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

arXiv: https://arxiv.org/abs/2605.04046 · HTML: https://mlsanity.com/paper/2605.04046/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 5 May 2026.

## Builds on (most cited first, 14 of 14)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [Learning metrics for persistence-based summaries and applications for graph classification](https://mlsanity.com/paper/1904.12189/) (2019)
- [Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams](https://mlsanity.com/paper/1802.03569/) (2018)
- [RetGK: Graph Kernels based on Return Probabilities of Random Walks](https://mlsanity.com/paper/1809.02670/) (2018)
- [Persformer: A Transformer Architecture for Topological Machine Learning](https://mlsanity.com/paper/2112.15210/) (2021)
- [DTM-based Filtrations](https://mlsanity.com/paper/1811.04757/) (2018)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification](https://mlsanity.com/paper/2605.02836/) (2026)

## Built on by (most cited first, 3 of 3)

- [Certified Topological Interaction in Neural Representations: Exact Tests and the Statistic They Require](https://mlsanity.com/paper/2609.08561/) (2026)
- [COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules](https://mlsanity.com/paper/2609.22012/) (2026)
- [Where Do Two Populations of Persistence Diagrams Differ? Calibrated Local Inference at a Fixed Budget](https://mlsanity.com/paper/2610.08292/) (2026)

## Abstract (arXiv)

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; ≤ 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;nu) on Dn under cross-diagram non-interference, with a (D/L)2 budget reduction over the uniform grid when diagrams concentrate. (ii) Equal weights wk = 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)√K/(γ√mmin)) with binary necessity threshold m = Omega(√ K/γ) from a matching Le Cam lower bound, and a closed-form filtration-selection rule. The kernel-Mahalanobis margin hatρMah is the strongest closed-form ranker across the chemical-graph pool (mean Spearman ρ ≈ +0.60); the isotropic surrogate hatγ/√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 ± 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× domain inflation, adaptive placement maintains 94% while the uniform grid collapses to chance (25% on 4-class data).