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

• URL: http://arxiv.org/abs/2605.02836v1
• Date: Mon, 04 May 2026 17:15:01 GMT
• Title: A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification
• Abstract summary: PLACE (Persistence-Landmark Analytic Classification Engine) is a closed-form pipeline for classifying point clouds and graphs.<n>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.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $λ(ν)$ (a Lipschitz lower bound on $\mathcal{D}_n$ under non-interference). (i) An $O(kR/(Δ\sqrt{m_{\min}}))$ margin bound, driven by class-mean separation $Δ$ 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 $ρ\approx +0.54$ across 10 benchmarks, positive on 9 of 10); the isotropic surrogate $Δ/\sqrt\ell$ 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 $\hatΔ/2$ 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.

Related papers

• Rand-SEMI-QAOA: Finite-Budget Depth-One MaxCut Ensembles on Compressed Quantum Registers [0.0]
  Rand-SEMI-QAOA is a finite-budget QRAO--QAOA ensemble for MaxCut.<n>The method samples labels uniformly without replacement from the product-$X$ family.<n>An exact gauge identity shows that product-family labels generate the orbit of the QRAO Hamiltonian.
  arXiv  Detail & Related papers  (2026-08-16T09:47:55Z)
• Robust Average-Reward Markov Decision Processes: Minimax-Optimal Learning via Plug-in Reductions [51.50375419691955]
  Distributionally robust Markov decision processes provide a principled framework for sequential decision making under model uncertainty.<n>We study how many samples are necessary and sufficient to learn an $varepsilon$-optimal robust policy under the average-reward criterion.
  arXiv  Detail & Related papers  (2026-08-06T19:49:48Z)
• Demixing Sparse Signals from Nonlinear Observations using Generalized Non-convex Regularization [0.0]
  We consider the recovery of a pair of sparse vectors from a limited number of nonlinear observations of their superposition.<n>We propose a regularization-based framework combining a Huberized data fidelity with generalized folded-concave penalties.<n>We derive estimation error bounds of order $sqrtslog(n)/m$ that hold at emphevery localized stationary point.
  arXiv  Detail & Related papers  (2026-07-12T07:29:39Z)
• Closed-form fractional radial links for elliptical Mahalanobis discriminant analysis [0.0]
  We derive the Bayes radial-link family from the within-class radius law.<n>The derived link is never significantly worse than the fitted GAM.<n>It is better on three of five heavy-tailed series plus the light-tailed control.
  arXiv  Detail & Related papers  (2026-07-07T10:02:33Z)
• Spectral Anatomy of Quantum Gaussian Process Kernels [38.264196157340216]
  We prove a Cauchy-Schwarz tail bound on Nystrm approximation error, a finite-sample variance-contraction identity, and a characterization of the emphtarget-dependent optimal entropy.<n>The diagnostic is kernel-agnostic: hardware-efficient, matchgate, IQP emphand RBF/Matérn/RFF/deep- kernel families all collapse onto identical $S/log n$ curves.
  arXiv  Detail & Related papers  (2026-05-29T07:41:14Z)
• Towards Scalable Persistence-Based Topological Optimization [44.16669776030478]
  Persistence-based topological optimization deforms a point cloud $X subset mathbbRd$ by minimizing objectives of the form $L(X) = ell(mathrmDgm(X))$, where $mathrmDgm(X)$ is a persistence diagram.<n>In practice, optimization is limited by two coupled issues: persistent homology is typically computed on subsamples, and the resulting topological gradients are highly sparse, with only a few anchor points receiving nonzero updates.
  arXiv  Detail & Related papers  (2026-05-09T15:47:20Z)
• A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classification [0.0]
  PALACE is the strongest closed-form diagram-based method on OrbitHS5k ($91.3pm 1.0%$, matching Persformer)<n>At $8$ domain inflation, placement maintains $94%$ while the uniform grid collapses to chance ($25%$ on 4-class data)<n> Empirically, PALACE is the strongest closed-form diagram-based method on OrbitHS5k ($91.3pm 1.0%$, matching Persformer)
  arXiv  Detail & Related papers  (2026-05-05T17:59:18Z)
• High-Probability Convergence in Decentralized Stochastic Optimization with Gradient Tracking [69.90407799170687]
  We study high-probability (HP) convergence guarantees in decentralized optimization.<n>Our results establish the same conditions on the horizon as in achieving comparable times.
  arXiv  Detail & Related papers  (2026-04-30T22:45:21Z)
• Adaptive Robust Confidence Intervals in Efron's Gaussian Two-Groups Model [28.636700996382558]
  Motivated by noise-oblivious nature of adversaries, we study confidence intervals for the null location parameter $$ in Efron's Gaussian two-groups fraction model.<n>We characterize the minimax-optimal length among confidence intervals with a prescribed coverage level uniformly over the unknown proportion contamination and all noise-oblivious adversaries.
