{
 "artifacts": null,
 "category": "dynamics",
 "description": "Treat the empirical hidden-state distribution of a recurrent or state-space model as a Wasserstein-space state and estimate the linearized pushforward operator on perturbation vector fields. Penalize tangent modes whose estimated transfer gains exceed one, while retaining near-unit fixed modes that represent robust invariant distributional structure.",
 "formulas_latex": [
  "$$T_\\mu\\mathcal P_2(M)=\\overline{\\{\\nabla\\psi:\\psi\\in C^\\infty(M)\\}}^{L^2(\\mu;TM)},\\qquad \\mu_t=(\\exp_x(tv(x)))_\\ast\\mu.$$",
  "$$D(F_\\ast)_\\mu[v]=P_\\mu\\mathcal L_Fv,\\qquad (\\mathcal L_Fv)(y)=\\mathbb E[D F(X)v(X)\\mid F(X)=y].$$",
  "$$\\rho(P_\\mu\\mathcal L_F)\u003c1\\ \\Longrightarrow\\ \\|D(F_\\ast)_\\mu^k v\\|_{L^2(\\mu)}\\lesssim C\\rho^k\\|v\\|_{L^2(\\mu)}.$$",
  "$$\\mathcal R_{\\mathrm{tan}}=\\sum_{j=1}^{r}\\left[\\max(0,\\widehat\\lambda_j-1)\\right]^2,\\qquad \\widehat\\lambda_j=\\frac{\\|\\widehat{P_\\mu\\mathcal L_F}q_j\\|_{L^2(\\mu)}}{\\|q_j\\|_{L^2(\\mu)}}.$$"
 ],
 "id": 2972,
 "implementation": "Integrate the method at the recurrent transition of a small GRU, SSM, or world model. Let hidden states $h_i$ be sampled from a minibatch and let $F_\\theta(h_i,u_i)$ produce the next state; initially use fixed inputs or identical input sequences so the transition is approximately autonomous. Create $r$ probe fields $q_j(h)$ as gradients of random Fourier features or small probe potentials. For every probe and state, use automatic differentiation to compute $a_{ij}=J_F(h_i)q_j(h_i)$, where $J_F$ is the Jacobian of the transition with respect to hidden state. Estimate conditional averaging using destination-state kernel regression: $\\widehat{\\mathcal L_Fq_j}(y)=\\sum_iK_\\tau(y,F(h_i))a_{ij}/\\sum_iK_\\tau(y,F(h_i))$. Estimate $P_\\mu$ by least-squares projection of these vectors onto a basis of gradient probes evaluated at sampled states. Compute leading singular values or Rayleigh gains, add $\\gamma\\mathcal R_{\\mathrm{tan}}$ to the task loss, and optionally reduce the learning rate when the largest gain exceeds one. The paper supplies the transfer derivative and fixed-space interpretation; Jacobians, kernels, projections, and spectral estimates are empirical approximations. First test on a GRU predicting a noisy linear dynamical system and on sequential MNIST, against an unregularized GRU and ordinary Jacobian spectral normalization. Multiply the recurrent matrix by a scalar gain $g$ to sweep stability. The prediction is a measurable boundary near $\\widehat\\rho=1$: below it, tangent perturbation norms decay geometrically; above it, they grow approximately like $\\widehat\\rho^k$. The rollout-divergence threshold should agree with the estimated spectral boundary within 20 percent.",
 "math_summary": "Let $\\mu$ be the hidden-state distribution and $F:M\\to M$ one recurrent transition satisfying $F_\\ast\\mu=\\mu$. A Wasserstein tangent perturbation is a gradient vector field $v\\in T_\\mu\\mathcal P_2(M)=\\overline{\\{\\nabla\\psi\\}}^{L^2(\\mu)}$. Define the vector-field transfer operator by $\\mathcal L_Fv(y)=\\mathbb E[D F(X)v(X)\\mid F(X)=y]$ for $X\\sim\\mu$. The paper's linearization is $D(F_\\ast)_\\mu[v]=P_\\mu\\mathcal L_Fv$, where $P_\\mu$ is the $L^2(\\mu)$ orthogonal projection onto gradient vector fields. The distributional perturbation is linearly stable when the spectral radius $\\rho(P_\\mu\\mathcal L_F)\u003c1$; fixed tangent modes satisfy $P_\\mu\\mathcal L_Fv=v$ and are first-order invariant deformations. In a neural implementation, estimate leading gains $\\widehat\\lambda_j$ on probe fields $q_j$ and regularize gains above one.",
 "math_tags": [
  "optimal-transport",
  "geometry",
  "dynamical-systems",
  "linear-algebra",
  "probability"
 ],
 "ml_areas": [
  "world-model",
  "ssm",
  "regularization",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2609.00451",
  "arxiv_url": "https://arxiv.org/abs/2609.00451",
  "summary_what_math_gives_to_ml": "The paper develops differentiable pushforward dynamics on Wasserstein space. At an invariant measure, the derivative is a transfer operator acting on perturbation vector fields followed by orthogonal projection onto the Wasserstein tangent space, and fixed tangent vectors represent first-order invariant deformations. This provides a transferable diagnostic and regularizer for recurrent neural networks and world models: estimate how hidden-state distribution perturbations are transported, then penalize gains above the stability boundary. The key falsifiable signature is a transition near spectral radius one, with perturbation norms decaying or growing geometrically on the corresponding side.",
  "title": "Pushforward dynamics on Wasserstein spaces and measure rigidity",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "stability",
  "accuracy",
  "generalization"
 ],
 "title": "Wasserstein Tangent-Space Stability Monitor",
 "url": "https://synthcore.org/idea/2972/wasserstein-tangent-space-stability-monitor",
 "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
   }
  }
 }
}