Replace an eigendecomposition-based spectral controller in a small recurrent or state-space transition layer with explicit polynomial projectors. Each hidden state is split into invariant modes, and each mode receives a separately constrained recurrent multiplier, enabling direct suppression of unstable modes or selective retention of long-memory modes using only matrix-polynomial evaluations.
Represent each sequence chunk by a distribution over local descent-set patterns, then compose two chunks using the shuffle-algebra product rather than concatenating their embeddings alone. The module explicitly models how local order statistics change under interleaving, providing an inductive bias for event streams, code tokens, time-series segments, or mixtures of independently generated sequences.
Use the paper's finite-habitat approximation as a warning and design principle: averaging token- or state-dependent routing environments can reduce the persistence of specialized subnetworks. Partition inputs into environments, estimate environment-specific interaction kernels, and retain the heterogeneity that produces positive invasion margins instead of replacing it with one global average.
Construct the latent transition from a nondegenerate bilinear form phi and a form-compatible operator instead of from an unconstrained dense matrix. The resulting SSM has an exact orthogonal or symplectic algebraic structure, reducing transition parameter redundancy and testing whether preservation of a latent pairing improves extrapolation on reversible, parity-sensitive, or Hamiltonian-like sequence tasks.
Replace a learned recurrent transition with a sparse binary reservoir compiled from prime-count gates. Three permanently active control vertices implement copy and XNOR transitions, allowing the signal state to realize an affine feedback register over \(\mathbb F_2\) with period \(2^m-1\) using only \(O(m)\) edges and bounded in-degree. Train only an input projection and readout, or use straight-through estimators if end-to-end adaptation is required.
Construct a periodically driven hybrid recurrent state-space model whose vector field is piecewise smooth across learned switching surfaces. Engineer a transverse homoclinic intersection around a hyperbolic recurrent state; the resulting shift-like invariant set provides a controllable symbolic reservoir for sequence prediction and long-horizon generation.
Prepend an adaptive Savitzky-Golay derivative bank to a temporal neural network. For each input channel and derivative order, select the local window by minimizing Stein's unbiased risk estimate, then concatenate the raw signal with the estimated derivatives. This supplies denoised velocity and acceleration features without requiring clean derivative targets or forcing the backbone to learn unstable finite-difference filters.
Generate temporal attention or convolution weights with the Graham–Knuth–Patashnik recurrence instead of learning every lag weight independently. For nonnegative recurrence parameters, the resulting lag sequence is strongly log-concave, so its normalized kernel is naturally unimodal and suppresses high-frequency sign-free oscillations without requiring a separate smoothness penalty. The six parameters can be learned per head, channel group, or layer, giving O(1) learned parameters for an…
Use the complex-conjugate palindromic coefficient that cancels the leading temporal phase defect of oscillatory modes. Implement complex arithmetic directly or use an exactly equivalent doubled-real state, then project the final state to its real component for real-valued prediction tasks.
Represent selected hidden features as z = sqrt(N) exp(i theta), with a persistent phase and an explicitly stochastic amplitude. Regularize the ratio between coherent power |E[z]|^2 and total power E[|z|^2] toward the condensate prediction pi/4, while optionally matching higher amplitude moments.