Robustifying State-space Models for Long Sequences via Approximate Diagonalization

State-space models (SSMs) have recently emerged as a framework for learning long-range sequence tasks. An example is the structured state-space sequence (S4) layer, which uses the diagonal-plus-low-rank structure of the HiPPO initialization framework. However, the complicated structure of the S4 layer poses challenges; and, in an effort to address these challenges, models such as S4D and S5 have considered a purely diagonal structure. This choice simplifies the implementation, improves computational efficiency, and allows channel communication. However, diagonalizing the HiPPO framework is itself an ill-posed problem. In this paper, we propose a general solution for this and related ill-posed diagonalization problems in machine learning. We introduce a generic, backward-stable "perturb-then-diagonalize" (PTD) methodology, which is based on the pseudospectral theory of non-normal operators, and which may be interpreted as the approximate diagonalization of the non-normal matrices defining SSMs. Based on this, we introduce the S4-PTD and S5-PTD models. Through theoretical analysis of the transfer functions of different initialization schemes, we demonstrate that the S4-PTD/S5-PTD initialization strongly converges to the HiPPO framework, while the S4D/S5 initialization only achieves weak convergences. As a result, our new models show resilience to Fourier-mode noise-perturbed inputs, a crucial property not achieved by the S4D/S5 models. In addition to improved robustness, our S5-PTD model averages 87.6% accuracy on the Long-Range Arena benchmark, demonstrating that the PTD methodology helps to improve the accuracy of deep learning models.

A Quadrature Perspective on Frequency Bias in Neural Network Training with Nonuniform Data

Small generalization errors of over-parameterized neural networks (NNs) can be partially explained by the frequency biasing phenomenon, where gradient-based algorithms minimize the low-frequency misfit before reducing the high-frequency residuals. Using the Neural Tangent Kernel (NTK), one can provide a theoretically rigorous analysis for training where data are drawn from constant or piecewise-constant probability densities. Since most training data sets are not drawn from such distributions, we use the NTK model and a data-dependent quadrature rule to theoretically quantify the frequency biasing of NN training given fully nonuniform data. By replacing the loss function with a carefully selected Sobolev norm, we can further amplify, dampen, counterbalance, or reverse the intrinsic frequency biasing in NN training.

Arbitrary-Depth Universal Approximation Theorems for Operator Neural Networks

The standard Universal Approximation Theorem for operator neural networks (NNs) holds for arbitrary width and bounded depth. Here, we prove that operator NNs of bounded width and arbitrary depth are universal approximators for continuous nonlinear operators. In our main result, we prove that for non-polynomial activation functions that are continuously differentiable at a point with a nonzero derivative, one can construct an operator NN of width five, whose inputs are real numbers with finite decimal representations, that is arbitrarily close to any given continuous nonlinear operator. We derive an analogous result for non-affine polynomial activation functions. We also show that depth has theoretical advantages by constructing operator ReLU NNs of depth $2k^3+8$ and constant width that cannot be well-approximated by any operator ReLU NN of depth $k$, unless its width is exponential in $k$.