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A Novel Schur-Decomposition-Based Weight Projection Method for Stable State-Space Neural-Network Architectures

NeurIPS 2026 · Learning theory and algorithms › Matrix and tensor methods · approach: Geometry of low-rank and SPD structure · 0 citations (OpenAlex)

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Proposes a Schur-decomposition-based projection that clips the quasi-triangular factor of a state matrix to be Schur-stable at each training step (plus a pre-factorized variant), enabling backprop-compatible stable state-space layers with fewer parameters than SIMBa.

Problem

Learning black-box state-space models of dynamical systems with guaranteed asymptotic (Schur) stability while keeping parameter count low and training convergence fast.

Approach

Apply a truncated nearest-stable projection to the quasi-triangular Schur factor of the state matrix (single pass of the bi-level nearest Schur-stable matrix algorithm, fixing the orthogonal factor), either dynamically per update or via a pre-factorized parameterization (SVD-orthogonal Z plus stabilized T), trading compute for weight efficiency.

Key results

• On synthetic benchmarks, Schur (Proj.) reaches NMSE comparable to SIMBa and Regularized methods (e.g., Smaller setup NMSE 5.91e-4 vs SIMBa 3.78e-4, within uncertainty) while needing fewer weights (64 vs 164).
• Regularized method needed the most epochs in several setups (e.g., Large: 3.45e4 vs Schur 3.48e4 comparable, Extended: 1.91e4 vs 1.68e4 for Schur), indicating faster convergence for Schur methods.
• Truncated projection is orders of magnitude faster than the full bi-level Schur-stable projection (about 1e-5 s vs 1e3 s for n=10 in the reported tables), though the NSFE is somewhat higher.

Surprising findings

• The bi-level projection's advantage in NSFE vanishes for larger matrices, and the truncated projection gives better eigenvalue matching (NSSR) in several cases.
• The Schur-based methods are on par with SIMBa in NMSE despite using far fewer parameters, with no clear accuracy advantage for any method beyond uncertainty.

Limitations

• Projection is restricted to Schur stability (unit disk); generalizing to arbitrary eigenvalue-magnitude bounds is future work.
• Schur decomposition is only implemented on CPU in JAX, so GPU implementation remains a hurdle.
• Accuracy gains over SIMBa and regularization are not significant within reported uncertainty; the truncated projection is a single-iteration approximation without optimality guarantees beyond the quasi-triangular factor.

Experimental setup

Datasets: Synthetic random stable LTI systems (from SIMBa benchmark) Silverbox CED EMPS Industrial Robot Fine-Steering Mirror

Benchmarks: Synthetic LTI system identification (Original, Extended, Small, Smaller, Large setups) Silverbox CED EMPS Industrial Robot Fine-Steering Mirror

Metrics: NMSE NSSR MSVR NSFE epoch time epoch count

Baselines: SIMBa Regularized stability (spectral-norm penalty) Bi-level nearest Schur-stable optimization (Manopt)

Builds on: Nearest Omega-stable matrix bi-level optimization (Schur-based) SIMBa regularized stability identification AdamW Extended Kalman filter with RTS smoother

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