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

arXiv: https://arxiv.org/abs/2605.02836 · HTML: https://mlsanity.com/paper/2605.02836/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 47 references that are in the map.
- First posted on arXiv 4 May 2026.

## Builds on (most cited first, 18 of 18)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Selective Classification for Deep Neural Networks](https://mlsanity.com/paper/1705.08500/) (2017)
- [Distribution-Free Predictive Inference For Regression](https://mlsanity.com/paper/1604.04173/) (2016)
- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [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)
- [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)
- [Differentiable Euler Characteristic Transforms for Shape Classification](https://mlsanity.com/paper/2310.07630/) (2023)
- [DTM-based Filtrations](https://mlsanity.com/paper/1811.04757/) (2018)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [Embeddings of Persistence Diagrams into Hilbert Spaces](https://mlsanity.com/paper/1905.05604/) (2019)
- [On the Metric Distortion of Embedding Persistence Diagrams into separable Hilbert spaces](https://mlsanity.com/paper/1806.06924/) (2018)

## Built on by (most cited first, 4 of 4)

- [A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classification](https://mlsanity.com/paper/2605.04046/) (2026)
- [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 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 wk2 propto (dk+12 - dk2)/Rk2 maximizes the distortion slope in Mitra-Virk's affine certificate under nu-coherence. (i) An O(kR/(Delta√mmin)) margin bound, driven by class-mean separation Delta and embedding radius R, matched in the sample-starved regime m lesssim R/Delta 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 Delta/√ℓ 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.