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

TL;DR

PLACE is a novel, closed-form pipeline for classifying point clouds and graphs using persistent-homology signatures, providing guarantees and strong empirical performance without learned weights.

Contribution

It introduces a fully analytical classification method with theoretical guarantees and competitive results, avoiding learned parameters or calibration.

Findings

Achieves a margin bound proportional to class separation and embedding radius.

Outperforms existing diagram-based methods on Orbit5k and matches topology-based baselines on MUTAG and COX2.

Provides a certificate with no per-prediction overhead, though not yet operational at large scales.

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 $\lambda(\nu)$ (a Lipschitz lower bound on $\mathcal{D}_n$ under non-interference). (i) An $O(kR/(\Delta\sqrt{m_{\min}}))$ margin bound, driven by class-mean separation $\Delta$ 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 $\rho \approx +0.54$ across 10 benchmarks, positive on 9 of 10); the isotropic surrogate $\Delta/\sqrt\ell$ 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 $\hat\Delta/2$ 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.