{
 "artifacts": null,
 "category": "sampling",
 "description": "Represent an image or spatial latent state in an orthogonal multiresolution wavelet basis and learn the conditional distribution of detail coefficients at each scale given all coarser coefficients. At inference time, sample coarse coefficients first and recursively sample finer coefficients, using a small conditional network or a few local Langevin steps at each level instead of running a long pixel-space Markov chain. The mechanism should remove critical slowing down caused by long-range correlations, with the number of scale transitions growing as \\(\\log_2 L\\) for linear image size \\(L\\).",
 "download_zip": "https://synthcore.org/download/1129",
 "formulas_latex": [
  "$$p_0(\\phi_0)=Z^{-1}e^{-E_0(\\phi_0)},\\qquad E_0(\\phi_0)=\\frac{1}{2T}\\sum_{i,j}\\phi_0(i)K_{ij}\\phi_0(j)+\\sum_i\\left[\\phi_0^2(i)+\\lambda\\left(\\phi_0^2(i)-1\\right)^2\\right].$$",
  "$$w=W\\phi,\\qquad p(w)=p(c_J)\\prod_{j=J}^{1}p\\!\\left(d_j\\mid c_J,d_J,\\ldots,d_{j+1}\\right).$$",
  "$$q_\\theta(w)=q_{\\theta_0}(c_J)\\prod_{j=J}^{1}q_{\\theta_j}\\!\\left(d_j\\mid h_j\\right),\\qquad h_j=\\operatorname{Context}(c_J,d_J,\\ldots,d_{j+1}).$$",
  "$$\\tau_{\\mathrm{total}}\\approx\\sum_{j=1}^{J}\\tau_j,\\qquad \\tau_j\\approx\\tau_*=O(1)\\ \\Longrightarrow\\ \\tau_{\\mathrm{total}}=O(J)=O(\\log_2 L).$$"
 ],
 "id": 2918,
 "implementation": "Integrate this as an alternative inference sampler for a spatial neural energy model or score model. Start with a 2-D convolutional energy model \\(E_\\psi(x)\\), and choose an orthogonal Haar or Daubechies wavelet transform with \\(J=\\log_2 L\\) levels. During training, transform every training image into \\(w=(c_J,d_J,\\ldots,d_1)\\). Train one conditional density model per scale, such as a small U-Net, masked convolution, conditional normalizing flow, or diagonal Gaussian mixture, using the negative log likelihood \\(-\\log q_{\\theta_j}(d_j\\mid h_j)\\). At inference, sample \\(c_J\\) from the coarsest model, then iterate from \\(j=J\\) down to \\(1\\): construct \\(h_j\\), sample \\(d_j\\), and concatenate it with the already sampled coarser coefficients before moving to the next level. Reconstruct the image with \\(x=W^{-1}w\\). If exact conditional learning is difficult, run one to five Langevin or Metropolis updates within each conditional model rather than thousands of pixel-space updates. Pseudocode is: `w=sample_coarse(); for j=J,...,1: h=Context(w_coarse); d=ConditionalNet[j].sample(h); w.append(d); x=InverseWavelet(w)`. The wavelet transform and chain-rule factorization come from the paper; conditional networks, likelihood calibration, and optional MCMC corrections are empirical choices. First test on 32x32, 64x64, and 128x128 Ising-like fields or CIFAR-10 with a convolutional energy model, comparing pixel Langevin, multiscale Langevin, and the learned sampler at equal network evaluations. Measure integrated autocorrelation time for low-frequency magnetization or image mean, Fourier structure factors, and held-out likelihood or FID. The prediction is that \\(\\tau_j\\) remains approximately constant across levels and total decorrelation cost grows linearly with \\(J\\), rather than with \\(L\\). A log-linear fit of cost versus \\(L\\) should therefore be compatible with \\(O(\\log_2 L)\\). Reducing conditional-network width should increase observable bias and KL divergence while leaving per-level mixing time comparatively unchanged.",
 "math_summary": "The source model uses the dimensionless Gibbs density \\(p_0(\\phi_0)=Z^{-1}\\exp[-E_0(\\phi_0)]\\), with \\(E_0(\\phi_0)=\\frac{1}{2T}\\sum_{i,j}\\phi_0(i)K_{ij}\\phi_0(j)+\\sum_i[\\phi_0^2(i)+\\lambda(\\phi_0^2(i)-1)^2]\\). Here \\(\\phi_0(i)\\) is the field at lattice site \\(i\\), \\(K\\) is the interaction kernel, \\(T\\) is the temperature-like parameter, \\(\\lambda\\) controls the local quartic potential, and \\(Z\\) normalizes the density. Let \\(W\\) be an orthogonal wavelet transform and write \\(w=(c_J,d_J,d_{J-1},\\ldots,d_1)\\), where \\(c_J\\) is the coarsest coefficient block and \\(d_j\\) is the detail block at scale \\(j\\). The exact density factorizes by the chain rule as \\(p(w)=p(c_J)\\prod_{j=J}^{1}p(d_j\\mid c_J,d_J,\\ldots,d_{j+1})\\). WCRG replaces each exact conditional by a learned conditional \\(q_{\\theta_j}(d_j\\mid h_j)\\), where \\(h_j\\) contains coarser coefficients and local spatial context. If the conditional decorrelation time \\(\\tau_j\\) is bounded independently of scale, the total scale complexity is \\(O(J)=O(\\log_2 L)\\).",
 "math_tags": [
  "statistical-mechanics",
  "renormalization",
  "probability",
  "stochastic-processes",
  "harmonic-analysis"
 ],
 "ml_areas": [
  "diffusion-sampling",
  "vae",
  "diffusion",
  "inference-speedup"
 ],
 "paper": {
  "arxiv_id": "2608.31114",
  "arxiv_url": "https://arxiv.org/abs/2608.31114",
  "summary_what_math_gives_to_ml": "The paper provides a constructive multiscale sampling mechanism for frustrated systems: transform configurations into wavelet coefficients, learn conditional distributions of fine-scale coefficients given coarser coefficients, and sample recursively from coarse to fine. Its key transferable asset is a factorization of a difficult high-dimensional distribution into scale-conditioned sampling problems whose conditional dynamics remain decorrelated in O(1) sweeps per scale, giving O(log L) total scale complexity. For neural networks, the most direct use is a learned wavelet-coordinate sampler for spatial energy-based models or diffusion-like inference, replacing pixel-space local MCMC with hierarchical conditional generation. The falsifiable tradeoff is that mixing should become nearly scale-independent while distributional error decreases as the conditional model becomes more expressive.",
  "title": "Overcoming critical slowing down in frustrated spin systems by learned multiscale sampling",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 6,
  "novelty": 6,
  "usefulness": 7
 },
 "solves": [
  "speedup",
  "stability",
  "accuracy"
 ],
 "title": "Wavelet Conditional Sampler for Neural EBMs",
 "url": "https://synthcore.org/idea/2918/wavelet-conditional-sampler-for-neural-ebms",
 "verification": {
  "peer_reviewed": false,
  "status": "queued_mechanism",
  "status_label": "Queued — mechanism check",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}