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.

翻译：我们提出PLACE（持续同调-地标分析分类引擎），一种通过持续同调签名对点云与图进行分类的闭式流水线。三项定量保证——基于间隔的过量风险率、闭式描述符选择规则以及逐预测认证——完全源自训练标签，无需任何学习权重或保留校准。该嵌入将Mitra-Virk单点坐标函数稀疏地求和于地标网格上；闭式权重最大化结构畸变常数$λ(ν)$（即$\mathcal{D}_n$在无干扰条件下的Lipschitz下界）。(i) $O(kR/(Δ\sqrt{m_{\min}}))$间隔界，由类均值分离度$Δ$与嵌入半径$R$驱动，并匹配样本匮乏的极小极大下界。(ii) Ledoit-Wolf收缩协方差下的马氏间隔，在异构的64描述符化学图池上表现最优（10个基准中平均斯皮尔曼$ρ\approx +0.54$，其中9个为正）；各向同性代理$Δ/\sqrt\ell$在同质（14-15描述符）蛋白质/社交池上具有闭式选择一致性率。(iii) 无需逐预测开销的训练时决定认证，有非渐近Pinelis与渐近高斯插件两种形式。实验表明，PLACE在Orbit5k上是最强的图基方法，在MUTAG与COX2上则在统计噪声范围内匹敌最强拓扑基线。其余差距归为两类可诊断情形：NCI1/NCI109上的描述符盲区，以及其他池上的覆盖范围限制。在训练集规模下，每个基准上所有半径均超过激发阈值$\hatΔ/2$，主要由多元正态界的$\sqrt\ell$缩放主导；逐预测认证虽具构造性，但在此规模下尚不具备操作性。