{
 "artifacts": null,
 "category": "architecture",
 "description": "Build a fixed binary-sign feature layer whose evaluation can be restricted to N carefully selected sign patterns while preserving the squared L2 norm of every feature vector in the first-N Rademacher subspace exactly. Use a Hadamard matrix when order N exists; otherwise use N+1 patterns and positive weights satisfying the same isotropy equation.",
 "formulas_latex": [
  "$$\\int_{\\Omega}f^{2}\\,d\\mu=\\sum_{j=1}^{m}\\lambda_{j}f^{2}(\\xi^{j})$$",
  "$$A\\Lambda A^{\\mathsf T}=I_N,\\qquad A_{l,k}=r_l(\\Delta_{i_k}),\\qquad \\Lambda=\\operatorname{diag}(\\lambda_1,\\ldots,\\lambda_m)$$",
  "$$\\int_0^1\\left(\\sum_{l=0}^{N-1}c_l r_l(x)\\right)^2dx=\\sum_{k=1}^{m}\\lambda_k\\left(\\sum_{l=0}^{N-1}c_l A_{l,k}\\right)^2$$",
  "$$HH^{\\mathsf T}=NI_N\\quad\\Longrightarrow\\quad H\\left(\\frac{1}{N}I_N\\right)H^{\\mathsf T}=I_N$$"
 ],
 "id": 154,
 "implementation": "Integrate this at an embedding or MLP bottleneck that currently evaluates a large binary-sign feature bank. Let the input be u in R^d, choose an integer N, and construct a fixed sign code A in {+1,-1}^{N times m} from a Hadamard matrix when possible. The module computes coefficients c=g_theta(u) in R^N and represents the feature function by sign evaluations z_k=sum_l A_{l,k}c_l. Retain only the m=N Hadamard columns and set z_tilde_k=z_k divided by sqrt(N); then sum_k z_tilde_k squared equals the coefficient norm exactly. Pseudocode: `c = encoder(u)`; `z = transpose(A) @ c`; `z = sqrt(lambda) * z`; `energy = (z*z).sum()`; `y = decoder(z)`. If N is not a Hadamard order, search over N+1 sign columns and solve A diag(lambda) A^T=I_N for nonnegative weights, using softplus weights and a matrix-residual penalty if necessary. The exact part comes from the isotropy identity; decoder reconstruction error is empirical. First test on CIFAR-10 with a small MLP replacing a 2^N-wide binary-feature layer for N=8, 12, and 16. Compare dense, random-sign, and Hadamard variants at equal parameter count. Success means lower feature memory and wall-clock cost with no accuracy loss on data represented by the learned coefficient subspace.",
 "math_summary": "Let r_0,...,r_{N-1} be the first N Rademacher functions on [0,1], each constant with value plus or minus 1 on dyadic intervals Delta_0,...,Delta_{2^N-1}. For f(x)=sum_{l=0}^{N-1} c_l r_l(x), define A in {+1,-1}^{N times m} by A_{l,k}=r_l(Delta_{i_k}), where i_k are selected cells, and Lambda=diag(lambda_1,...,lambda_m). Exact discretization is A Lambda A^T=I_N, equivalently the integral of f squared equals sum_k lambda_k f(xi_k)^2 for every f in the Rademacher span. If H is a Hadamard matrix with H H^T=N I_N, choosing A=H and lambda_k=1/N gives exact norm preservation with m=N. The paper states that the minimum number of nodes is N when a Hadamard matrix of order N exists, and N+1 otherwise.",
 "math_tags": [
  "linear-algebra",
  "harmonic-analysis",
  "combinatorics"
 ],
 "ml_areas": [
  "mlp",
  "embedding",
  "memory",
  "inference-speedup"
 ],
 "paper": {
  "arxiv_id": "2608.25058",
  "arxiv_url": "https://arxiv.org/abs/2608.25058",
  "summary_what_math_gives_to_ml": "The paper gives a constructive exact quadrature principle for the span of the first N Rademacher functions: an L2 norm can be recovered from only N or N+1 evaluations rather than all 2^N dyadic sign patterns. The transferable object is the weighted binary design A Lambda A^T = I_N, which is an exactly norm-preserving sketch for a known N-dimensional sign-feature subspace. This can replace dense or exhaustive evaluations of binary-feature modules with a fixed Hadamard-derived subset of rows, reducing memory and inference cost while preserving feature energy exactly. Exactness applies only to activations in the modeled Rademacher span, so experiments should measure both savings and approximation error outside that subspace.",
  "title": "On exact discretization of the $L_2$-norm in the space spanned by the first $N$ Rademacher functions",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 5,
  "usefulness": 7
 },
 "solves": [
  "memory",
  "speedup",
  "accuracy"
 ],
 "title": "Hadamard-exact binary feature sketch",
 "url": "https://synthcore.org/idea/154/hadamard-exact-binary-feature-sketch",
 "verification": {
  "peer_reviewed": false,
  "status": "unverified",
  "status_label": "Unverified",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}