Zehui Zhou

Bangti Jin, Zehui Zhou, Jun Zou

Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in several applications. The universal approximation properties of INNs have also been established recently. However, the approximation rate of INNs is largely missing. In this work, we provide an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain, and the result shows that it can well approximate both forward and inverse maps simultaneously. Furthermore, we develop an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map, and we analyze the overall approximation error of the approach. Preliminary numerical results show the feasibility of the approach for approximating the solution operator for parameterized second-order elliptic problems.

Bangti Jin, Zehui Zhou

Stochastic variance reduced gradient (SVRG) is an accelerated version of stochastic gradient descent based on variance reduction, and is promising for solving large-scale inverse problems. In this work, we analyze SVRG and a regularized version that incorporates a priori knowledge of the problem, for solving linear inverse problems in Hilbert spaces. We prove that, with suitable constant step size schedules and regularity conditions, the regularized SVRG can achieve optimal convergence rates in terms of the noise level without any early stopping rules, provided that the truncation level is chosen suitably, and standard SVRG is also optimal for problems with nonsmooth solutions under a priori stopping rules. The analysis is based on an explicit error recursion and suitable a priori estimates on the inner loop updates with respect to the anchor point. Numerical experiments are provided to complement the theoretical analysis.