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Žiga Virk

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math.AT math.GT Machine Learning math.MG math.AP

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preprint2026arXiv

A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classification

We introduce PALACE (Persistence Adaptive-Landmark Analytic Classification Engine), the data-adaptive companion to PLACE, paying a small cross-validation tier on three knobs (budget, radii, bandwidth; $\leq 5$ choices each). A cover-theoretic core (Lebesgue-number criterion on the landmark cover) yields four closed-form guarantees. (i) A structural lower distortion bound $λ(τ;ν)$ on $\mathcal{D}_n$ under cross-diagram non-interference, with a $(D/L)^2$ budget reduction over the uniform grid when diagrams concentrate. (ii) Equal weights $w_k = K^{-1/2}$ maximizing $λ$ , and farthest-point-sampling positions $2$ -approximating the optimal $k$ -center covering radius; both derived from training labels alone, no gradient training. (iii) A kernel-RKHS classification rate $O((k-1)\sqrt{K}/(γ\sqrt{m_{\min}}))$ with binary necessity threshold $m = Ω(\sqrt K/γ)$ from a matching Le Cam lower bound, and a closed-form filtration-selection rule. The kernel-Mahalanobis margin $\hatρ_{\mathrm{Mah}}$ is the strongest closed-form ranker across the chemical-graph pool (mean Spearman $ρ\approx +0.60$ ); the isotropic surrogate $\hatγ/\sqrt{K}$ admits a selection-consistency rate, and $\widehatλ$ from (i) provides an independent data-level signal (positive on COX2 and PTC). (iv) A per-prediction certificate, in non-asymptotic Pinelis and asymptotic Gaussian forms, with no calibration split. Empirically, PALACE is the strongest closed-form diagram-based method on Orbit5k ( $91.3 \pm 1.0\%$ , matching Persformer), leads every diagram-based competitor on COX2 and MUTAG, and is competitive on DHFR (within 1 pp of ECP). At $8\times$ domain inflation, adaptive placement maintains $94\%$ while the uniform grid collapses to chance ( $25\%$ on 4-class data).

preprint2026arXiv

A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification

We introduce PLACE (Persistence-Landmark Analytic Classification Engine), a closed-form pipeline for classifying point clouds and graphs through their persistent-homology signatures. Three quantitative guarantees -- a margin-based excess-risk rate, a closed-form descriptor-selection rule, and a per-prediction certificate -- are derived from training labels alone, with no learned weights or held-out calibration. The embedding sums Mitra-Virk single-point coordinate functions over a sparse landmark grid; closed-form weights maximize a structural distortion constant $λ(ν)$ (a Lipschitz lower bound on $\mathcal{D}_n$ under non-interference). (i) An $O(kR/(Δ\sqrt{m_{\min}}))$ margin bound, driven by class-mean separation $Δ$ and embedding radius $R$ , matched by a sample-starved minimax lower bound. (ii) The Mahalanobis margin under Ledoit-Wolf-shrunk covariance is the strongest closed-form descriptor selector on a heterogeneous 64-descriptor chemical-graph pool (mean Spearman $ρ\approx +0.54$ across 10 benchmarks, positive on 9 of 10); the isotropic surrogate $Δ/\sqrt\ell$ admits a closed-form selection-consistency rate on homogeneous (14-15 descriptor) protein/social pools. (iii) A training-time-decided certificate with no per-prediction overhead, in non-asymptotic Pinelis and asymptotic Gaussian plug-in forms. Empirically, PLACE is the strongest diagram-based method on Orbit5k and matches the strongest topology-based baseline within statistical noise on MUTAG and COX2. The remaining gaps fall into two diagnosable regimes: descriptor blindness on NCI1/NCI109, and pool-coverage limits elsewhere. Both radii exceed the firing threshold $\hatΔ/2$ on every benchmark at our training-set sizes, dominated by the $\sqrt\ell$ scaling of the multivariate-norm bound; the per-prediction certificate is constructive but not yet operational at these sizes.

