{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace an unconstrained recurrent transition matrix with a J-selfadjoint matrix A, where J is a fixed diagonal signature matrix with only a small number of negative entries. Add a sampled Kreiss-resolvent penalty to suppress transient amplification while preserving the expressive dimension of the hidden state. The paper's bound predicts that worst finite-time amplification depends on the smaller inertia index rather than the full hidden dimension.",
 "formulas_latex": [
  "$$A^*J=JA,\\qquad J=J^*=J^{-1},\\qquad \\operatorname{inertia}(J)=(n-q,q).$$",
  "$$\\mathcal{M}(A):=\\sup_{k\\geq 0}\\lVert A^k\\rVert_2,\\qquad \\mathcal{K}(A):=\\sup_{|z|\u003e1}(|z|-1)\\left\\lVert(zI-A)^{-1}\\right\\rVert_2.$$",
  "$$\\mathcal{M}(A)\\leq e\\min\\{d(A),2q+1,2(n-q)+1\\}\\mathcal{K}(A),\\qquad \\sigma(A)\\subset\\overline{\\mathbb{D}}.$$",
  "$$A=JH,\\quad H=H^*,\\qquad \\mathcal{L}_{\\mathrm{K}}(A)=\\frac{1}{|\\mathcal{Z}|}\\sum_{z\\in\\mathcal{Z}}\\left[(|z|-1)\\,\\sigma_{\\max}\\left((zI-A)^{-1}\\right)\\right]^2.$$"
 ],
 "id": 2853,
 "implementation": "(1) Integration point: use this matrix as the transition operator of a vanilla RNN or linear state-space layer, h_(t+1)=A h_t+B u_t. Choose hidden width n and a small inertia index q, for example q=4, and fix J=diag(+1,...,+1,-1,...,-1). Store an unconstrained real matrix H0, symmetrize it as H=(H0+H0^T)/2, and set A=JH. For complex-valued models use conjugate transpose instead. (2) Pseudocode: construct A=J(H0+H0^T)/2; estimate the spectral radius rho_hat(A) with several Arnoldi or power iterations; set A_s=A*min(1,(1-epsilon)/max(rho_hat(A),epsilon)); sample z=rho_i exp(i theta_j), with rho_i in {1.01,1.05,1.2,1.5,2.0}; estimate sigma_max((zI-A_s)^(-1)) using a few power iterations, each requiring linear solves with zI-A_s; compute L_K as the mean squared value of (|z|-1) times this singular-value estimate; optimize task_loss+lambda_K L_K. Use A_s in the forward pass. (3) The J-selfadjoint parameterization and inertia factor come from the paper. The spectral radius, sampled resolvent maximum, and finite-horizon power growth max over k less than or equal to T of ||A_s^k||_2 are empirical estimates. For a generic dense H, use r at most 2q+1 rather than estimating d(A). (4) First experiment: compare a width-128 tanh RNN, a spectrally rescaled unconstrained RNN, and this q=4 model on sequential MNIST and the adding or copying task. Match initialization, optimizer, and parameter count as closely as possible. Sweep lambda_K and measure validation accuracy, gradient norm variance, loss after long sequences, sampled K(A), and max over k less than or equal to 500 of ||A_s^k||_2. A successful result is lower transient amplification and fewer exploding gradients with equal or better long-sequence accuracy at comparable training loss and FLOPs. Cache the resolvent penalty every 10 to 50 optimizer steps to reduce overhead.",
 "math_summary": "The main theorem states that if A is an n by n matrix with spectrum in the closed unit disk, J is a fundamental symmetry with inertia (n-q,q), and A*J=JA, then M(A) is at most e times r(A) times K(A), where M(A)=sup over k greater than or equal to zero of ||A^k||_2, K(A)=sup over |z| greater than 1 of (|z|-1)||(zI-A)^(-1)||_2, and r(A)=min{d(A),2q+1,2(n-q)+1}. Here d(A) is the degree of A's minimal polynomial, n is the hidden dimension, and q is the smaller inertia index. The constraint is implemented by A=JH with H=H*, because (JH)*J=HJ J=H=JA. The theorem's transferable property is that small q yields r(A) at most 2q+1 independently of n. In a neural implementation, K(A) is approximated on a finite grid of complex points outside the unit disk; this is a practical surrogate rather than a rigorous continuous supremum.",
 "math_tags": [
  "spectral-theory",
  "linear-algebra",
  "dynamical-systems"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "training-dynamics",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.29823",
  "arxiv_url": "https://arxiv.org/abs/2608.29823",
  "summary_what_math_gives_to_ml": "The paper gives a dimension-sensitive certificate for transient growth of discrete-time linear dynamics with an indefinite selfadjointness constraint. For a recurrent or state-space layer whose transition matrix satisfies A*J=JA, the usual ambient-dimension factor in the Kreiss bound is replaced by the smaller of the minimal-polynomial degree and twice the smaller inertia index of J. This suggests a structured recurrent layer and a resolvent-based regularizer that directly suppresses nonnormal transient amplification rather than merely penalizing eigenvalue magnitude. The main practical opportunity is improved long-horizon stability at fixed state size, although the spectral-disk and resolvent conditions must be estimated numerically.",
  "title": "Inertia-Sensitive Kreiss Bounds for $J$-Selfadjoint Matrices",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy"
 ],
 "title": "Inertia-Constrained Kreiss RNN",
 "url": "https://synthcore.org/idea/2853/inertia-constrained-kreiss-rnn",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}