General distance-like functions on the Wasserstein space

Viscosity solutions to the eikonal equations play a fundamental role to study the geometry, topology and geodesic flows. The classical definition of viscosity solution depends on the differential structure and can not extend directly to a general metric space. However, the distance-like functions, which are exactly viscosity solutions of the eikonal equation on a Riemannian manifold, is independent of the differential structure and well-defined on a non-compact, complete, separable metric space. In this paper, we study the viscosity solutions of the eikonal equation on Wasserstein space Pp(X) (p > 1), whose ambient space X is complete, separable, non-compact, locally compact length space. But the Wasserstein space is not locally compact, the co-rays (or calibrated curves) of the distance-like functions may not exist. Fortunately, we show that if the distance-like function is induced by a sequence of closed subset diverging to infinity consisting of Dirac probability measures in the Wasserstein space, the co-rays do indeed exist. The concrete conditions of the existence of the co-rays are also provided. We also show that a distance-like function on the ambient space can induce a distance-like function on the associated Wasserstein space.Comment: 19 page

Simultaneous Active and Passive Information Transfer for RIS-Aided MIMO Systems: Iterative Decoding and Evolution Analysis

This paper investigates the potential of reconfigurable intelligent surface (RIS) for passive information transfer in a RIS-aided multiple-input multiple-output (MIMO) system. We propose a novel simultaneous active and passive information transfer (SAPIT) scheme. In SAPIT, the transmitter (Tx) and the RIS deliver information simultaneously, where the RIS information is carried through the RIS phase shifts embedded in reflected signals. We introduce the coded modulation technique at the Tx and the RIS. The main challenge of the SAPIT scheme is to simultaneously detect the Tx signals and the RIS phase coefficients at the receiver. To address this challenge, we introduce appropriate auxiliary variables to convert the original signal model into two linear models with respect to the Tx signals and one entry-by-entry bilinear model with respect to the RIS phase coefficients. With this auxiliary signal model, we develop a message-passing-based receiver algorithm. Furthermore, we analyze the fundamental performance limit of the proposed SAPIT-MIMO transceiver. Notably, we establish the state evolution to predict the receiver performance in a large-size system. We further analyze the achievable rates of the Tx and the RIS, which provides insight into the code design for sum-rate maximization. Numerical results validate our analysis and show that the SAPIT scheme outperforms the passive beamforming counterpart in achievable sum rate of the Tx and the RIS.Comment: 15 pages, 7 figure

A Quasi-Newton Subspace Trust Region Algorithm for Least-square Problems in Min-max Optimization

The first-order optimality conditions of convexly constrained nonconvex-nonconcave min-max optimization problems formulate variational inequality problems, which are equivalent to a system of nonsmooth equations. In this paper, we propose a quasi-Newton subspace trust region (QNSTR) algorithm for the least-square problem defined by the smoothing approximation of the nonsmooth equation. Based on the structure of the least-square problem, we use an adaptive quasi-Newton formula to approximate the Hessian matrix and solve a low-dimensional strongly convex quadratic program with ellipse constraints in a subspace at each step of QNSTR algorithm. According to the structure of the adaptive quasi-Newton formula and the subspace technique, the strongly convex quadratic program at each step can be solved efficiently. We prove the global convergence of QNSTR algorithm to an $\epsilon$-first-order stationary point of the min-max optimization problem. Moreover, we present numerical results of QNSTR algorithm with different subspaces for the mixed generative adversarial networks in eye image segmentation using real data to show the efficiency and effectiveness of QNSTR algorithm for solving large scale min-max optimization problems

Joint estimation of covariance matrix via Cholesky Decomposition

Ph.DDOCTOR OF PHILOSOPH