Deep Equilibrium Models are Almost Equivalent to Not-so-deep Explicit
  Models for High-dimensional Gaussian Mixtures
Zenan Ling Longbo Li Zhanbo Feng Yixuan Zhang Feng Zhou Robert C. Qiu Zhenyu Liao
Deep equilibrium models (DEQs), as a typical implicit neural network, have
demonstrated remarkable success on various tasks. There is, however, a lack of
theoretical understanding of the connections and differences between implicit
DEQs and explicit neural network models. In this paper, leveraging recent
advances in random matrix theory (RMT), we perform an in-depth analysis on the
eigenspectra of the conjugate kernel (CK) and neural tangent kernel (NTK)
matrices for implicit DEQs, when the input data are drawn from a
high-dimensional Gaussian mixture. We prove, in this setting, that the spectral
behavior of these Implicit-CKs and NTKs depend on the DEQ activation function
and initial weight variances, but only via a system of four nonlinear
equations. As a direct consequence of this theoretical result, we demonstrate
that a shallow explicit network can be carefully designed to produce the same
CK or NTK as a given DEQ. Despite derived here for Gaussian mixture data,
empirical results show the proposed theory and design principle also apply to
popular real-world datasets.