arXiv subjects

Xiaoping Xue

Publications and source records attributed to Xiaoping Xue.

8 recordsLinked to original sources

arXiv2026

Wasserstein geometry of nonnegative measures on finite Markov chains I: Gradient flow

We investigate a Benamou--Brenier type transportation metric for nonnegative measures on a finite reversible Markov chain, which endows the space of measures with a Riemannian structure. Using this geometric framework, we identify a generalized heat equation with source as the gradient flow of the discrete entropy. Moreover, by means of a local \L{}ojasiewicz inequality, we prove exponential convergence of the flow to a unique equilibrium. Our results clarify the role of the Benamou--Brenier formulation in discrete optimal transport for nonnegative measures and provide a coherent geometric interpretation of generalized diffusion equations with source terms.

math.AP↗

arXiv2026

Wasserstein geometry of nonnegative measures on finite Markov chains II: Geodesic and duality formulae

In this paper, we investigate the geodesic structure and the associated Kantorovich-type duality for a Benamou-Brenier-type transportation metric defined on the space of nonnegative measures over a finite reversible Markov chain. The metric is introduced through a dynamic formulation that combines transport and source costs along solutions of a nonconservative continuity equation, where mass variation is constrained to occur along a fixed strictly positive reference direction. We show that geodesics associated with this metric exhibit a non-locality property: almost every time, they are supported on the whole state space, independently of the choice of endpoints. Moreover, along optimal curves, the source term displays a characteristic temporal profile, with mass creation occurring at early times and subsequent decay as the curve approaches the target measure. As an application of this property, we compare our metric with the shift-transport distance and prove that the latter is always bounded above by our metric. Finally, we establish a Kantorovich-type duality formula in terms of Hamilton-Jacobi subsolutions, which provides a characterization of the metric and highlights the role of the momentum associated with geodesic curves.

math.AP↗

arXiv2022

Robust Secure Transmission Design for IRS-Assisted mmWave Cognitive Radio Networks

Cognitive radio networks (CRNs) and millimeter wave (mmWave) communications are two major technologies to enhance the spectrum efficiency (SE). Considering that the SE improvement in the CRNs is limited due to the interference temperature imposed on the primary user (PU), and the severe path loss and high directivity in mmWave communications make it vulnerable to blockage events, we introduce an intelligent reflecting surface (IRS) into mmWave CRNs. Due to the estimation mismatch and the passivity of Eavesdroppers (Eves), perfect channel state information (CSI) of wiretap links is challenging to obtain, which promotes our research on robust secure beamforming (BF) design in the IRS-assisted mmWave CRNs. This paper considers the collaborate scenario of Eves, which allows us to investigate the BF design in the harsh eavesdropping environment. Specifically, by using a uniform linear array (ULA) at the cognitive base station (CBS) and a uniform planar array (UPA) at the IRS, and supposing that imperfect CSIs of angle-of-departures for wiretap links are known, we formulate a constrained problem to maximize the worst-case achievable secrecy rate (ASR) of the secondary user (SU) by jointly designing the transmit BF at the CBS and reflect BF at the IRS. To solve the non-convex problem with coupled variables, an efficient alternating optimization algorithm is proposed. Finally, simulation results indicate that the ASR performance of our proposed algorithm has a small gap with that of the optimal solution with perfect CSI compared with the other benchmarks.

cs.IT↗

arXiv2021

Joint space-time analyticity of mild solutions to the Navier-Stokes equations

In this paper, we show the optimal decay rate estimates of the space-time derivatives and the joint space-time analyticity of solutions to the Navier-Stokes equations. As it is known from the Hartogs's theorem, for a complex function with two complex variables, the joint analyticity with respect to two variables can be derived from combining of analyticity with respect to each variable. However, as a function of two real variables for space and time, the joint space-time analyticity of solutions to the Navier-Stokes equations cannot be directly obtained from the combination of space analyticity and time analyticity. Our result seems to be the first quantitative result for the joint space-time analyticity of solutions to the Navier-Stokes equations, and the proof only involves real variable methods. Moreover, the decay rate estimates also yield the bounds on the growth (in time) of radius of space analyticity, time analyticity, and joint space-time analyticity of solutions.

math.AP↗

arXiv2021

Secure and Energy Efficient Transmission for IRS-Assisted Cognitive Radio Networks

