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

Sushovan Majhi; Atish Mitra; Žiga Virk; Pramita Bagchi

cs.LGMay 5, 2026

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

Authors: Sushovan Majhi, Atish Mitra, Žiga Virk, Pramita Bagchi

Organizations: Data Science, George Washington University, USA · Department of Mathematical Sciences, Montana Technological University, USA · Faculty of Computer and Information Science, University of Ljubljana, Slovenia · Biostatistics and Bioinformatics, George Washington University, USA

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).

Explore similar work

May 4, 2026cs.LG

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

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; the closed-form weight rule

w_k^2 \propto (d_{k+1}^2 - d_k^2)/R_k^2

maximizes the distortion slope in Mitra-Virk's affine certificate under

ν

-coherence. (i) An

O(kR/(Δ\sqrt{m_{\min}}))

margin bound, driven by class-mean separation

Δ

and embedding radius

R

, matched in the sample-starved regime

m \lesssim R/Δ

by a Le Cam minimax lower bound. (ii) The Mahalanobis margin under Ledoit-Wolf-shrunk covariance is the strongest closed-form ranker on a 64-descriptor chemical-graph pool (mean Spearman

ρ= +0.56

across 11 benchmarks, positive on 10 of 11); the isotropic surrogate

Δ/\sqrt{\ell}

admits a closed-form selection-consistency rate on the homogeneous protein/social pools. (iii) A training-time-decided certificate, with no per-prediction overhead, in three concrete radii (Pinelis, Gaussian plug-in, and variance-aware Pinelis-Bernstein). Empirically, PLACE is the strongest diagram-based method on Orbit5k and matches the strongest topology-based baseline within statistical noise on MUTAG and COX2; remaining gaps fall into two diagnosable regimes (descriptor blindness on NCI1/NCI109; pool-coverage limits elsewhere). The Pinelis-Bernstein radius fires on 8 of the 12 benchmarks; on MUTAG the empirical and population nearest-centroid rules agree on every one of 940 held-out test predictions, validating the certificate's mechanism.

Sushovan Majhi, Atish Mitra, Žiga Virk +1

Point Clouds Persistent Homology Image Classification Closed-Form

Ask PDF

Jun 30, 2026cs.GR

Benchmarking Federated Learning and Knowledge Distillation for Point Cloud Classification

Deploying 3D point cloud analysis in privacy-sensitive, resource-constrained settings faces two barriers: data cannot be centralized, and models must run on limited edge hardware. We present a multi-seed benchmark jointly evaluating federated learning (FL) and knowledge distillation (KD) for 3D point cloud classification. It spans 13 FL algorithms and 10 KD objectives (a 130-pair cross-product) across 504 training runs, evaluated on ModelNet40 and a clinical craniosynostosis dataset. We report three findings. First, under extreme non-IID label skew, standalone FL degrades sharply: on ModelNet40, the strongest method reaches 76.32% against a 92.26% centralized reference; on clinical data, the best reaches 75.83% against 100%. Second, distillation successfully compresses the teacher into a student 74.51% smaller and roughly twice as fast at inference, often matching or surpassing the teacher. Third, the combined pipeline exposes an evaluation pitfall: when distillation keeps a hard-label cross-entropy term on a labeled proxy split, a collapsed federated teacher (8.50%) paired with Logit-MSE still yields a 92.94% student. This 84.4-point gap reflects the proxy labels rather than the federated model, reusing the very labels whose privacy motivated federation. Objectives without hard labels instead track teacher quality (

r \approx 0.99

) and collapse when the teacher does. We therefore recommend evaluating FL-KD pipelines with label-free distillation so reported accuracy reflects the federated teacher, not the proxy.

Aizierjiang Aiersilan

Point Clouds Heterogeneous Knowledge Distillation Privacy-Preserving Framework Cross-Dataset Benchmark Geometry-Aware Uncertainty Coresets

Ask PDF

Sep 6, 2026cs.LG

Learning Kernels by Alignment for Multiclass Bayes Classification

Kernel methods separate data representation from decision-making, but typically require the kernel to be chosen in advance. We show that this kernel can instead be learned by alignment, and develop the resulting framework through the recently introduced Collaborative Learning and Inference (CLaI). We show that Collaborative Learning can be viewed as a kernel alignment process, in which an embedding is trained so that its induced similarity matches a label-derived target kernel. We also prove that Collaborative Inference is equivalent to kernel Bayes classification with Parzen-window density estimation. Motivated by these perspectives, we generalise CLaI by replacing cosine similarity with a learned Mahalanobis distance and extend it to multiclass classification. On CIFAR-10, PathMNIST, and SleepEDF, the Mahalanobis formulation improves accuracy, converges faster, and yields lower calibration error than the cosine-based variant. Auxiliary experiments further support these connections, showing that CLaI produces latent signals of the same form as a Gaussian process, while achieving competitive calibration on sepsis prediction. Together, these results establish a principled learned-kernel framework that unifies representation learning, kernel alignment, and Bayesian classification, and extends naturally to the multiclass setting.

Hollan Haule, Javier Escudero

Multiclass Classification Bayesian Learning Representation Learning

Ask PDF