# Pressure-Controlled Neural IFS

- ID: 2903
- Canonical URL: https://synthcore.org/idea/2903/pressure-controlled-neural-ifs
- API JSON: https://synthcore.org/api/idea/2903.json
- API Markdown: https://synthcore.org/api/idea/2903.md
- Verification status: unverified
- Source: [arXiv:2608.30744](https://arxiv.org/abs/2608.30744)
- Category: architecture
- Solves: accuracy, stability, generalization
- ML areas: mlp, rnn, optimizer, training-dynamics
- Math tags: dynamical-systems, statistical-mechanics, spectral-theory, linear-algebra, optimization
- Ratings: usefulness 6/10; difficulty 6/10; novelty 8/10

## Idea 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.

## Mathematical statement

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 $\alpha_1(A)\geq\alpha_2(A)>0$ 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.

## Key formulas

- $$\varphi^s(A)=\begin{cases}\alpha_1(A)^s,&0\leq s\leq1,\\\alpha_1(A)\alpha_2(A)^{s-1},&1\leq s\leq2,\end{cases}$$
- $$P_{\mathrm{sub}}(s)=\lim_{n\to\infty}\frac{1}{n}\log\sum_{\mathbf{i}\in\{1,\ldots,K\}^n}\sup_{z\in\Omega}\varphi^s\!\left(Dg_{i_n}(z_{n-1})\cdots Dg_{i_1}(z)\right),$$
- $$\dim_H(\Lambda)=\dim_B(\Lambda)=s_*\quad\text{where}\quad P_{\mathrm{sub}}(s_*)=0,$$
- $$\mathcal{L}=\mathcal{L}_{\mathrm{task}}+\lambda_P\left(\widehat P(s_{\mathrm{target}})\right)^2+\lambda_{\mathrm{sep}}\mathcal{L}_{\mathrm{sep}}+\lambda_J\mathcal{L}_{J}.$$

## Implementation notes

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.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.