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

Forum topic · 小凯 · 2026-09-22

Summary

COMPLEX is a closed-form, training-free embedding method for multiparameter persistence modules, addressing a key gap in topological machine learning: existing multiparameter persistence vectorizations only offer one-sided Lipschitz upper bounds without lower bounds, so feature faithfulness cannot be guaranteed. The method slices a multiparameter module along a fixed near-diagonal net, embeds each slice's barcode via the certified PLACE/PALACE landmark map, and concatenates the results. Under a checkable witnessing-slice coherence condition (verified on 100% of audited pairs on Orbit5k), a single slice carries a closed-form lower gauge so that separated modules remain separated in the embedding. Combined with a standard upper bound, this yields the first two-sided distortion bound for multiparameter feature maps. Empirically, COMPLEX achieves state-of-the-art results on Orbit benchmarks (Orbit5k 91.95%, Orbit100k 92.98%), matches or exceeds Euler characteristic surfaces, outperforms transformers and graphcode, and beats GRIL on all four shared molecular benchmarks. arXiv:2609.22012.

Paper Overview

Field: Machine Learning
Authors: Sushovan Majhi, Atish Mitra, Žiga Virk, Pramita Bagchi
Published: 2026-09-18
arXiv: 2609.22012

Abstract

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:

1. Slice the module along a fixed near-diagonal net. 2. Embed each slice barcode by the certified PLACE/PALACE landmark map. 3. Concatenate the embeddings.

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. Combined with the standard upper bound, this gives, to the authors' knowledge, the first two-sided distortion bound for multiparameter feature maps, making faithfulness measurable.

Findings

The lower bound is tight within a small factor of realized distances, but operationally local: an RBF-SVM reaches 91% while 1-NN on the same features reaches only 78%. Per-prediction local certification therefore fails for a structural reason shared by all landmark embeddings whose lower bounds are witnessed by a single coordinate.
COMPLEX achieves state-of-the-art on two Orbit benchmarks (Orbit5k 91.95%, Orbit100k 92.98%) with no learned embedding and no held-out calibration — only one cross-validated SVM head — matching or beating Euler characteristic surfaces, and above transformer and graphcode baselines.
On graphs, a single fixed configuration beats GRIL on all four shared molecular benchmarks, including being the only multiparameter method to surpass the COX2 majority-class baseline by more than a third.
Closed-form choices (landmark radii, certificate-preserving kernel, bifiltration subset) bring further accuracy; adaptivity via gradient shaping yields no gains.

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