Daily Papers

We require decomposition methods for the ABCD-matrix formulation in rotationally symmetric paraxial geometric optics when designing a multi-component optical system from a given single paraxial specification (represented by an ABCD matrix) to optimize non-paraxial specifications (e.g., optical aberrations). In this study, we propose two kinds of three-matrix decomposition of ABCD matrices by focusing on the fact that the ABCD matrices have three real-number degrees of freedom. In addition, we formulate a transformation between the two kinds of decomposition for a single matrix, which can increase or decrease the number of refraction surfaces in the optical configuration while keeping the paraxial specifications fixed. This nature is useful for the optical design of multi-component systems with optimized non-paraxial characteristics.

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.