express the $K$-theory in terms of $H_*(H;\ell^\infty(G,\mathbb Z))$, which we then compute.

math.KT↗

arXiv2026

On the Spectral Synthesis of Lipschitz Persistence Diagram Vectorizations

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.

math.FA↗

arXiv2026

Square Metric Spaces

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 $\ell^\infty$-prime factorizations, we use prime multiplicities to determine the existence and classification of roots. We also study metric spaces satisfying $X\cong X\times_\infty X$, 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 $X\cong X\times_\infty X$. As an application to persistent homology, we show how to recover filtration parameters whose products or powers form a given space of intervals.

math.AT↗

arXiv2026

Higher-order Persistence Diagrams

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.

cs.CG↗

arXiv2026

Cross-attentive Cohesive Subgraph Embedding to Mitigate Oversquashing in GNNs

Graph neural networks (GNNs) have achieved strong performance across various real-world domains. Nevertheless, they suffer from oversquashing, where long-range information is distorted as it is compressed through limited message-passing pathways. This bottleneck limits their ability to capture essential global context and decreases their performance, particularly in dense and heterophilic regions of graphs. To address this issue, we propose a novel graph learning framework that enriches node embeddings via cross-attentive cohesive subgraph representations to mitigate the impact of excessive long-range dependencies. This framework enhances the node representation by emphasizing cohesive structure in long-range information but removing noisy or irrelevant connections. It preserves essential global context without overloading the narrow bottlenecked channels, which further mitigates oversquashing. Extensive experiments on multiple benchmark datasets demonstrate that our model achieves consistent improvements in classification accuracy over standard baseline methods.

cs.LG↗

arXiv2026

Random Walks on Virtual Persistence Diagrams

In the uniformly discrete case of virtual persistence diagram groups $K(X,A)$, we construct a translation-invariant heat semigroup. The kernels are supported on a countable subgroup $H$, and the restriction to $H$ has Fourier exponent $\lambda_H$ satisfying $\lambda_H(\theta)=\sum_{\kappa\in H\setminus\{0\}}\bigl(1-\Re\theta(\kappa)\bigr)\nu(\kappa),$ for a symmetric $\nu\in\ell^1(H\setminus\{0\})$. This gives a symmetric jump process on $H$. The exponent $\lambda_H$ determines heat kernels, which define reproducing kernel Hilbert spaces and their associated semimetrics. Convex orders on the mixing measures give monotonicity for the kernels, Hilbert spaces, and semimetrics.

math.PR↗

arXiv2026

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

Persistent homology maps a simplicial complex filtered by elements in $\mathbb R$ to finite formal sums of elements of $\mathbb R_{\leq}^{2} = \{ (b,d) \in \mathbb R^2 \cup \{ \infty \} \mid b < d \}$ called (finite) persistence diagrams. This map is stable with respect to the $p$--Wasserstein distance for all $p \in \left[1, + \infty \right]$. Bubenik and Elchesen extend the free translation-invariant commutative Lipschitz monoid of finite persistence diagrams $D(X,A) = D(X)/D(A)$ on arbitrary metric pairs $(X,d,A)$ with $A \subset X$ onto the free translation-invariant abelian Lipschitz group of virtual persistence diagrams $K(X,A) = K(X)/K(A)$ as an isometric embedding $D(X,A) \hookrightarrow K(X,A)$ via the Grothendieck group completion. They prove that the $p$-Wasserstein distance is translation invariant on $D(X,A)$ if and only if $p=1$ and define the unique translation-invariant embedding of $W_1[d]$ into $K(X,A)$ as $\rho.$ When $K(X,A)$ is locally compact abelian, translation-invariant kernels can be constructed via positive-definite functions and Bochner's theorem on the Pontryagin dual. We prove that, for the metric topology induced by $\rho$, the group $(K(X,A),\rho)$ is locally compact if and only if it is discrete, equivalently when the pointed metric space $(X/A,d_1,[A])$ is uniformly discrete, and hence this approach fails outside that case. Assuming instead that $(X/A,d_1,[A])$ is separable and not uniformly discrete, we develop a translation-invariant kernel theory for non--locally compact virtual persistence diagram groups. The group $K(X,A)$ embeds isometrically into its canonical Banach-space linearization $B=\widehat V(X,A)\cong\mathcal F(X/A,d_1)$, and each bounded symmetric positive operator $Q\colon B\to B^\ast$ determines a translation-invariant Gaussian kernel $k(x,y)=\exp\!\left(-\tfrac12\,\langle Q(x-y),x-y\rangle_{B,B^\ast}\right).$

math.FA↗

arXiv2025

Topological Conditioning for Mammography Models via a Stable Wavelet-Persistence Vectorization

Breast cancer is the most commonly diagnosed cancer in women and a leading cause of cancer death worldwide. Screening mammography reduces mortality, yet interpretation still suffers from substantial false negatives and false positives, and model accuracy often degrades when deployed across scanners, modalities, and patient populations. We propose a simple conditioning signal aimed at improving external performance based on a wavelet based vectorization of persistent homology. Using topological data analysis, we summarize image structure that persists across intensity thresholds and convert this information into spatial, multi scale maps that are provably stable to small intensity perturbations. These maps are integrated into a two stage detection pipeline through input level channel concatenation. The model is trained and validated on the CBIS DDSM digitized film mammography cohort from the United States and evaluated on two independent full field digital mammography cohorts from Portugal (INbreast) and China (CMMD), with performance reported at the patient level. On INbreast, augmenting ConvNeXt Tiny with wavelet persistence channels increases patient level AUC from 0.55 to 0.75 under a limited training budget.

cs.CV↗

arXiv2025

Reproducing Kernel Hilbert Spaces for Virtual Persistence Diagrams

A persistence diagram is a finite multiset of birth-death pairs representing the lifetimes of topological features across a filtration. Existing functional and kernel representations of persistence diagrams are typically constructed extrinsically through embeddings into auxiliary spaces. For filtrations with finite indexing sets, the associated virtual persistence diagram group obtained by Grothendieck completion of the persistence diagram monoid is a finitely generated lattice. We define a phase map sending each persistence interval to a circular coordinate and a character map aggregating the phases of intervals in a virtual persistence diagram. We introduce heat damping on characters of virtual persistence diagram groups to suppress the unstable frequencies. We derive Lipschitz bounds for the resulting kernels and apply them in a synthetic segmentation experiment.

math.AT↗