Narayana-stable polynomial layer
Implementation & benchmark of arXiv:2607.01572 — The Narayana transformation
Source paper: The Narayana transformation arXiv:2607.01572 ⓘ · analyzed Aug 30, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Replace a monomial polynomial feature block by a fixed Narayana basis transformation. If the input polynomial has nonnegative coefficients and only real roots, the transformed polynomial is guaranteed to have only real roots as well, giving a certified stability-preserving coordinate change for polynomial neural networks.
Formulas
Mathematical statement
For an integer m >= 0 and degree n >= 0, define the Narayana polynomial N_{n,m}(x) = {}_2F_1(-n,-n-m;m+1;x). Its coefficient expansion is N_{n,m}(x) = sum_{j=0}^n c_{n,j}^{(m)}x^j, with c_{n,0}^{(m)}=1 and c_{n,j+1}^{(m)} = c_{n,j}^{(m)}((n-j)(n+m-j))/((m+1+j)(j+1)); therefore all coefficients are nonnegative. For p(z)=sum_{k=0}^d a_k z^k, define the basis transformation T_{N_m}(p)(x)=sum_{k=0}^d a_kN_{k,m}(x). The paper proves that if every a_k >= 0 and p is real-rooted, then T_{N_m}(p) is real-rooted. In the proposed neural layer, generate p from nonnegative root parameters r_i >= 0 as p(z)=s product_{i=1}^d(z+r_i), where s >= 0. Its roots are -r_i, so the theorem applies; the transformed coefficient vector is b_j=sum_{k=j}^d a_kc_{k,j}^{(m)}.
Implementation notes
Integrate this at the coefficient-feature interface of a polynomial MLP, rather than inside an ordinary dense layer. For each sample and channel, let a small neural subnetwork output unconstrained values u_1,...,u_d and v. Set r_i=softplus(u_i) and s=softplus(v), then construct the degree-d coefficient vector a of p(z)=s product_i(z+r_i) by repeated polynomial convolution. Precompute the fixed triangular matrix C for the chosen m and degree d using C[k][0]=1 and the recurrence C[k][j+1]=C[k][j](k-j)(k+m-j)/((m+1+j)(j+1)). Apply the basis change b[j]=sum_{k=j}^d a[k]C[k][j]. Normalize b by b/(sum_j b[j]+epsilon), and pass it through a learned linear projection followed by the next nonlinear block. Pseudocode is: u,v=RootNet(h); r=softplus(u); s=softplus(v); a=[s]; for i in 1..d: a=conv1d(a,[r[i],1]); b=C^T a; h_next=MLP(normalize(b)). The exact mathematical quantity is the real-rootedness-preserving map; no spectral estimate is required for training. During evaluation, form the companion matrix of p and of the transformed polynomial and measure the maximum absolute imaginary part of their eigenvalues. Begin with d=4 or 6 and m in {0,1,2} on synthetic polynomial regression, where targets are generated from known nonpositive roots, then test a small real dataset with polynomial feature prediction. Compare against a monomial coefficient MLP with equal parameter count and against the same root-constrained layer without the Narayana transform. Success is lower validation error or better extrapolation at equal parameter count, fewer coefficient explosions, and transformed-root imaginary parts at numerical precision. Also ablate fixed m against a learned soft mixture of several precomputed Narayana matrices.
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
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