1,109 research outputs found
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 -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
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