{
 "artifacts": null,
 "category": "architecture",
 "description": "Construct a generative or recurrent neural architecture with several contractive or mildly expanding branches, and explicitly control the geometric complexity of its invariant set using the sub-additive singular-value pressure of branch-Jacobian products. Instead of regularizing only the operator norm, the model can preserve anisotropic directions while targeting a desired attractor dimension, potentially improving coverage of structured data without uncontrolled folding or collapse.",
 "formulas_latex": [
  "$$\\varphi^s(A)=\\begin{cases}\\alpha_1(A)^s,\u00260\\leq s\\leq1,\\\\\\alpha_1(A)\\alpha_2(A)^{s-1},\u00261\\leq s\\leq2,\\end{cases}$$",
  "$$P_{\\mathrm{sub}}(s)=\\lim_{n\\to\\infty}\\frac{1}{n}\\log\\sum_{\\mathbf{i}\\in\\{1,\\ldots,K\\}^n}\\sup_{z\\in\\Omega}\\varphi^s\\!\\left(Dg_{i_n}(z_{n-1})\\cdots Dg_{i_1}(z)\\right),$$",
  "$$\\dim_H(\\Lambda)=\\dim_B(\\Lambda)=s_*\\quad\\text{where}\\quad P_{\\mathrm{sub}}(s_*)=0,$$",
  "$$\\mathcal{L}=\\mathcal{L}_{\\mathrm{task}}+\\lambda_P\\left(\\widehat P(s_{\\mathrm{target}})\\right)^2+\\lambda_{\\mathrm{sep}}\\mathcal{L}_{\\mathrm{sep}}+\\lambda_J\\mathcal{L}_{J}.$$"
 ],
 "id": 2903,
 "implementation": "1. Exact integration point: define $K$ neural maps $g_i(z)=c_i+\\rho\\,\\mathrm{MLP}_i(z)$ on a two-dimensional latent state, or use the same maps as a multi-branch recurrent cell $h_{t+1}=g_{i_t}(h_t)$. Choose branch centers $c_i$ so their image regions are initially disjoint. Apply spectral normalization or explicit Jacobian scaling to keep each branch in the intended contraction or expansion regime. 2. Pseudocode: sample latent states $z$ and branch words of length $n$; initialize $A=I$ and $z_t=z$; for each branch, compute the Jacobian $Dg_{i_t}(z_t)$ with autodiff, update $A\\leftarrow Dg_{i_t}(z_t)A$, and accumulate $\\log\\varphi^s(A)$. Estimate $\\widehat P(s)$ with log-sum-exp over all words when $K^n$ is small, or with Monte Carlo importance sampling and beam search otherwise. Add the squared pressure penalty, a branch-image separation penalty, and a Jacobian-distortion penalty to the task loss. 3. Computed versus estimated: Jacobian-vector products and singular values are computed by autodiff; the infinite-time pressure, spatial supremum, and attractor dimension are estimated empirically. The paper supplies the pressure-to-dimension mechanism, while the implementation must verify separation, bounded distortion, and contraction or expansion numerically. 4. First cheap experiment: use four two-dimensional affine maps with trainable small MLP residuals, fit samples from a Sierpinski-like or anisotropic synthetic fractal, and compare ordinary training against pressure-controlled training. Sweep target dimensions $s_{\\mathrm{target}}\\in\\{0.5,1.0,1.5\\}$ and word lengths $n\\in\\{2,4,8,16\\}$. The quantitative prediction is that the empirical pressure curve crosses zero near $s_{\\mathrm{target}}$, and the measured box-counting dimension agrees with that crossing within about $0.1$ when separation holds. Removing the pressure penalty should cause systematic drift of the pressure zero and a corresponding dimension mismatch; deliberately overlapping branch images should break the prediction.",
 "math_summary": "For branch maps $g_i:\\mathbb{R}^2\\to\\mathbb{R}^2$, let $A_{i_1\\ldots i_n}(z)=Dg_{i_n}(z_{n-1})\\cdots Dg_{i_1}(z)$ be the Jacobian product along a branch sequence, where $z_t=g_{i_t}(z_{t-1})$. Let $\\alpha_1(A)\\geq\\alpha_2(A)\u003e0$ be the singular values of a matrix $A$. For $0\\leq s\\leq2$, define the singular-value potential $\\varphi^s(A)=\\alpha_1(A)^s$ when $0\\leq s\\leq1$, and $\\varphi^s(A)=\\alpha_1(A)\\alpha_2(A)^{s-1}$ when $1\\leq s\\leq2$. The sub-additive pressure is $P_{\\mathrm{sub}}(s)=\\lim_{n\\to\\infty}n^{-1}\\log\\sum_{i_1,\\ldots,i_n}\\sup_z\\varphi^s(A_{i_1\\ldots i_n}(z))$. Under uniform contraction for the inverse dynamics or the corresponding repeller assumptions, strong separation, and weak irreducibility, the attractor dimension is the unique $s_*$ satisfying $P_{\\mathrm{sub}}(s_*)=0$. A neural loss can penalize the pressure at a desired dimension while separate terms enforce separation and bounded Jacobian distortion.",
 "math_tags": [
  "dynamical-systems",
  "statistical-mechanics",
  "spectral-theory",
  "linear-algebra",
  "optimization"
 ],
 "ml_areas": [
  "mlp",
  "rnn",
  "optimizer",
  "training-dynamics"
 ],
 "paper": {
  "arxiv_id": "2608.30744",
  "arxiv_url": "https://arxiv.org/abs/2608.30744",
  "summary_what_math_gives_to_ml": "The paper provides a non-conformal Bowen formula: under genericity, separation, irreducibility, and expansion assumptions, the Hausdorff and box dimensions of a planar repeller or nonlinear IFS attractor are the unique zero of a sub-additive singular-value pressure. The transferable mechanism is to treat products of neural Jacobians as non-conformal cocycles and use pressure, rather than only the largest singular value, to measure multiscale geometric complexity. A concrete neural application is a multi-branch recurrent or generative architecture whose attractor dimension is controlled by a pressure penalty, with the predicted dimension obtained from an empirical pressure zero-crossing. This is most credible for expanding neural IFSs with explicit branch separation, rather than arbitrary feed-forward networks.",
  "title": "Dimensions of surface repellers and attractors of non-linear planar IFSs",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 8,
  "usefulness": 6
 },
 "solves": [
  "accuracy",
  "stability",
  "generalization"
 ],
 "title": "Pressure-Controlled Neural IFS",
 "url": "https://synthcore.org/idea/2903/pressure-controlled-neural-ifs",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}