{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace a recurrent, SSM, or residual linear operator W by an exact affine image of a contractive matrix sigma. The resulting W lies in a prescribed matrix ball and, for the positive-real choice of parameters, satisfies a sector inequality such as W+W^* \u003e= 0. This gives an anisotropic, learnable alternative to plain spectral normalization: the allowable operator set is convex and its center and left/right radii can encode known directional scales.",
 "formulas_latex": [
  "$$\\begin{pmatrix}F(z)\u0026I_m\\end{pmatrix}M\\begin{pmatrix}F(z)^*\\\\ I_m\\end{pmatrix}\\geq 0,$$",
  "$$F(z)+F(z)^*\\geq 0\\quad\\text{and}\\quad I_m-F(z)F(z)^*\\geq 0,$$",
  "$$F(z)=T^{-1}+(T^{-1}+tT)^{1/2}\\,\\sigma(z)\\,T^{-1/2},\\qquad T=T^*\u003e0,\\quad T^{-1}+tT\u003e0,$$",
  "$$W(\\theta)=T^{-1}+(T^{-1}+tT)^{1/2}\\,B(I+B^*B)^{-1/2}\\,T^{-1/2}.$$"
 ],
 "id": 3103,
 "implementation": "(1) Integration point: use this parameterization for the recurrent transition matrix in a small RNN, the state matrix of a linear/nonlinear SSM, or the linear map inside a residual block. Let the feature width be m. Start with fixed diagonal T=diag(exp(q_i)) and fixed t\u003c0 satisfying T^{-1}+tT\u003e0; do not apply the transform to the nonlinear activation or bias. (2) Pseudocode: sample unconstrained B from trainable parameters; compute C=B^T B+epsilon I; compute R=C^{-1/2} by eigendecomposition or 5--10 Newton--Schulz iterations; set sigma=B R; set A=(T^{-1}+tT)^{1/2}; set W=T^{-1}+A sigma T^{-1/2}; run h_next=phi(W h+x U+b), or use W as the SSM transition. The singular values of sigma are s/sqrt(1+s^2)\u003c1, so the contraction condition is enforced without clipping. (3) The paper supplies the affine map and the exact admissibility condition. The contraction construction, inverse-square-root approximation, and optional real-valued transpose implementation are engineering adaptations. Estimate the largest singular value of W empirically with 5-step power iteration, and monitor the minimum eigenvalue of W+W^T when positive-real behavior is desired. (4) First experiment: compare vanilla RNN, spectral-normalized RNN, and this layer on permuted-MNIST or sequential CIFAR-10 using equal hidden width and FLOPs. Train at learning rates where the vanilla model becomes unstable, and record gradient norms, exploding-state frequency, validation accuracy, and loss versus update count. A successful result is no divergence at larger learning rates, lower Jacobian products over long sequences, and equal-or-better accuracy without per-step projection.",
 "math_summary": "The paper defines the matrix inequality [F(z) I_m] M [F(z)^*; I_m] \u003e= 0, with M=M^* in C^{2m x 2m}; M=[[0,I_m],[I_m,0]] gives the positive-real condition F+F^* \u003e= 0, while M=[[-I_m,0],[0,I_m]] gives contractivity I_m-FF^* \u003e= 0. For T=T^*\u003e0 and real t satisfying A^2=T^{-1}+tT\u003e0, Corollary 3.2 states F(z)=T^{-1}+A sigma(z)T^{-1/2}. Theorem 4.5 states that F belongs to the paper's class HP_{T,t} if and only if sigma is bounded/contractive. Thus every admissible sigma produces an F in the matrix ball with center T^{-1}, left radius A, and right radius T^{-1/2}. In the scalar case T=beta, the center and radius are C(beta)=1/beta and R(beta)=sqrt(t+1/beta^2), with C(beta)^2-R(beta)^2=-t. Here m is the feature dimension, T controls anisotropic scaling, t controls the sector width, A is the principal positive-definite square root, and sigma is the learned contractive core.",
 "math_tags": [
  "convex-analysis",
  "linear-algebra",
  "geometry",
  "dynamical-systems"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2609.03403",
  "arxiv_url": "https://arxiv.org/abs/2609.03403",
  "summary_what_math_gives_to_ml": "The paper gives an explicit matrix-valued parameterization of sector-bounded analytic functions by contractive functions: an affine map converts any contraction sigma into a function F satisfying a prescribed indefinite quadratic inequality. The transferable asset is not the complex analysis itself, but the exact matrix-ball geometry and the ability to enforce positive-real or sector constraints without projecting a learned matrix after every update. This suggests constrained neural layers and recurrent/state-space transitions whose weights are generated from unconstrained parameters through a contractive core and a tunable anisotropic affine transform. The construction is especially promising when stability, bounded gain, or directional conditioning matters more than maximizing unconstrained layer expressivity.",
  "title": "Hyperpositive functions, sector bounded functions and a trace formula",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "generalization",
  "accuracy"
 ],
 "title": "Sector-Parametrized Neural Operator",
 "url": "https://synthcore.org/idea/3103/sector-parametrized-neural-operator",
 "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
   }
  }
 }
}