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A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classification
Sushovan Maj · 2026-05-06 · via cs.LG updates on arXiv.org

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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 $\lambda(\tau;\nu)$ 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 $\widehat{\lambda}$ 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).
Subjects: Machine Learning (cs.LG); Algebraic Topology (math.AT)
MSC classes: 55N31 (Primary), 62H30, 68T05 (Secondary)
Cite as: arXiv:2605.04046 [cs.LG]
  (or arXiv:2605.04046v1 [cs.LG] for this version)
  https://doi.org/10.48550/arXiv.2605.04046

arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Sushovan Majhi [view email]
[v1] Tue, 5 May 2026 17:59:18 UTC (71 KB)

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