{
 "artifacts": null,
 "category": "dynamics",
 "description": "Use diagonal similarity scaling to reduce the induced infinity norm of a sparse recurrent transition or companion operator before or during training. Similarity preserves eigenvalues while changing coordinate-wise amplification, so the method can suppress non-normal transient growth without changing the represented linear dynamics.",
 "formulas_latex": [
  "$$\\rho(\\mathbf{H}_{n})=\\max\\{|\\lambda|:\\lambda\\text{ is an eigenvalue of }\\mathbf{H}_{n}\\}$$",
  "$$\\|\\mathbf{H}_{n}\\|_{S}=\\|S^{-1}\\mathbf{H}_{n}S\\|_{\\infty}=\\max_{1\\le i\\le n}\\left\\{\\sum_{j=1}^{n}|(S^{-1}\\mathbf{H}_{n}S)_{i,j}|\\right\\}$$",
  "$$S=\\operatorname{diag}(d_{1},\\ldots,d_{n}),\\qquad \\|\\mathbf{H}_{n}\\|_{S}=\\max_i\\sum_j|(\\mathbf{H}_{n})_{i,j}|\\left|\\frac{d_j}{d_i}\\right|$$",
  "$$\\rho(\\mathbf{H}_{n})\\le \\|\\mathbf{H}_{n}\\|_{S}$$"
 ],
 "id": 3085,
 "implementation": "Integrate this into the transition operator of a delayed RNN, SSM, or companion-form polynomial layer. Form the augmented state z_t = [h_t; h_(t-1); ...; h_(t-r)], whose transition is a sparse block matrix T containing the learned A block, the delayed block -gamma I, and identity shift blocks. Introduce positive diagonal coordinates D = diag(exp(s_1), ..., exp(s_m)) over the augmented state and execute the transition in balanced coordinates as ztilde_(t+1) = D^(-1) T D ztilde_t; decode with z_t = D ztilde_t. In pseudocode: `T = build_augmented_transition(A, gamma)`; `D = diag(exp(s))`; `Tbal = solve_left(D, T @ D)`; `rowbound = max(sum(abs(Tbal), dim=1))`; `loss += lambda_bal * softplus(rowbound - target)^2`; `z = Tbal @ z + input`. For large widths, do not materialize T: compute the row sums from its sparse blocks and optimize s with 10 to 20 gradient steps every 100 training steps, or update s using a detached copy of T. The exact paper-derived quantity is the similarity-transformed infinity norm and its spectral-radius upper bound. The actual reduction in transient growth must be measured because non-normal powers can behave differently from a one-step bound. Begin with width 64, delay r = 2, and a synthetic linear system plus sequential MNIST. Compare no scaling, ordinary spectral normalization, and learned diagonal balancing at equal wall-clock budget. Track maximum hidden norm, gradient clipping frequency, Jacobian products ||T^k||, spectral radius, and task loss. Success means lower peak activation and gradient variance with unchanged validation accuracy, fewer training failures at the same learning rate, or faster convergence at equal FLOPs.",
 "math_summary": "The paper represents the recurrence through a banded non-self-adjoint Hessenberg matrix H_n whose eigenvalues are the zeros of P_n. For an invertible matrix S, it defines the similarity-scaled norm ||H_n||_S = ||S^(-1) H_n S||_infinity, where ||M||_infinity = max_i sum_j |M_(i,j)|. With diagonal S = diag(d_1, ..., d_n) and nonzero d_i, the transformed entry is (S^(-1) H_n S)_(i,j) = d_i^(-1) H_(i,j) d_j, so the row-sum bound is ||H_n||_S = max_i sum_j |H_(i,j)| |d_j/d_i|. The spectral radius satisfies rho(H_n) \u003c= ||H_n||_S. Apply this to the augmented transition matrix of a delayed neural recurrence, learning positive diagonal scales d_i to minimize the maximum row sum. The scale changes coordinates only; it does not change eigenvalues or the exact linear map after decoding.",
 "math_tags": [
  "spectral-theory",
  "linear-algebra",
  "numerical-analysis"
 ],
 "ml_areas": [
  "ssm",
  "rnn",
  "training-dynamics",
  "initialization"
 ],
 "paper": {
  "arxiv_id": "2609.02801",
  "arxiv_url": "https://arxiv.org/abs/2609.02801",
  "summary_what_math_gives_to_ml": "The paper provides a constructive family of sparse delayed recurrences whose characteristic polynomials have exact (r+1)-fold rotational symmetry and whose eigenvalues lie on an (r+1)-ray star. This structure can be transferred into neural state-space or polynomial-filter layers as a parameter-efficient alternative to unconstrained deep recurrence: periodic positive coefficients impose predictable spectral geometry while reducing the number of learned parameters. A second transferable tool is diagonal similarity scaling of the associated Hessenberg operator, which gives a cheap way to control non-normal amplification through an induced infinity-norm bound. The strongest first experiments are small recurrent layers and graph polynomial filters, comparing stability and accuracy against untied delayed residual blocks at equal parameter count.",
  "title": "Ratio and limiting zero distribution asymptotics for symmetric multiple orthogonal polynomials",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 5
 },
 "solves": [
  "stability",
  "speedup"
 ],
 "title": "Diagonal-Balanced Hessenberg Dynamics",
 "url": "https://synthcore.org/idea/3085/diagonal-balanced-hessenberg-dynamics",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}