# Dilation-Matched Metropolized Dynamics - ID: 2517 - Canonical URL: https://synthcore.org/idea/2517/dilation-matched-metropolized-dynamics - API JSON: https://synthcore.org/api/idea/2517.json - API Markdown: https://synthcore.org/api/idea/2517.md - Verification status: mech_ok_no_baseline - Source: [arXiv:2608.18365](https://arxiv.org/abs/2608.18365) - Category: sampling - Solves: stability, sample-efficiency, scalability - ML areas: diffusion-sampling, optimizer, world-model - Math tags: dynamical-systems, numerical-analysis, measure-theory - Ratings: usefulness 7/10; difficulty 5/10; novelty 8/10 ## Idea description Replace the unstable classical derivative of a discretized rough energy component with a matched dilation quotient derived from its intrinsic scale recursion. Use this field inside kick-drift-kick proposals and apply an exact Metropolis correction, allowing the proposal field to be measurable and nonconservative rather than an exact neural-energy gradient. The experiment should test whether acceptance rates and posterior samples remain stable as the rough-energy resolution increases. ## Mathematical statement The rough component is the finite Weierstrass truncation $W_N(x)=\sum_{n=0}^{N}a^n\phi(b^n x)$, where $01$, and $\phi:\mathbb{R}\to\mathbb{R}$ is bounded and one-periodic. The potential is $U_N(x)=V(x)+\kappa W_N(x)$, where $V$ is a baseline potential, $\kappa\in\mathbb{R}$ is the rough amplitude, and $x>0$. Instead of the classical derivative $W_N'(x)$, which can grow geometrically with $N$, use the dilation quotient $D_{a,b}W_N(x):=[W_N(x)-aW_N(bx)]/x$. Substitution gives the exact finite-scale closure $D_{a,b}W_N(x)=[\phi(x)-a^{N+1}\phi(b^{N+1}x)]/x$, so the field does not require differentiating the increasingly oscillatory truncation. Define $G_N(x)=V'(x)+\kappa D_{a,b}W_N(x)$. A kick $p\mapsto p-(\varepsilon/2)G_N(x)$ is a triangular shear with unit Jacobian. Consequently, a reversible kick-drift-kick composition can be Metropolized even if $G_N$ is not the gradient of $U_N$. Here $p$ is momentum, $\varepsilon$ is the step size, and $M$ is a positive-definite mass matrix. ## Key formulas - $$W_N(x)=\sum_{n=0}^{N}a^n\phi\left(b^n x\right),\qquad U_N(x)=V(x)+\kappa W_N(x),\qquad 01.$$ - $$W_N(x)-aW_N(bx)=\phi(x)-a^{N+1}\phi\left(b^{N+1}x\right).$$ - $$D_{a,b}W_N(x):=\frac{W_N(x)-aW_N(bx)}{x}=\frac{\phi(x)-a^{N+1}\phi\left(b^{N+1}x\right)}{x},\qquad G_N(x):=V'(x)+\kappa D_{a,b}W_N(x).$$ - $$\begin{aligned}p_{1/2}&=p_0-\frac{\varepsilon}{2}G_N(x_0),\\x_1&=x_0+\varepsilon M^{-1}p_{1/2},\\p_1&=p_{1/2}-\frac{\varepsilon}{2}G_N(x_1),\\\alpha&=\min\left\{1,\exp\left[-H_N(x_1,p_1)+H_N(x_0,p_0)\right]\right\},\\H_N(x,p)&=U_N(x)+\frac12p^{\mathsf T}M^{-1}p.\end{aligned}$$ ## Implementation notes Integrate the method at the sampler force-evaluation point, rather than inside the neural-network forward pass. Begin with a scalar or low-dimensional latent target so the dilation operation $x\mapsto bx$ is well defined. Let the target energy be $U_N(x)=E_\theta(x)+\kappa W_N(x)$, where $E_\theta$ is a differentiable neural energy and $W_N$ is the explicit rough feature. At each leapfrog step, use autodiff to compute $g_\theta=\nabla E_\theta(x)$. Compute the rough field outside autodiff as $d_N=[W_N(x)-aW_N(bx)]/x$, or use the finite-closure expression $d_N=[\phi(x)-a^{N+1}\phi(b^{N+1}x)]/x$. Set $G_N=g_\theta+\kappa d_N$, then apply the kick-drift-kick equations above. At the end, optionally negate momentum to make reversibility explicit, and accept using the full target energy $U_N$, not the surrogate field. Restrict the initial test to $x\geq x_{\min}>0$ to avoid the quotient singularity; separately test an additive quotient near zero. The paper determines the scale recursion and closure, while acceptance rate, integrated autocorrelation time, effective sample size, and stable step size are measured empirically. Use a 2D target with a small MLP energy and resolutions $N\in\{4,8,12,16\}$. Compare classical autodiff HMC, the matched-quotient sampler, and random-walk Metropolis at equal energy evaluations. Success means that quotient-sampler acceptance, posterior moments, and ESS per evaluation vary substantially less with $N$ than classical HMC, whose step size should need resolution-dependent shrinking. ## Verification - Status: mech_ok_no_baseline - Mechanism evidence: yes - Mechanism confirmed: yes - Practical verdict: harms - Verdict: Built a scalar dilation-matched kick-drift-kick sampler with exact Metropolis correction and numerical checks. Closure error stayed below 6.5e-9; the classical derivative grew geometrically while the quotient remained bounded, and fixed-step acceptance stayed high for the quotient but collapsed for classical dynamics at high resolution. This demonstrates a real stability signal, but not a fully validated sampling-quality improvement because short-chain quotient posterior means and ESS were unstable. ### Mechanism check - Verdict: Built a scalar dilation-matched kick-drift-kick sampler with exact Metropolis correction and numerical checks. Closure error stayed below 6.5e-9; the classical derivative grew geometrically while the quotient remained bounded, and fixed-step acceptance stayed high for the quotient but collapsed for classical dynamics at high resolution. This demonstrates a real stability signal, but not a fully validated sampling-quality improvement because short-chain quotient posterior means and ESS were unstable. - Confidence: 8/10 - Limitations: Only a one-dimensional positive-domain toy target was tested. No 2D neural energy, GPU implementation, random-walk baseline, equal-FLOP accounting, long-chain diagnostics, or independent reference posterior was included. ### Practical benchmark - Paired seeds: 8 - Baseline mean: 0 - Idea mean: 0 - p-value: 0.0081 - Paired wins: 0/8 - Benchmark verdict: idea worse (significant) ## Artifacts - [bench_report.json](https://synthcore.org/code/967/bench_report.json) - [dilation_experiment.py](https://synthcore.org/code/967/dilation_experiment.py) - [report.md](https://synthcore.org/code/967/report.md) - [report_bench_2026-09-01T170548.md](https://synthcore.org/code/967/report_bench_2026-09-01T170548.md) - [rough_track.py](https://synthcore.org/code/967/rough_track.py) - [stage2_bench.py](https://synthcore.org/code/967/stage2_bench.py) - [Download all files as ZIP](https://synthcore.org/download/967) ## Disclaimer AI-generated research hypothesis, automatically tested. Not peer-reviewed.