# Toda-Krylov adaptive polynomial layer - ID: 1811 - Canonical URL: https://synthcore.org/idea/1811/toda-krylov-adaptive-polynomial-layer - API JSON: https://synthcore.org/api/idea/1811.json - API Markdown: https://synthcore.org/api/idea/1811.md - Verification status: mech_ok_no_baseline - Source: [arXiv:2608.04850](https://arxiv.org/abs/2608.04850) - Category: architecture - Solves: accuracy, stability, scalability - ML areas: ssm, graph-nn, mlp, inference-speedup - Math tags: linear-algebra, dynamical-systems, geometry - Ratings: usefulness 7/10; difficulty 5/10; novelty 7/10 ## Idea description Replace a fixed-order polynomial or recurrent state-space block by an Arnoldi basis built from a learned operator and the current input, and use subdiagonal coefficients as geometry-aware gates over Krylov depth. The gates quantify how much genuinely new direction each operator application contributes, allowing the layer to stop early near Krylov breakdown and suppress redundant or unstable directions. ## Mathematical statement For a seed vector x in C^D, define the Krylov matrix K_n(x) = [x, Hx, ..., H^(n-1)x], where H is a learned square operator. Its Gram determinant is tau_n = det(K_n(x)^* K_n(x)), with tau_0 = 1. Arnoldi orthogonalization produces an orthonormal basis u_0, ..., u_(d_K-1) and an upper-Hessenberg matrix A satisfying H U = U A. The subdiagonal entries are residual norms b_(n+1) >= 0. The Toda/Krylov identity is b_n^2 = tau_(n+1) tau_(n-1) / tau_n^2, with indexing shifted depending on whether the seed is numbered zero or one. The paper's Arnoldi matrix has diagonal entries a_n, upper entries h_(m,n), and only nearest-neighbor forward entries b_(n+1). On the real section, tau_n > 0 is equivalent to rank(K_n) = n; when tau_n = 0, the corresponding Krylov geometry is undefined rather than safely assignable a zero coefficient. The ML adaptation uses gates g_n = b_(n+1)^2 / (b_(n+1)^2 + epsilon) to weight successive basis features. ## Key formulas - $$\mathsf{K}_n(x)=[x,\mathsf{H}x,\ldots,\mathsf{H}^{n-1}x],\qquad \tau_n=\det\!\left(\mathsf{K}_n(x)^*\mathsf{K}_n(x)\right),\qquad \tau_0=1.$$ - $$b_n^2=\frac{\tau_{n+1}\tau_{n-1}}{\tau_n^2},\qquad \tau_n>0\Longleftrightarrow \operatorname{rank}(\mathsf{K}_n)=n.$$ - $$\mathsf{H}\mathsf{U}=\mathsf{U}\mathsf{A},\qquad \mathsf{A}=\begin{pmatrix}a_0&h_{0,1}&\cdots&h_{0,D-1}\\ b_1&a_1&\ddots&\vdots\\ \vdots&\ddots&\ddots&h_{D-2,D-1}\\ 0&\cdots&b_{D-1}&a_{D-1}\end{pmatrix}.$$ - $$g_n=\frac{b_{n+1}^2}{b_{n+1}^2+\epsilon},\qquad k=\min\{n:b_{n+1}^2<\delta\},\qquad y=\sum_{n=0}^{k-1}g_n u_n f_n(\mathsf{A}).$$ ## Implementation notes Integrate this at the linear transformation inside an SSM, graph-NN propagation block, or polynomial MLP layer. Given a minibatch tensor X and a learned operator H, choose a probe vector x from each example or use a batch-shared normalized probe. Build v_0 = x and v_(j+1) = H v_j. Run modified Gram-Schmidt with reorthogonalization: set r_j = v_j - U_(