  State space models (SSM) have recently been shown to be very effective as a
deep learning layer as a promising alternative to sequence models such as RNNs,
CNNs, or Transformers. The first version to show this potential was the S4
model, which is particularly effective on tasks involving long-range
dependencies by using a prescribed state matrix called the HiPPO matrix. While
this has an interpretable mathematical mechanism for modeling long
dependencies, it introduces a custom representation and algorithm that can be
difficult to implement. On the other hand, a recent variant of S4 called DSS
showed that restricting the state matrix to be fully diagonal can still
preserve the performance of the original model when using a specific
initialization based on approximating S4's matrix. This work seeks to
systematically understand how to parameterize and initialize such diagonal
state space models. While it follows from classical results that almost all
SSMs have an equivalent diagonal form, we show that the initialization is
critical for performance. We explain why DSS works mathematically, by showing
that the diagonal restriction of S4's matrix surprisingly recovers the same
kernel in the limit of infinite state dimension. We also systematically
describe various design choices in parameterizing and computing diagonal SSMs,
and perform a controlled empirical study ablating the effects of these choices.
Our final model S4D is a simple diagonal version of S4 whose kernel computation
requires just 2 lines of code and performs comparably to S4 in almost all
settings, with state-of-the-art results for image, audio, and medical
time-series domains, and averaging 85\% on the Long Range Arena benchmark.
