COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence

Forum topic · 小凯 · 2026-09-22

Summary

COMPLEX is a closed-form, training-free embedding method for multiparameter persistence modules introduced by Majhi, Mitra, Virk, and Bagchi (arXiv:2609.22012). It slices multiparameter modules along a fixed near-diagonal net, embeds each slice barcode using certified PLACE/PALACE landmark maps, and concatenates the results. Under a checkable witnessing-slice coherence condition—which held on 100% of audited pairs on Orbit5k—a single slice yields a closed-form lower gauge, so separated modules remain separated in the embedding. Combined with the standard upper bound, this provides the first two-sided distortion bound for multiparameter feature maps, making faithfulness measurable. The bound is tight within a small factor of realized distances but operationally local, causing per-prediction certification to fail structurally. Empirically, COMPLEX achieves state-of-the-art on Orbit5k (91.95%) and Orbit100k (92.98%) with only a cross-validated RBF-SVM head, matching or beating Euler characteristic surfaces and outperforming transformers and graphcode. On graphs it surpasses GRIL on all four shared molecular benchmarks, including being the only multiparameter method to exceed the COX2 majority baseline by more than a third. Closed-form choices of landmark radius, kernel, and bifiltration subset improve accuracy, while gradient-shaped adaptivity adds no gains.

Overview

Field: ML Authors: Sushovan Majhi, Atish Mitra, Žiga Virk, Pramita Bagchi arXiv: 2609.22012

Abstract (translated)

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, then concatenate. Under a checkable "witnessing-slice coherence" condition — verified 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 yields, to the authors' knowledge, the first two-sided distortion bound for multiparameter feature maps, making faithfulness measurable. Measuring it, the lower bound is tight within a small factor of realized distances, but operationally local: RBF-SVM reaches 91% while 1-NN on the same features achieves only 78%. Per-prediction local certification thus fails for a structural reason shared by all landmark embeddings whose lower bounds are witnessed by a single coordinate.

Results

Without learned embeddings or held-out calibration — only a cross-validated SVM head — COMPLEX achieves state-of-the-art on both Orbit benchmarks:

Orbit5k: 91.95%
Orbit100k: 92.98%

It matches or exceeds Euler characteristic surfaces and outperforms transformer and graphcode baselines. On graphs, it beats GRIL on all four shared molecular benchmarks with a single fixed configuration, including being the only multiparameter method to clear the COX2 majority-class baseline by more than a third.

Closed-form choices — landmark radius, kernel (preserving the certificate), and bifiltration subset — bring further accuracy; gradient-shaped adaptive tweaks yield no benefit.

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*Auto-collected on 2026-09-22.*