  arXiv  Detail & Related papers  (2026-04-28T22:40:54Z)
• Minimax Optimality and Spectral Routing for Majority-Vote Ensembles under Markov Dependence [6.908972852063454]
  Majority-vote ensembles achieve variance reduction by averaging over diverse, approximately independent base learners.<n>We provide a minimax characterization of this phenomenon for discrete classification in a fixed-dimensional Markov setting.<n>Experiments on synthetic Markov chains, 2D spatial grids, the 128-dataset UCR archive, and Atari DQN ensembles validate the theoretical predictions.
  arXiv  Detail & Related papers  (2026-04-15T02:32:30Z)
• Topological Characterization of Churn Flow and Unsupervised Correction to the Wu Flow-Regime Map in Small-Diameter Vertical Pipes [0.0]
  We introduce the first topology-based characterization using Characteristic Surfaces (ECS)<n>This work provides the first mathematical definition of churn flow and demonstrates that unsupervised topological descriptors can challenge and correct mechanistic models.
  arXiv  Detail & Related papers  (2026-04-07T17:59:15Z)
• Adaptive Lipschitz-Free Conditional Gradient Methods for Stochastic Composite Nonconvex Optimization [23.28384210732827]
  We propose ALFCG, the first textitadaptive projection-free framework for composite nontracking variants.<n>Unlike prior methods that use openloop diminishing steps, ALFCG maintains self-normalized accumulators.<n>ALFCG-FS attains $tildemathcalO(2+-2$ and $tildemathcalO(-3+-2) $to 0$, while ALFCG-MVR2 achieve $tildemathcalO(2+
  arXiv  Detail & Related papers  (2026-03-06T15:20:52Z)
• Optimal Unconstrained Self-Distillation in Ridge Regression: Strict Improvements, Precise Asymptotics, and One-Shot Tuning [61.07540493350384]
  Self-distillation (SD) is the process of retraining a student on a mixture of ground-truth and the teacher's own predictions.<n>We show that for any prediction risk, the optimally mixed student improves upon the ridge teacher for every regularization level.<n>We propose a consistent one-shot tuning method to estimate $star$ without grid search, sample splitting, or refitting.
  arXiv  Detail & Related papers  (2026-02-19T17:21:15Z)
• Anchor-MoE: A Mean-Anchored Mixture of Experts For Probabilistic Regression [0.0]
  We present an Anchored Mixture of Experts (Anchor-MoE) model that handles both probabilistic and point regression.<n>We show that Anchor-MoE attains the minimax-optimal $L2$ risk rate.<n>It consistently matches or surpasses the strong NG baseline in RMSE and NLL.
  arXiv  Detail & Related papers  (2025-08-22T21:12:41Z)
• Near-Optimal Clustering in Mixture of Markov Chains [74.3828414695655]
  We study the problem of clustering $T$ trajectories of length $H$, each generated by one of $K$ unknown ergodic Markov chains over a finite state space of size $S$.<n>We derive an instance-dependent, high-probability lower bound on the clustering error rate, governed by the weighted KL divergence between the transition kernels of the chains.<n>We then present a novel two-stage clustering algorithm.
  arXiv  Detail & Related papers  (2025-06-02T05:10:40Z)
• Entangled Mean Estimation in High-Dimensions [36.97113089188035]
  We study the task of high-dimensional entangled mean estimation in the subset-of-signals model.<n>We show that the optimal error (up to polylogarithmic factors) is $f(alpha,N) + sqrtD/(alpha N)$, where the term $f(alpha,N)$ is the error of the one-dimensional problem and the second term is the sub-Gaussian error rate.
  arXiv  Detail & Related papers  (2025-01-09T18:31:35Z)
• Large Deviation Upper Bounds and Improved MSE Rates of Nonlinear SGD: Heavy-tailed Noise and Power of Symmetry [47.653744900375855]
  We provide a framework of optimal convex gradient methods in the presence of heavy-tailed noise.<n>We show an explicit dependence on the choice of step-size-tailed nonlinearity, noise and problem parameters.<n>We provide an MSE $widetmathcalO(t-1)$ that can be made close to the optimal $mathcalO(t-1)$.
  arXiv  Detail & Related papers  (2024-10-21T04:50:57Z)
• Optimal and instance-dependent guarantees for Markovian linear stochastic approximation [47.912511426974376]
  We show a non-asymptotic bound of the order $t_mathrmmix tfracdn$ on the squared error of the last iterate of a standard scheme. We derive corollaries of these results for policy evaluation with Markov noise.
  arXiv  Detail & Related papers  (2021-12-23T18:47:50Z)
• Sample Complexity of Asynchronous Q-Learning: Sharper Analysis and Variance Reduction [63.41789556777387]
  Asynchronous Q-learning aims to learn the optimal action-value function (or Q-function) of a Markov decision process (MDP) We show that the number of samples needed to yield an entrywise $varepsilon$-accurate estimate of the Q-function is at most on the order of $frac1mu_min (1-gamma)5varepsilon2+ fract_mixmu_min (1-gamma)$ up to some logarithmic factor.
  arXiv  Detail & Related papers  (2020-06-04T17:51:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.