# COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules

arXiv: https://arxiv.org/abs/2609.22012 · HTML: https://mlsanity.com/paper/2609.22012/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 22 of its 33 references that are in the map.
- First posted on arXiv 18 Sep 2026.

## Builds on (most cited first, 22 of 22)

- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Sliced Wasserstein Kernel for Persistence Diagrams](https://mlsanity.com/paper/1706.03358/) (2017)
- [Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams](https://mlsanity.com/paper/1802.03569/) (2018)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Interactive Visualization of 2-D Persistence Modules](https://mlsanity.com/paper/1512.00180/) (2015)
- [Persformer: A Transformer Architecture for Topological Machine Learning](https://mlsanity.com/paper/2112.15210/) (2021)
- [Stable Vectorization of Multiparameter Persistent Homology using Signed Barcodes as Measures](https://mlsanity.com/paper/2306.03801/) (2023)
- [TopoGCL: Topological Graph Contrastive Learning](https://mlsanity.com/paper/2406.17251/) (2024)
- [Euler Characteristic Tools For Topological Data Analysis](https://mlsanity.com/paper/2303.14040/) (2023)
- [GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning](https://mlsanity.com/paper/2304.04970/) (2023)
- [On the Metric Distortion of Embedding Persistence Diagrams into separable Hilbert spaces](https://mlsanity.com/paper/1806.06924/) (2018)
- [Graphcode: Learning from multiparameter persistent homology using graph neural networks](https://mlsanity.com/paper/2405.14302/) (2024)
- [A Framework for Fast and Stable Representations of Multiparameter Persistent Homology Decompositions](https://mlsanity.com/paper/2306.11170/) (2023)
- [Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions](https://mlsanity.com/paper/2107.06800/) (2021)
- [D-GRIL: End-to-End Topological Learning with 2-parameter Persistence](https://mlsanity.com/paper/2406.07100/) (2024)
- [The rank invariant stability via interleavings](https://mlsanity.com/paper/1412.3374/) (2014)
- [A Closed-Form Persistence-Landmark Pipeline for Certified Point-Cloud and Graph Classification](https://mlsanity.com/paper/2605.02836/) (2026)
- [Statistical Inference for Persistence Diagrams via Landmark Embeddings: Minimax Theory and Finite Approximation](https://mlsanity.com/paper/2609.07691/) (2026)
- [Computing the interleaving distance is NP-hard](https://mlsanity.com/paper/1811.09165/) (2018)
- [Exact computation of the matching distance on 2-parameter persistence modules](https://mlsanity.com/paper/1812.09085/) (2018)
- [A Closed-Form Adaptive-Landmark Kernel for Certified Point-Cloud and Graph Classification](https://mlsanity.com/paper/2605.04046/) (2026)

## Abstract (arXiv)

Every multiparameter persistence vectorization we know of carries a one-sided Lipschitz upper bound and nothing below it: without a lower gauge there is no sense in which the features are faithful, and no per-prediction guarantee can be built on them. This paper supplies the missing side. COMPLEX is a closed-form, training-free embedding of multiparameter modules -- slice the module along a fixed near-diagonal net, embed each slice barcode by the certified PLACE/PALACE landmark map, concatenate. Under a checkable witnessing-slice coherence condition, holding on 100% of audited pairs on Orbit5k, a single slice carries a closed-form lower gauge: separated modules stay separated in the embedding. With the standard upper bound this gives, to our knowledge, the first two-sided distortion bound for a multiparameter feature map, making faithfulness measurable. Measuring it, we find the floor tight within a small factor of realized distances yet operationally local: an RBF-SVM reaches 91% where 1-NN reaches 78% on the same features. Local per-prediction certification therefore fails for a structural reason common to every landmark embedding whose lower gauge is witnessed by one coordinate. With no learned embedding and no held-out calibration -- only a cross-validated SVM head -- COMPLEX sets the state of the art on both Orbit benchmarks (91.95% on Orbit5k, 92.98% on Orbit100k), level with or above Euler-characteristic surfaces and above transformers and graphcode. On graphs it exceeds GRIL on all four shared molecular benchmarks with one fixed configuration, including the only multiparameter method to clear COX2's majority baseline by more than three points. Closed-form selection -- of the landmark radius, the kernel (certificate-preserving), and the bifiltration set -- buys further accuracy; gradient-shaped adaptation buys none.