# Uncovering the Spectral Bias in Diagonal State Space Models

## TL;DR

This paper investigates the spectral biases of diagonal state space models, proposing a Fourier domain initialization method that improves training efficiency and achieves state-of-the-art results on long-range tasks.

## Contribution

It introduces a Fourier domain initialization scheme for diagonal SSMs, revealing spectral biases and enabling scalable training with superior performance.

## Key findings

- Proposed S4D-DFouT initialization improves training efficiency.
- Achieved state-of-the-art results on Long Range Arena benchmark.
- Enabled training on large datasets from scratch.

## Abstract

Current methods for initializing state space models (SSMs) parameters mainly rely on the \textit{HiPPO framework}, which is based on an online approximation of orthogonal polynomials. Recently, diagonal alternatives have shown to reach a similar level of performance while being significantly more efficient due to the simplification in the kernel computation. However, the \textit{HiPPO framework} does not explicitly study the role of its diagonal variants. In this paper, we take a further step to investigate the role of diagonal SSM initialization schemes from the frequency perspective. Our work seeks to systematically understand how to parameterize these models and uncover the learning biases inherent in such diagonal state-space models. Based on our observations, we propose a diagonal initialization on the discrete Fourier domain \textit{S4D-DFouT}. The insights in the role of pole placing in the initialization enable us to further scale them and achieve state-of-the-art results on the Long Range Arena benchmark, allowing us to train from scratch on very large datasets as PathX-256.