{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace a dense directional feature map with a finite stable-zonotope layer: a small set of large random vectors creates a hyperplane arrangement over the input sphere, while a cheaper small-jump residual gives continuous features inside each cell. The layer exposes an explicit compute-versus-resolution knob through the jump cutoff and can be used either as an embedding before an MLP or as a lightweight router for mixture-of-experts blocks.",
 "formulas_latex": [
  "$$X(u)=\\sum_{k=1}^{\\infty}\\Gamma_k^{-1/\\alpha}\\varepsilon_k\\mathbf{1}_{\\{\\langle u,\\varepsilon_k\\rangle\u003e0\\}},\\qquad u\\in S^{d-1},\\quad 0\u003c\\alpha\u003c1.$$",
  "$$X_{\\mathrm{large}}(u)=\\sum_{k=1}^{K}w_k\\varepsilon_k\\mathbf{1}_{\\{\\langle u,\\varepsilon_k\\rangle\u003e0\\}},\\qquad X_{\\mathrm{small}}(u)=\\sum_{k=K+1}^{K+R}w_k\\varepsilon_k\\mathbf{1}_{\\{\\langle u,\\varepsilon_k\\rangle\u003e0\\}},\\quad w_k=\\Gamma_k^{-1/\\alpha}.$$",
  "$$\\dim_H\\overline{X(S^{d-1})}\\le (d-1)\\alpha,\\qquad \\dim_H\\ext Z_\\alpha=(d-1)\\alpha\\quad\\text{almost surely}.$$",
  "$$T_n=C_N\\sum_{Q\\in\\mathcal Q_n}(1+M_Q)^N V_Q^{\\,s},\\qquad \\eta_n=\\max_{Q\\in\\mathcal Q_n}V_Q.$$"
 ],
 "id": 2786,
 "implementation": "Integrate this as a residual embedding or router immediately before an MLP or MoE block. Given an input vector $h\\in\\mathbb R^d$, compute $u=h/(\\|h\\|_2+10^{-6})$. Sample and freeze $K+R$ unit vectors $\\varepsilon_k$; generate ordered positive weights using $\\Gamma_k=\\sum_{j=1}^kE_j$ with $E_j\\sim\\operatorname{Exp}(1)$, then set $w_k=\\Gamma_k^{-1/\\alpha}$. Clip the weights to $[-w_{\\max},w_{\\max}]$ or normalize the resulting code by its RMS to avoid rare Poisson outliers. Compute half-space gates $g_k=\\mathbf 1[\\langle u,\\varepsilon_k\\rangle\u003e0]$. Form the large code $z_L=\\sum_{k=1}^K w_kg_k\\varepsilon_k$ and small code $z_S=\\sum_{k=K+1}^{K+R}w_kg_k\\varepsilon_k$. Concatenate $[u,z_L,z_S]$, apply a learned linear projection, and add the result as a residual to the hidden state. For MoE routing, feed $[g_1,\\ldots,g_K,\\langle u,\\varepsilon_1\\rangle,\\ldots,\\langle u,\\varepsilon_K\\rangle]$ to a small router and cache the large gates for repeated tokens. The mathematical quantities computed exactly are the Poisson-ordered heavy-tailed weights and hyperplane signs; estimate practical cell complexity by counting distinct $K$-bit gate patterns per minibatch. First test a parameter-matched MLP and a small Transformer on CIFAR-10, TinyImageNet, or a language-modeling subset, against a dense projection and random Fourier features. Sweep $\\alpha\\in\\{0.3,0.5,0.7,0.9\\}$ and $K/R\\in\\{1/4,1/2,1\\}$. Record validation accuracy, activation RMS, number of occupied angular cells, router FLOPs, and parameter count. Success is equal or better accuracy at lower learned-parameter count or router cost, with stable activation statistics and useful cell diversity; failure is gate collapse, exploding activations, or no improvement over the dense baseline at equal compute.",
 "math_summary": "The paper parametrizes exposed points using $X(u)=\\sum_{k=1}^{\\infty}\\Gamma_k^{-1/\\alpha}\\varepsilon_k\\mathbf{1}_{\\{\\langle u,\\varepsilon_k\\rangle\u003e0\\}}$, where $u\\in S^{d-1}$ is the normalized input direction, $\\Gamma_k$ is the $k$th arrival time of a unit-rate Poisson process, $\\varepsilon_k$ are independent uniform vectors on $S^{d-1}$, and $0\u003c\\alpha\u003c1$. The heavy-tailed weights $\\Gamma_k^{-1/\\alpha}$ are ordered from large to small. For a finite cutoff, split $X=X_{\\mathrm{large}}+X_{\\mathrm{small}}$: the signs of the large terms define a finite arrangement of hyperplanes $\\langle u,\\varepsilon_k\\rangle=0$, hence angular cells on which the large contribution is constant. The upper-bound construction uses $T_n=C_N\\sum_{Q\\in\\mathcal Q_n}(1+M_Q)^N V_Q^s$ and $\\eta_n=\\max_{Q\\in\\mathcal Q_n}V_Q$, where $N=d-1$, $Q$ are cells, $M_Q$ counts relevant large-jump hyperplanes, and $V_Q$ bounds the image diameter of the small-jump residual on $Q$. The theorem states $\\dim_H\\overline{X(S^{d-1})}\\le N\\alpha$, with a matching lower bound, motivating $\\alpha$ as a controllable feature-complexity parameter.",
 "math_tags": [
  "probability",
  "stochastic-processes",
  "geometry",
  "metric-geometry"
 ],
 "ml_areas": [
  "embedding",
  "mlp",
  "moe-routing"
 ],
 "paper": {
  "arxiv_id": "2608.28004",
  "arxiv_url": "https://arxiv.org/abs/2608.28004",
  "summary_what_math_gives_to_ml": "The paper constructs a random directional field whose output is a sum of heavy-tailed vector increments gated by spherical half-spaces, and proves that its image has almost-sure Hausdorff dimension exactly $(d-1)\\alpha$. The transferable asset is the explicit multiscale separation between a finite set of large jumps, which induces a hyperplane arrangement, and a small-jump residual that controls local image diameters. This suggests a finite neural feature or routing layer in which large random vectors define discrete angular cells while a truncated stable residual supplies within-cell variation, with the truncation level controlling compute, memory, and approximation error.",
  "title": "Hausdorff Dimension of the Set of Extreme Points of a Random Countable Stable Zonotope",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 6
 },
 "solves": [
  "scalability",
  "generalization",
  "memory"
 ],
 "title": "Stable Hyperplane Feature Layer",
 "url": "https://synthcore.org/idea/2786/stable-hyperplane-feature-layer",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)"
 }
}