{
 "artifacts": null,
 "category": "architecture",
 "description": "Replace raw polynomial or Fourier features for a scalar coordinate x with iterated Stieltjes-integral features generated by a monotone data-adaptive coordinate g. Use a Newton-form basis so the model can add higher-order terms or new interpolation knots without recomputing all previous coefficients.",
 "formulas_latex": [
  "$$d_g(x,y)=\\left\\lvert g(x)-g(y)\\right\\rvert$$",
  "$$g_{y_0,0}(x)=1,\\qquad g_{y_0,k+1}(x)=\\int_{y_0}^{x}g_{y_0,k}(t)\\,dg(t),\\qquad \\operatorname{P}_n=\\operatorname{span}\\{1,g_{y_0,1},\\ldots,g_{y_0,n}\\}$$",
  "$$\\{1,N[x_1],N[x_1,x_2],\\ldots,N[x_1,\\ldots,x_n]\\}\\text{ is a basis of }\\operatorname{P}_n$$",
  "$$h_{m+1}(x)=h_m(x)+c_{m+1}N[x_1,\\ldots,x_m,x],\\qquad c_{m+1}=f[x_1,\\ldots,x_{m+1}]$$"
 ],
 "id": 148,
 "implementation": "Integrate this as a small feature head before an MLP, or as the scalar-coordinate branch of a coordinate network. For each scalar input x, first fit or learn a monotone coordinate g. A cheap fixed choice is the clipped empirical CDF g(x)=rank(x)/(N+1); a learnable choice is a monotone rational-quadratic spline with positive bin widths and heights, initialized to the identity. Let z=g(x), choose ordered knots z_1\u003c...\u003cz_K, and compute generalized Newton features recursively. In the continuous case use B_0(z)=1 and B_{j+1}(z)=(z-z_{j+1})B_j(z). For a genuinely Stieltjes version, approximate the integral recurrence by quadrature over sorted training values and include jump increments Δg(t). Concatenate B_0,...,B_K with the original hidden representation and apply a learned linear projection. Initialize the constant coefficient and add one feature every few epochs using the Newton update h ← h+c_j B_j, freezing old coefficients during the first validation interval. Normalize every feature by its training-set RMS and regularize the spline derivative toward a positive bounded range. The paper supplies the basis construction and lower-triangular incremental update; estimate g, knot locations, feature scales, and coefficients empirically. First test on one-dimensional Friedman regression and a UCI tabular regression dataset with a two-layer MLP baseline at equal parameter count. Compare the raw MLP, ordinary polynomial features, Fourier features, and Stieltjes-Newton features. Success means lower validation MSE at equal parameters, faster loss reduction at equal FLOPs, and an advantage when samples are highly nonuniform or contain sharp distributional jumps.",
 "math_summary": "The paper defines the pseudometric d_g(x,y)=|g(x)-g(y)|, so inputs are close when their g-coordinates are close. Define Stieltjes monomials recursively by g_{y0,0}(x)=1 and g_{y0,k+1}(x)=∫_{y0}^{x} g_{y0,k}(t) dg(t), where y0 is a base point and dg is the Stieltjes measure induced by g. The generalized polynomial space is P_n=span{1,g_{y0,1},...,g_{y0,n}}. The paper states that {1,N[x1],N[x1,x2],...,N[x1,...,xn]} is a basis of P_n, where N denotes generalized Newton basis functions, and that this basis induces a lower-triangular interpolation matrix. Thus a new node can be added without recomputing all previous coefficients. The interpolation matrix uses rows (1,g_{y0,1}(x_i),...,g_{y0,n}(x_i)); for g-ordered distinct nodes its determinant is positive. In the neural adaptation, g is a nondecreasing scalar function, z=g(x), and the continuous special case gives g_{y0,k}(x)=(z-z0)^k/k!. Jumps can be retained by numerically evaluating the Stieltjes recurrence.",
 "math_tags": [
  "approximation-theory",
  "measure-theory",
  "linear-algebra"
 ],
 "ml_areas": [
  "mlp",
  "embedding",
  "initialization"
 ],
 "paper": {
  "arxiv_id": "2608.24884",
  "arxiv_url": "https://arxiv.org/abs/2608.24884",
  "summary_what_math_gives_to_ml": "The paper builds a polynomial-like function space from iterated Stieltjes integrals, while preserving interpolation, Newton-coordinate, Hermite, and best-approximation properties. Its transferable asset is an adaptive one-dimensional geometry: the pseudometric d_g(x,y)=|g(x)-g(y)| measures distance in a transformed coordinate, allowing basis resolution to follow the data distribution rather than the raw coordinate. A practical neural-network adaptation is a compact Stieltjes-Newton feature head using a fixed or learned monotone g, with incremental basis expansion when additional capacity is needed.",
  "title": "Stieltjes polynomial interpolation",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 6,
  "usefulness": 5
 },
 "solves": [
  "accuracy",
  "scalability",
  "sample-efficiency"
 ],
 "title": "Stieltjes-Newton Feature Head",
 "url": "https://synthcore.org/idea/148/stieltjes-newton-feature-head",
 "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
   }
  }
 }
}