Unverified 2026

Krasikov-Normalized Jacobi Feature Layer

Usefulness5/10

Difficulty4/10

Novelty6/10

Architecture Stability Accuracy Generalization Mlp Initialization Regularization Approximation theory Functional analysis Spectral theory

Source paper: The Erdélyi--Magnus--Nevai and Krasikov Conjectures for Jacobi Polynomials arXiv:2608.30304 ⓘ · analyzed Sep 1, 2026

AI-generated research hypothesis, automatically tested. Not peer-reviewed.

Idea description

Replace raw polynomial features in a scalar MLP expansion with endpoint-weighted orthonormal Jacobi features. The paper's envelope gives a degree- and parameter-aware scale for each feature, preventing high-degree terms or endpoint behavior from dominating gradients while preserving a richer approximation basis than low-degree monomials.

Formulas

$$w_{\alpha,\beta}(x)=(1-x)^\alpha(1+x)^\beta,\qquad \int_{-1}^{1}p_n^{(\alpha,\beta)}(x)p_m^{(\alpha,\beta)}(x)w_{\alpha,\beta}(x)\,dx=\delta_{nm}.$$

$$(1-x)^{\alpha+1/2}(1+x)^{\beta+1/2}\left|p_n^{(\alpha,\beta)}(x)\right|^2\leq C\max\left\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\right\},\qquad S=\alpha+\beta+1.$$

$$q_n(x)=(1-x)^{\alpha/2+1/4}(1+x)^{\beta/2+1/4}p_n^{(\alpha,\beta)}(x),\qquad |q_n(x)|\leq \sqrt{C B_n},\quad B_n=\max\left\{1,S^{1/3},S^{1/2}(n+1)^{-1/6}\right\}.$$

$$h_n^{(\alpha,\beta)}=\int_{-1}^{1}\left[P_n^{(\alpha,\beta)}(x)\right]^2w_{\alpha,\beta}(x)\,dx=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma(n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+1)\Gamma(n+\alpha+\beta+1)},\qquad p_n^{(\alpha,\beta)}(x)=P_n^{(\alpha,\beta)}(x)/\sqrt{h_n^{(\alpha,\beta)}}.$$

Mathematical statement

For alpha,beta >= -1/2, define the Jacobi weight w_{alpha,beta}(x)=(1-x)^alpha(1+x)^beta on [-1,1], and let p_n^{(alpha,beta)} be degree-n polynomials orthonormal under the weighted inner product integral from -1 to 1 of p_n(x)p_m(x)w_{alpha,beta}(x) dx = delta_nm. The paper proves the uniform estimate (1-x)^{alpha+1/2}(1+x)^{beta+1/2}|p_n^{(alpha,beta)}(x)|^2 <= C B_n, where S=alpha+beta+1 and B_n=max{1,S^{1/3},S^{1/2}(n+1)^{-1/6)}. Define q_n(x)=(1-x)^{alpha/2+1/4}(1+x)^{beta/2+1/4}p_n^{(alpha,beta)}(x). The theorem implies |q_n(x)| <= sqrt(C B_n) uniformly on [-1,1]. The universal constant C is absorbed into an empirical safety factor. The S^{1/3} term must remain because the paper's Bessel turning-point argument shows that it cannot generally be omitted.

Implementation notes

Integrate the construction at the scalar-feature expansion inside an MLP, immediately after a linear projection and a bounded map into [-1,1]. For an input tensor u, compute x=tanh(linear(h)). Choose fixed alpha,beta >= -1/2 and maximum degree N. Precompute S=alpha+beta+1 and B[n]=max(1,S^(1/3),S^(1/2)*(n+1)^(-1/6)). Evaluate the orthonormal Jacobi polynomials p[n](x) with a stable three-term recurrence or a trusted special-function routine. Form q[n]=(1-x)^(alpha/2+1/4)*(1+x)^(beta/2+1/4)*p[n], then normalize phi[n]=q[n]/sqrt(B[n]+epsilon). Compute each output channel as y[j]=sum over n of a[j,n]*phi[n], followed by the ordinary residual connection and normalization. In pseudocode: x=tanh(linear(h)); for n=0,...,N: p[n]=jacobi_orthonormal(n,alpha,beta,x); q[n]=(1-x)^(alpha/2+0.25)*(1+x)^(beta/2+0.25)*p[n]; phi[n]=q[n]/sqrt(B[n]+epsilon); y=sum_n a[n]*phi[n]. The theorem determines the functional form of B[n]; estimate the unknown constant C by evaluating max over a dense grid of q[n](x)^2/B[n], then multiply the normalization by a safety factor of 2 to 4. First test on CIFAR-10 using a four-layer width-256 MLP, comparing ReLU, raw Jacobi features, and normalized Jacobi features at equal parameter count. Record loss versus training FLOPs, gradient-norm percentiles, activation maxima, and validation accuracy. The method succeeds if it reduces gradient spikes and either improves loss descent or accuracy at degrees N=8 or 16, particularly when inputs are concentrated near x=1 or x=-1.

Verification

This idea has not been verified yet.

Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.

Artifacts

Artifacts unavailable.