{
 "artifacts": null,
 "category": "dynamics",
 "description": "Replace a recurrent update by a time-inhomogeneous random choice among candidate maps, and regulate the candidate Jacobian gains so that the expected product of gains contracts geometrically. This should make hidden-state distributions forget their initial state even when the map family and selection probabilities vary over time, improving long-horizon stability without requiring every individual candidate map to be strongly contractive.",
 "formulas_latex": [
  "$$X_{t+1}=F_{t,I_t}(X_t),\\quad P(I_t=i)=p_{t,i},\\quad ||F_{t,i}(x)-F_{t,i}(y)||\\le L_{t,i}||x-y||.$$",
  "$$E[\\prod_{t=0}^{n-1}L_{t,I_t}]\\le C\\rho^n,\\quad 0\u003c\\rho\u003c1;\\qquad \\prod_{t=0}^{n-1}\\left(\\sum_i p_{t,i}L_{t,i}\\right)\\le C\\rho^n.$$",
  "$$d_{BL}(\\mu_n,\\nu_n)\\le C\\rho^n d_{BL}(\\mu_0,\\nu_0),\\quad d_{BL}(\\mu,\\nu)=\\sup_{||f||_\\infty\\le1,\\operatorname{Lip}(f)\\le1}|E_\\mu f-E_\\nu f|.$$",
  "$$L_{contract}=\\frac{1}{T}\\sum_{t=0}^{T-1}\\operatorname{softplus}\\left(\\log\\left(\\sum_i p_{t,i}\\widehat L_{t,i}\\right)-\\log\\rho_*\\right)^2.$$"
 ],
 "id": 2803,
 "implementation": "Integrate the mechanism into an RNN or state-space block with K candidate maps, such as Fₜ,ᵢ(hₜ,xₜ) = tanh(Wₜ,ᵢhₜ + Uₜ,ᵢxₜ + bₜ,ᵢ), and a time-dependent categorical gate pₜ,ᵢ. At each training step, estimate each candidate gain L̂ₜ,ᵢ using one to three power-iteration steps on the hidden-to-hidden Jacobian. A cheaper upper bound is ||Wₜ,ᵢ||₂ because tanh is 1-Lipschitz. Pseudocode: compute pₜ; sample a candidate or use the probability-weighted mixture; update hₜ₊₁ = Fₜ,ᴵₜ(hₜ,xₜ); estimate all L̂ₜ,ᵢ; compute gₜ = log(Σᵢ pₜ,ᵢL̂ₜ,ᵢ + epsilon); add λ times the squared positive part of gₜ − log(ρ*) to the task loss. Track γ̂ₜ = T⁻¹Σₜgₜ, and optionally rescale recurrent matrices when γ̂ₜ exceeds zero. The paper supplies the contraction mechanism; Jacobian estimates, routing probabilities, and finite-horizon exponents are measured empirically. First test on sequential MNIST and the adding problem against a GRU and an unconstrained random-map RNN with matched parameter counts. Run two identical input sequences from hidden states separated by unit norm, and record dₜ = ||hₜ − h′ₜ|| for 200 steps. The prediction is a stability boundary near γ̂ = 0: below it, log dₜ has negative slope approximately γ̂; above it, distances do not decay geometrically. The measured slope should agree with the accumulated log-gain within about 20 percent.",
 "math_summary": "Let Xₜ₊₁ = Fₜ,ᴵₜ(Xₜ), where Fₜ,ᵢ is candidate map i at time t and Iₜ is sampled with probability pₜ,ᵢ. Let Lₜ,ᵢ satisfy ||Fₜ,ᵢ(x) − Fₜ,ᵢ(y)|| ≤ Lₜ,ᵢ ||x − y||. A sufficient geometric-attraction condition is E[Πₜ₌₀ⁿ⁻¹ Lₜ,ᴵₜ] ≤ Cρⁿ for constants C \u003c infinity and 0 \u003c ρ \u003c 1. For independent selections this reduces to Πₜ₌₀ⁿ⁻¹ Σᵢ pₜ,ᵢLₜ,ᵢ ≤ Cρⁿ. If μₙ and νₙ are the state distributions generated from two initial distributions, their bounded-Lipschitz distance should decay geometrically: d_BL(μₙ,νₙ) ≤ Cρⁿd_BL(μ₀,ν₀). In the neural-network implementation, Lₜ,ᵢ is estimated by the spectral norm of the candidate hidden-state Jacobian, and the logarithmic average gain is regularized below a target log contraction rate.",
 "math_tags": [
  "dynamical-systems",
  "probability",
  "control-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "rnn",
  "ssm",
  "training-dynamics",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2608.29022",
  "arxiv_url": "https://arxiv.org/abs/2608.29022",
  "summary_what_math_gives_to_ml": "The paper develops geometric-attraction criteria for iterated function systems whose transformation families and selection probabilities change with time. Its transferable mechanism is a quantitative contraction condition on products of time-dependent random-map gains, guaranteeing that state distributions forget their initialization at a geometric rate in bounded-Lipschitz distance. A direct neural-network use is a stochastic recurrent or state-space architecture with time-varying candidate layers, where the candidate Jacobian gains and routing probabilities are constrained so that their cumulative expected gain decays geometrically. The central falsifiable prediction is that the initialization-distance decay rate is controlled by the accumulated logarithmic contraction exponent.",
  "title": "On the Existence of Geometrically Attracting Measures for Iterated Function Systems with Varying Sets of Transformations",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 7
 },
 "solves": [
  "stability",
  "generalization"
 ],
 "title": "Geometrically Attracting Random Recurrent Layer",
 "url": "https://synthcore.org/idea/2803/geometrically-attracting-random-recurrent-layer",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}