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

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; $\le 5$ choices each). A cover-theoretic core (Lebesgue-number criterion on the landmark cover) yields four closed-form guarantees. (i) A structural lower distortion bound $\lambda (\tau ;\nu )$ 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 $\lambda$, 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 $O((k-1)\sqrt{K}/(\gamma \sqrt{m_{\min }}))$ with binary necessity threshold $m=\Omega (\sqrt{K}/\gamma )$ from a matching Le Cam lower bound, and a closed-form filtration-selection rule. The kernel-Mahalanobis margin ${\hat{\rho }}_{\mathrm{Mah}}$ is the strongest closed-form ranker across the chemical-graph pool (mean Spearman $\rho \approx +0.60$); the isotropic surrogate $\hat{\gamma }/\sqrt{K}$ admits a selection-consistency rate, and $\hat{\lambda }$ 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).