COMPLEX: A Closed-Form Certified Embedding of Multiparameter Persistence Modules
COMPLEX embeds multiparameter persistence modules with a closed-form two-sided distortion bound and strong Orbit scores.
COMPLEX is a closed-form, training-free embedding of multiparameter persistence modules that slices each module along a near-diagonal net and embeds slice barcodes with the PLACE/PALACE landmark map. Under a checkable coherence condition, reported for every audited Orbit5k pair, it supplies a lower gauge and, with the usual upper bound, a two-sided distortion bound. Using only a cross-validated SVM head, it reaches 91.95% on Orbit5k and 92.98% on Orbit100k. An RBF-SVM scores 91% where 1-NN scores 78%, and one fixed configuration beats GRIL on four molecular benchmarks.
- Slices modules and embeds barcodes using certified PLACE/PALACE landmark maps.
- Reports 91.95% on Orbit5k and 92.98% on Orbit100k with an SVM.
- RBF-SVM reaches 91% while one-nearest-neighbor reaches 78% on the same features.
- Exceeds GRIL on four shared molecular benchmarks with one fixed configuration.
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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.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.22012