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

2026-05-05

AI Summary

This work addresses the lack of provable performance guarantees and adaptive kernel design in point cloud and graph classification by proposing PALACE, a novel method that achieves closed-form, label-dependent adaptive landmark selection without requiring gradient-based training or calibration sets. PALACE determines landmark locations via the Lebesgue number covering criterion combined with farthest point sampling and derives an analytical solution for optimal weights within a reproducing kernel Hilbert space (RKHS) framework. The approach provides non-asymptotic prediction certificates, theoretical lower bounds on structural distortion, and guaranteed classification error rates. Empirical results demonstrate that PALACE achieves 91.3% accuracy on Orbit5k, significantly outperforming baseline methods on graph datasets such as COX2 and MUTAG, while retaining 94% of its performance under an eightfold domain expansion.

Abstract

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).

Problem

• point-cloud classification
• graph classification
• persistent homology
• kernel methods
• certified prediction

Innovation

• adaptive landmark selection
• closed-form kernel
• topological data analysis
• certified classification
• persistence diagrams