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

2026-05-04

AI Summary

This work addresses the problem of certifiable classification for point cloud and graph data without requiring learned weights or calibration sets. It proposes PLACE, a closed-form analytical pipeline that leverages persistent homology features to embed data via summation of Mitra–Virk coordinate functions over a sparse landmark grid, followed by efficient classification through a closed-form optimization of the distortion constant. Relying solely on training labels, the method provides three theoretical guarantees: a margin-based excess risk bound, a closed-form descriptor selection rule, and zero-overhead prediction certificates from a single sample. Experiments show that PLACE outperforms the strongest graph neural networks on Orbit5k and matches state-of-the-art topological methods on MUTAG and COX2; across ten benchmarks, nine exhibit significantly positive correlations in descriptor selection (average Spearman ρ ≈ +0.54).

Abstract

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.

Problem

• persistent homology
• point-cloud classification
• graph classification
• certified prediction
• topological data analysis

Innovation

• persistent homology
• closed-form classification
• landmark embedding
• prediction certificate
• topological data analysis