Fast computation of eigenvalues of periodic CMV matrices
arXiv:2607.06400
2026
Architecture
1 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The paper's transferable asset is not the eigenvalue algorithm itself, but the factored representation of a structured unitary operator as alternating products of local 2x2 core transformations. Such a representation stores only O(n) parameters under periodic tying, applies the operator through local operations, and preserves norm exactly, making it attractive for long-horizon recurrent or state-space dynamics. The turnover identity provides a stable way to reorder or refactor overlapping local unitary factors without materializing a dense matrix. This suggests compact trainable unitary recurrent layers with periodic parameter sharing and explicit control of hidden-state stability.
Ideas from this paper
Unverified
2026
Replace a dense recurrent transition matrix with a periodic CMV-style product of alternating local 2x2 unitary cores. The transition is exactly norm-preserving, has O(n) trainable parameters under periodic tying, and can be applied through local factor operations rather than stored as an n-by-n matrix. Use turnover refactorization when changing the ordering or boundary connection of cores, enabling a compact cyclic unitary state-space layer.
Useful6/10
Difficulty5/10
Novelty5/10