arXiv subjects

John Rick Manzanares

Publications and source records attributed to John Rick Manzanares.

5 recordsLinked to original sources

arXiv2026

Frame-invariant topological representations of trabecular bone microarchitecture for strength prediction

Directional topological representations of trabecular bone should retain interpretable structural information without depending on an arbitrary transverse coordinate frame. We develop a frame-invariant directional filtration and compare its strength prediction with signed distance persistent homology and conventional morphometry. Twenty-four human trabecular cores were analyzed using persistence images, directional Betti tensors, and nested ridge regression over 13 validation groups. The directional construction combines cone occupancy, directional covariance, and principal axis degeneracy. Equal bone removal experiments and a pair closely matched in morphometry were used to examine whether topological differences tracked changes in simulated elastic stiffness. The original combined persistence image model had a root mean squared error (RMSE) of 1.952 MPa, compared with 2.001 MPa for morphometry; the paired difference was $-0.049$ MPa with a 95% bootstrap interval of $[-0.491,0.393]$ MPa. Exploratory signed distance persistent homology in dimension zero gave an RMSE of 1.639 MPa. The post hoc frame-invariant directional dimension zero model gave 1.610 MPa, compared with 1.877 MPa for dimension one and 1.630 MPa for a harmonized signed distance dimension zero model. The paired RMSE difference between the invariant and signed distance models was $-0.021$ MPa with a 95% interval of $[-0.234,0.210]$ MPa. Localized removal reduced stiffness more than diffuse removal in 19 of 24 cores despite producing smaller $H_0$ persistence image changes. Frame invariance removes the dependence of directional topology on an arbitrary transverse coordinate frame. Connected component representations warrant external evaluation for strength prediction, while the mechanical experiments limit their interpretation as scalar stiffness surrogates.

q-bio.QM↗

arXiv2026

Scalable persistence pairing graphs on scalar fields with an application to porous media

We present a scalable method for computing persistence pairing graphs from large cubical filtrations and use it to test whether graph organization among persistent classes contains information beyond persistence intervals alone. The method cancels equal-value pairs before global reduction, constructs the compressed complex in streaming form, and recovers graph transitions from one persistence reduction through projected relations and triangular event solves. We apply the method to 119 signed distance volumes from the DRP-372 porous media collection for permeability prediction. Each 256-cubed volume induces more than 135 million cubical cells before contraction, whereas the median compressed complex contains 37,471 generators. In validation grouped by material family, adding persistence pairing graph descriptors improves prediction over persistence summaries under both Ridge regression and partial least squares regression. In the stricter experiment that holds out one source project at a time, Ridge retains a modest improvement over persistence alone, although additional benefit beyond geometry and persistence is not consistent across projects. Two null models show that the observed persistence and graph coupling and edge target organization are strongly nonrandom, but neither uniquely explains the predictive improvement; the useful signal appears to reside mainly in coarser graph and event organization. Spatial analyses likewise distinguish persistence magnitude from graph prominence and reveal nonlocal organization among algebraically related persistence events. These results show that persistence pairing graphs provide structurally informative summaries beyond the barcode while exposing domain dependence and representative sensitivity that motivate more invariant relational descriptors.

math.AT↗

arXiv2026

Exact and Approximate Range Queries in Ball Mapper

Ball Mapper summarizes a finite metric dataset by covering the sample with closed balls centered at selected landmarks and connecting landmarks whose balls share observations. Its construction therefore depends critically on repeated fixed radius range queries, yet the effect of replacing exact queries by approximate search has not been systematically characterized. We formulate Ball Mapper through an abstract range query procedure that separates the mathematical construction from the search backend used to realize it. Under fixed ordering, exact procedures preserve the landmark sequence, cover, graph, and membership-based colorings. For approximate procedures, we derive deterministic bounds on covering radius and landmark separation under additive and multiplicative query errors, prove inclusions for the induced nerve, characterize edge survival through witness redundancy for conservative approximations, and bound perturbations of mean vertex colorings. The accompanying implementation provides independent exact reference backends together with exhaustive and approximate search methods under a common closed ball convention. Experiments on Gaussian, mixture, and noisy curve data across three seeds show that approximation fidelity depends strongly on geometry and that edges supported by multiple witnesses are substantially more robust to missed memberships. At 20,000 observations, the approximate indexes did not outperform exhaustive FAISS Flat search. The results therefore establish a framework for controlled approximation rather than a universal speed advantage, and identify the geometric and combinatorial quantities that govern when approximate range search preserves the Ball Mapper summary.

cs.CG↗

arXiv2026

Persistence Pairing Graphs: Tracking Changes of Selected Homology Bases

Persistent homology summarizes a filtration by recording when homological features appear and disappear, but the resulting barcode does not determine a preferred homology basis or preferred cycle representatives. We study the additional information carried by a basis selected through matrix reduction and introduce the \emph{persistence pairing graph}, a reduction-dependent directed graph that records how selected homology classes are re-expressed when their associated persistence intervals end. After fixing a filtered finite CW complex, a coefficient field, a filtration-compatible cell order, and a reduced factorization of the boundary matrix, the reduction determines selected birth cycles. When an associated persistence interval ends, the image of its selected birth class has unique coordinates in the basis formed by selected classes whose intervals survive beyond the same filtration value, and the nonzero coordinates define the graph edges. These coordinates can also be recovered through the Kronecker pairing with a dual cohomology basis. We prove that the active selected birth cycles form a homology basis at every cellwise filtration stage, characterize the kernel and image of each homological transition, and show that the graph is directed and acyclic, with every edge pointing toward an interval containing the source interval. In dimension zero, under the standard reduction convention, the graph agrees with the elder rule merge tree. We also prove that its edges can change while the persistence pairs and barcode remain fixed. Finally, we define a bounded fused graph discrepancy and separate the contribution controlled by ordinary persistence stability from the reduction-dependent graph structure. The graph is therefore a descriptor relative to a fixed reduction convention, rather than an invariant of the persistence module alone.

math.AT↗

arXiv2026

A Probabilistic Interpretation of the Ball Mapper Graph

We introduce Probabilistic Ball Mapper, a formulation of Ball Mapper in which each data point is assigned a probability distribution supported only on the metric balls that contain it. This assignment defines both a partition subordinate to the Ball Mapper cover and a Markov kernel from the finite data space to the cover. We study two assignment schemes: a uniform-on-support rule and a localized radial-basis rule that incorporates distance to landmarks while preserving the underlying cover. Pushing the empirical data distribution through the kernel produces a probability distribution over vertices. Drawing twice, conditionally and independently, from each pointwise distribution produces a soft-overlap matrix. This matrix is symmetric, nonnegative, positive semidefinite, and has the vertex distribution as both marginals. It therefore provides a mass-normalized refinement of classical Ball Mapper overlap rather than another unnormalized edge count. For graphs constructed on a common cover, the vertex and overlap distributions can be compared directly. For independently fitted covers, we formulate Wasserstein and fused Gromov--Wasserstein-type discrepancies that account for vertex mass, landmark geometry when a common ambient metric is available, and intrinsic graph relations. For a fixed cover, we derive explicit perturbation bounds controlled by the sensitivity of the assignment rule, the magnitude of the data perturbation, and the data mass near cover boundaries. When the cover is recomputed, landmark motion creates an additional source of variation, for which we state a transport-based stability principle rather than an unconditional theorem. The resulting framework turns Ball Mapper into a probability-valued representation suitable for quantitative comparison while retaining its geometric interpretability and computational simplicity.

cs.CG↗