  Toeplitz Neural Networks (TNNs) (Qin et. al. 2023) are a recent sequence
model with impressive results. They require O(n log n) computational complexity
and O(n) relative positional encoder (RPE) multi-layer perceptron (MLP) and
decay bias calls. We aim to reduce both. We first note that the RPE is a
non-SPD (symmetric positive definite) kernel and the Toeplitz matrices are
pseudo-Gram matrices. Further 1) the learned kernels display spiky behavior
near the main diagonals with otherwise smooth behavior; 2) the RPE MLP is slow.
For bidirectional models, this motivates a sparse plus low-rank Toeplitz matrix
decomposition. For the sparse component's action, we do a small 1D convolution.
For the low rank component, we replace the RPE MLP with linear interpolation
and use asymmetric Structured Kernel Interpolation (SKI) (Wilson et. al. 2015)
for O(n) complexity: we provide rigorous error analysis. For causal models,
"fast" causal masking (Katharopoulos et. al. 2020) negates SKI's benefits.
Working in the frequency domain, we avoid an explicit decay bias. To enforce
causality, we represent the kernel via the real part of its frequency response
using the RPE and compute the imaginary part via a Hilbert transform. This
maintains O(n log n) complexity but achieves an absolute speedup. Modeling the
frequency response directly is also competitive for bidirectional training,
using one fewer FFT. We set a speed state of the art on Long Range Arena (Tay
et. al. 2020) with minimal score degradation.
