提出可闭式求解的自适应地标核，提升点云与图分类的准确率和可信度。
A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classification
- 基于覆盖理论设计自适应地标位置，无需梯度训练
- 在多个数据集上达到91.3%准确率，且对分布外数据鲁棒
- 提供每条预测的闭式证书，无需校准分割
我们提出PALACE（Persistence Adaptive-Landmark Analytic Classification Engine），作为PLACE的数据自适应版本，仅需在三个参数（预算、半径、带宽；每个不超过5个取值）上进行小规模交叉验证。基于勒贝格数准则的覆盖理论核心，给出四项闭式保证：(i) 在交叉图无干扰条件下，对$$\mathcal{D}_n$$的结构下界失真度$λ(τ;ν)$，当图谱集中时比均匀网格节省$(D/L)^2$的预算；(ii) 等权$w_k = K^{-1/2}$最大化$λ$，最远点采样近似最优$k$-中心覆盖半径，均仅依赖训练标签，无需梯度训练；(iii) 核-RKHS分类率$O((k-1)$\sqrt{K}/(γ\sqrt{m_{\min}})$)$，二元必要阈值$m = Ω($\sqrt{K}/γ)$$来自匹配的Le Cam下界，以及闭式过滤选择规则。核-马哈兰诺比裕度$$\hatρ_{\mathrm{Mah}}$$在化学图库中表现最强（平均斯皮尔曼相关$ρ≈+0.60$）；各向同性代理$$\hatγ/\sqrt{K}$$具有选择一致性速率，而$\widehatλ$提供独立数据级信号（在COX2和PTC为正）。(iv) 提供非渐近Pinelis与渐近高斯形式的逐预测证书，无需校准分裂。实验表明，PALACE在Orbit5k上达$91.3 \pm 1.0\%$准确率（媲美Persformer），在COX2和MUTAG上超越所有图基方法，在DHFR上与ECP相差不足1个百分点。当域膨胀8倍时，自适应布局保持94%性能，而均匀网格退化至随机水平（4类数据25%）。
原文摘要 · Abstract (English)
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).
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