The spectrum efficiency (SE) and security of the secondary users (SUs) in the cognitive radio networks (CRNs) have become two main issues due to the limitation interference to the primary users (PUs) and the shared spectrum with the PUs. Intelligent reflecting surface (IRS) has been recently proposed as a revolutionary technique which can help to enhance the SE and physical layer security of wireless communications. This paper investigates the application of IRS in an underlay CRN, where a multi-antenna cognitive base station (CBS) utilizes spectrum assigned to the PU to communicate with a SU via IRS in the presence of multiple coordinated eavesdroppers (Eves). To achieve the trade-off between the secrecy rate (SR) and energy consumption, we investigate the secrecy energy efficiency (SEE) maximization problem by jointly designing the transmit beamforming at the CBS and the reflect beamforming at the IRS. To solve the non-convex problem with coupled variables, we propose an iterative alternating optimization algorithm to solve the sub-problems alternately, by utilizing an iterative penalty function based algorithm for sub-problem 1 and the difference of two-convex functions method for sub-problem 2. Furthermore, we provide a second-order-cone-programming (SOCP) approximation approach to reduce the computational complexity. Finally, the simulation results demonstrate that IRS can help significantly improve the SE and enhance the physical layer security in the CRNs. Moreover, the effectiveness and superiority of our proposed algorithm in achieving the trade-off between the SR and energy consumption are verified.

cs.IT↗

arXiv2021

Optimal decay rates and space-time analyticity of solutions to the Patlak-Keller-Segel equations

Based on some elementary estimates for the space-time derivatives of the heat kernel, we use a bootstrapping approach to establish the optimal decay rates for the $L^q(\mathbb{R}^d)$ ($1\leq q\leq\infty$, $d\in\mathbb{N}$) norm of the space-time derivatives of solutions to the (modified) Patlak-Keller-Segel equations with initial data in $L^1(\mathbb{R}^d)$, which implies the joint space-time analyticity of solutions. When the $L^1(\mathbb{R}^d)$ norm of the initial datum is small, the upper bound for the decay estimates is global in time, which yields a lower bound on the growth rate of the radius of space-time analyticity in time. As a byproduct, the space analyticity is obtained for any initial data in $L^1(\mathbb{R}^d)$. The decay estimates and space-time analyticity are also established for solutions bounded in both space and time variables. The results can be extended to a more general class of equations, including the Navier-Stokes equations.

math.AP↗

arXiv2021

Smoothing fast iterative hard thresholding algorithm for $\ell_0$ regularized nonsmooth convex regression problem

We investigate a class of constrained sparse regression problem with cardinality penalty, where the feasible set is defined by box constraint, and the loss function is convex, but not necessarily smooth. First, we put forward a smoothing fast iterative hard thresholding (SFIHT) algorithm for solving such optimization problems, which combines smoothing approximations, extrapolation techniques and iterative hard thresholding methods. The extrapolation coefficients can be chosen to satisfy $\sup_k β_k=1$ in the proposed algorithm. We discuss the convergence behavior of the algorithm with different extrapolation coefficients, and give sufficient conditions to ensure that any accumulation point of the iterates is a local minimizer of the original cardinality penalized problem. In particular, for a class of fixed extrapolation coefficients, we discuss several different update rules of the smoothing parameter and obtain the convergence rate of $O(\ln k/k)$ on the loss and objective function values. Second, we consider the case in which the loss function is Lipschitz continuously differentiable, and develop a fast iterative hard thresholding (FIHT) algorithm to solve it. We prove that the iterates of FIHT converge to a local minimizer of the problem that satisfies a desirable lower bound property. Moreover, we show that the convergence rate of loss and objective function values are $o(k^{-2})$. Finally, some numerical examples are presented to illustrate the theoretical results.

math.OC↗

arXiv2021

Global existence and spatial analyticity for a nonlocal flux with fractional diffusion

In this paper, we study a one dimensional nonlinear equation with diffusion $-ν(-\partial_{xx})^{\fracα{2}}$ for $0\leq α\leq 2$ and $ν>0$. We use a viscous-splitting algorithm to obtain global nonnegative weak solutions in space $L^1(\mathbb{R})\cap H^{1/2}(\mathbb{R})$ when $0\leqα\leq 2$. For subcritical $1<α\leq 2$ and critical case $α=1$, we obtain global existence and uniqueness of nonnegative spatial analytic solutions. We use a fractional bootstrap method to improve the regularity of mild solutions in Bessel potential spaces for subcritical case $1<α\leq 2$. Then, we show that the solutions are spatial analytic and can be extended globally. For the critical case $α=1$, if the initial data $ρ_0$ satisfies $-ν<\infρ_0<0$, we use the characteristics methods for complex Burgers equation to obtain a unique spatial analytic solution to our target equation in some bounded time interval. If $ρ_0\geq0$, the solution exists globally and converges to steady state.

math.AP↗