preprint2022arXiv

Contractions in persistence and metric graphs

We prove that the existence of a $1$ -Lipschitz retraction (a contraction) from a space $X$ onto its subspace $A$ implies the persistence diagram of $A$ embeds into the persistence diagram of $X$ . As a tool we introduce tight injections of persistence modules as maps inducing the said embeddings. We show contractions always exist onto shortest loops in metric graphs and conjecture on existence of contractions in planar metric graphs onto all loops of a shortest homology basis. Of primary interest are contractions onto loops in geodesic spaces. These act as ideal circular coordinates. Furthermore, as the Theorem of Adamaszek and Adams describes the pattern of persistence diagram of $S^1$ , a contraction $X \to S^1$ implies the same pattern appears in persistence diagram of $X$ .

preprint2021arXiv

Footprints of geodesics in persistent homology

Given a metric space $X$ and a subspace $A\subset X$ , we prove $A$ can generate various algebraic elements in persistent homology of $X$ . We call such elements (algebraic) footprints of $A$ . Our results imply that footprints typically appear in dimensions above the dimension of $A$ . Higher-dimensional persistent homology thus encodes lower-dimensional geometric features of $X$ . We pay special attention to a specific type of geodesics in a geodesic surface $X$ called geodesic circles. We explain how they may generate non-trivial odd-dimensional and two-dimensional footprints. In particular, we can detect even some contractible geodesics using two- and three-dimensional persistent homology. This provides a link between persistent homology and length spectrum in Riemannian geometry.

preprint2019arXiv

Rips complexes as nerves and a Functorial Dowker-Nerve Diagram

Using ideas of the Dowker duality we prove that the Rips complex at scale $r$ is homotopy equivalent to the nerve of a cover consisting of sets of prescribed diameter. We then develop a functorial version of the Nerve theorem coupled with the Dowker duality, which is presented as a Functorial Dowker-Nerve Diagram. These results are incorporated into a systematic theory of filtrations arising from covers. As a result we provide a general framework for reconstruction of spaces by Rips complexes, a short proof of the reconstruction result of Hausmann, and completely classify reconstruction scales for metric graphs. Furthermore we introduce a new extraction method for homology of a space based on nested Rips complexes at a single scale, which requires no conditions on neighboring scales nor the Euclidean structure of the ambient space.

preprint2018arXiv

Approximations of 1-Dimensional Intrinsic Persistence of Geodesic Spaces and Their Stability

A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first homology. We call the resulting object persistence. Recent results demonstrate that persistence of a compact geodesic locally contractible space $X$ carries a lot of geometric information. However, by definition the corresponding Rips complexes have uncountably many vertices. In this paper we show that nonetheless, the whole persistence of $X$ may be obtained by an appropriate finite sample (subset of $X$ ), and that persistence of any subset of $X$ is well interleaved with the persistence of $X$ . It follows that the persistence of $X$ is the minimum of persistences obtained by all finite samples. Furthermore, we prove a much improved Stability theorem for such approximations. As a special case we provide for each $r>0$ a density $s>0$ , so that for each $s$ -dense sample $S \subset X$ the corresponding fundamental group (and the first homology) of the Rips complex of $S$ is isomorphic to the one of $X$ , leading to an improved reconstruction result.

preprint2016arXiv

Multiple perturbations of a singular eigenvalue problem

We study the perturbation by a critical term and a $(p-1)$ -superlinear subcritical nonlinearity of a quasilinear elliptic equation containing a singular potential. By means of variational arguments and a version of the concentration-compactness principle in the singular case, we prove the existence of solutions for positive values of the parameter under the principal eigenvalue of the associated singular eigenvalue problem.

preprint2012arXiv

On Minc's sheltered middle path

This paper shows that a construction, which was introduced by Piotr Minc in connection with a problem that came from Helly type theorems and that allows to replace three PL-arcs with a "sheltered middle path", can in the case of general (non-PL) paths result in the topologist's sine curve.

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