{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace recursive adaptive coordinate traversal in a CPPN-based architecture generator with one static multiresolution candidate tensor and an active boolean mask. Evaluate all candidate coordinates in parallel, compute local CPPN-output variance over each parent cell's children, and retain only children whose parent is active and whose predicted variation exceeds a threshold. This preserves adaptive refinement while enabling population batching, JIT compilation, GPU vectorization, and memory-bounded streaming.",
 "formulas_latex": [
  "$$\\mathcal{G}_{D}=\\bigcup_{d=0}^{D}\\mathrm{Grid}_{d},\\qquad \\mathrm{Grid}_{d}=\\left\\{\\left(\\frac{2i+1}{2^{d+1}}-1,\\frac{2j+1}{2^{d+1}}-1\\right):i,j\\in[0,2^{d+1})\\right\\}$$",
  "$$N_{\\mathrm{total}}=\\frac{4^{D+2}-4}{3},\\qquad \\mathbf{W}=\\mathrm{vmap}(\\mathrm{CPPN})(s,T)$$",
  "$$V_p=\\frac{1}{|C(p)|}\\sum_{c\\in C(p)}\\left\\|y_{p,c}-\\bar y_p\\right\\|_2^2,\\qquad \\bar y_p=\\frac{1}{|C(p)|}\\sum_{c\\in C(p)}y_{p,c}$$",
  "$$a_{p,c}=a_p\\,\\mathbf{1}[V_p\u003e\\tau_d],\\qquad \\mathrm{output}=\\mathrm{CPPN}(T)\\odot a$$"
 ],
 "id": 2758,
 "implementation": "Integrate this at the topology-discovery or sparse-coordinate-generation stage of a CPPN-based network, neural field, adaptive CNN, or sparse graph generator; leave the CPPN architecture unchanged for the first experiment. First precompute tensors `coords[d]` for levels `d=0,...,D`, together with integer tensors `parent_of_child[d]` and `children_of_parent[d]`. Coordinates should have shape `[N_d, 2]`; mappings should use int32. For a population of genomes, evaluate the CPPN with a compiled batched call having shape `[population, N_d, coordinate_features]`. Reshape each level's child outputs to `[population, N_parent, 4, K]`, calculate `mean_child`, then calculate `V = mean((child_output - mean_child)^2, dim=(child, output))`. Propagate `active[d+1] = active[d][parent_of_child[d]] \u0026 (V[d] \u003e tau[d])`, forcing the root active. Apply the resulting mask to instantiate nodes, edges, sparse tokens, or neural-field samples. A practical threshold is `tau[d] = alpha * median(V[d])`; alternatively retain the top fixed fraction of parent cells to impose a compute budget. For large depths, evaluate coordinates in fixed-size chunks and write masks or CPPN outputs to host memory rather than materializing every activation on the GPU. The mathematical quantities to reproduce exactly are the coordinate grid, parent-child hierarchy, batched CPPN evaluation, variance score, and hierarchical mask; threshold schedule, chunk size, and CPPN-output decoding are tunable engineering choices. Start with XOR or a 2-D image-coordinate neural field, population 256, and depths 4–7. Compare against recursive traversal using identical CPPN parameters and thresholds. Record generation time, peak GPU memory, discovered-node count, and solve rate. Success means at least 5x lower topology-evaluation time at equal node count, no loss in solve rate, and stable scaling with population size without recompiling once per genome.",
 "math_summary": "The paper constructs the complete multiresolution grid $\\mathcal{G}_D=\\bigcup_{d=0}^{D}\\mathrm{Grid}_d$, where $D$ is the maximum refinement depth and $d$ is a grid level. Each level contains normalized cell centers $((2i+1)/2^{d+1}-1,(2j+1)/2^{d+1}-1)$ for integer indices $i,j\\in[0,2^{d+1})$. The total number of candidates is $N_{\\mathrm{total}}=(4^{D+2}-4)/3$. The batched evaluation $\\mathbf{W}=\\mathrm{vmap}(\\mathrm{CPPN})(s,T)$ evaluates a population of CPPNs with parameters or genomes $s$ at static coordinate tensor $T$. For each parent cell $p$, let $C(p)$ be its four children, let $y_{p,c}\\in\\mathbb{R}^{K}$ be the $K$ CPPN outputs at child $c$, and let $\\bar y_p$ be their mean. The local variation score is $V_p=|C(p)|^{-1}\\sum_{c\\in C(p)}\\|y_{p,c}-\\bar y_p\\|_2^2$. With $a_p\\in\\{0,1\\}$ denoting whether parent $p$ is active and $\\tau_d$ a threshold at depth $d$, define the child mask $a_{p,c}=a_p\\mathbf{1}[V_p\u003e\\tau_d]$. The root is forced active. High output variation therefore triggers refinement, while locally constant CPPN outputs suppress unnecessary nodes.",
 "math_tags": [
  "combinatorics",
  "numerical-analysis",
  "tensor-decomposition"
 ],
 "ml_areas": [
  "moe-routing",
  "graph-nn",
  "cnn",
  "inference-speedup"
 ],
 "paper": {
  "arxiv_id": "2608.27612",
  "arxiv_url": "https://arxiv.org/abs/2608.27612",
  "summary_what_math_gives_to_ml": "The paper's transferable contribution is a reformulation of irregular adaptive computation as static tensors plus boolean masks. A complete multiresolution coordinate set lets all candidate locations and genomes be evaluated by one compiled batched function, while hierarchical variance filtering preserves adaptive refinement without recursive control flow. This pattern can accelerate CPPN-based topology discovery, adaptive neural fields, sparse graph construction, and other models that currently perform per-example dynamic traversal. The most practical transfer is eager candidate generation combined with masked refinement and chunked streaming when the finest grid exceeds device memory.",
  "title": "Tensor-Accelerated Eager Multi-Resolution Grids for Evolving Large-Scale Substrates",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 4,
  "novelty": 5,
  "usefulness": 6
 },
 "solves": [
  "speedup",
  "scalability",
  "accuracy"
 ],
 "title": "Eager Masked Multiresolution Discovery",
 "url": "https://synthcore.org/idea/2758/eager-masked-multiresolution-discovery",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}