{"paper":{"title":"A Novel Schur-Decomposition-Based Weight Projection Method for Stable State-Space Neural-Network Architectures","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"Projecting the quasi-triangular factor from the real Schur decomposition of the state matrix onto its nearest stable peer keeps discrete-time state-space neural-network layers asymptotically stable during training.","cross_cats":["cs.SY","eess.SY"],"primary_cat":"cs.LG","authors_text":"Fredy Ruiz, Lasse Lensu, Sergio Vanegas","submitted_at":"2026-05-14T07:28:11Z","abstract_excerpt":"Building black-box models for dynamical systems from data is a challenging problem in machine learning, especially when asymptotic stability guarantees are required. In this paper, we introduce a novel stability-ensuring and backpropagation-compatible projection scheme based on the Schur decomposition for the state matrix of linear discrete-time state-space layers, as well as an alternative pre-factorized formulation of the methodology. The proposed methods dynamically project the quasi-triangular factor of the state matrix's real Schur decomposition onto its nearest stable peer, ensuring stab"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The proposed methods dynamically project the quasi-triangular factor of the state matrix's real Schur decomposition onto its nearest stable peer, ensuring stable dynamics with minimal overparameterization.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That repeatedly projecting the state matrix during training does not materially distort the optimization landscape or introduce bias that prevents reaching accurate models on real-world data.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A real Schur decomposition projection maps the state matrix of discrete-time state-space layers onto its nearest stable counterpart, delivering accuracy comparable to prior stable identification methods with fewer weights.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Projecting the quasi-triangular factor from the real Schur decomposition of the state matrix onto its nearest stable peer keeps discrete-time state-space neural-network layers asymptotically stable during training.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"59c2ece815cb79e0214cc2bb919513e08f63d86964d413104663ef831d0ba165"},"source":{"id":"2605.14489","kind":"arxiv","version":1},"verdict":{"id":"0f7f7b73-610c-404f-bca2-71244bc725af","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:33:53.095596Z","strongest_claim":"The proposed methods dynamically project the quasi-triangular factor of the state matrix's real Schur decomposition onto its nearest stable peer, ensuring stable dynamics with minimal overparameterization.","one_line_summary":"A real Schur decomposition projection maps the state matrix of discrete-time state-space layers onto its nearest stable counterpart, delivering accuracy comparable to prior stable identification methods with fewer weights.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That repeatedly projecting the state matrix during training does not materially distort the optimization landscape or introduce bias that prevents reaching accurate models on real-world data.","pith_extraction_headline":"Projecting the quasi-triangular factor from the real Schur decomposition of the state matrix onto its nearest stable peer keeps discrete-time state-space neural-network layers asymptotically stable during training."},"references":{"count":80,"sample":[{"doi":"","year":2002,"title":"Model reduction via balanced realizations: an extension and frequency weighting techniques.IEEE Transactions on Automatic Control, 33(7):687–692, 2002","work_id":"f1833826-61c3-4552-a1d6-e198a7ecbed6","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Efficiently Modeling Long Sequences with Structured State Spaces","work_id":"4150b761-b8bf-4d9b-a2f8-cb2d1b73d378","ref_index":2,"cited_arxiv_id":"2111.00396","is_internal_anchor":true},{"doi":"","year":2022,"title":"Simplified State Space Layers for Sequence Modeling","work_id":"d4f90830-6ceb-4c9e-b206-eb32ac063f45","ref_index":3,"cited_arxiv_id":"2208.04933","is_internal_anchor":true},{"doi":"","year":2023,"title":"Mamba: Linear-Time Sequence Modeling with Selective State Spaces","work_id":"4ee75248-1199-492c-a52f-6661e0f4adff","ref_index":4,"cited_arxiv_id":"2312.00752","is_internal_anchor":true},{"doi":"","year":2020,"title":"Hippo: Recurrent memory with optimal polynomial projections.Advances in neural information processing systems, 33:1474–1487, 2020","work_id":"0d6d46ae-5de3-4ecb-aa6f-923be89ee19a","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":80,"snapshot_sha256":"49a1d9a66f11aaebc5177e4a538166b3bff65c2342bf694ddfb513d92b874ade","internal_anchors":4},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}