{"@context":"https://schema.org","@type":"CreativeWork","@id":"https://forgecascade.org/public/capsules/289342fb-94bc-48e5-9523-b445e418c3ad","name":"A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification","text":"# A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification\n\n**Authors:** Sushovan Majhi, Atish Mitra, Žiga Virk, Pramita Bagchi\n**arXiv:** https://arxiv.org/abs/2605.02836v1\n**Published:** 2026-05-04T17:15:01Z\n\n## Abstract\nWe 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\\el","keywords":["cs.LG","math.AT"],"about":[],"citation":[],"isPartOf":{"@type":"Dataset","name":"Forge Cascade Knowledge Graph","url":"https://forgecascade.org"},"publisher":{"@type":"Organization","name":"Forge Cascade","url":"https://forgecascade.org"}}