This paper addresses some aspects of the general problem of information transfer and distributed function computation in wireless networks. Many applications of wireless technology foresee networks of autonomous devices executing tasks that can be posed as distributed function computation. In today's wireless networks, the tasks of communication and (distributed) computation are performed separately, although an efficient network operation calls for approaches in which the information transfer is dynamically adapted to time-varying computation objectives. Thus, wireless communications and function computation must be tightly coupled and it is shown in this paper that information theory may play a crucial role in the design of efficient computation-aware wireless communication and networking strategies. This is explained in more detail by considering the problem of computing $\ell_p$-norms over multiple access channels.
We consider a Gaussian multiple-access channel with random user activity where the total number of users $\ell_n$ and the average number of active users $k_n$ may be unbounded. For this channel, we characterize the maximum number of bits that can be transmitted reliably per unit-energy in terms of $\ell_n$ and $k_n$. We show that if $k_n\log \ell_n$ is sublinear in $n$, then each user can achieve the single-user capacity per unit-energy. Conversely, if $k_n\log \ell_n$ is superlinear in $n$, then the capacity per unit-energy is zero. We further demonstrate that orthogonal-access schemes, which…
This paper presents flexible storage codes, a class of error-correcting codes that can recover information from a flexible number of storage nodes. As a result, one can make a better use of the available storage nodes in the presence of unpredictable node failures and reduce the data access latency. Let us assume a storage system encodes $k\ell$ information symbols over a finite field $\mathbb{F}$ into $n$ nodes, each of size $\ell$ symbols. The code is parameterized by a set of tuples $\{(R_j,k_j,\ell_j): 1 \le j \le a\}$, satisfying $k_1\ell_1=k_2\ell_2=...=k_a\ell_a$ and $k_1>k_2>...>k_a = …
This paper considers a Gaussian multiple-access channel with random user activity where the total number of users $\ell_n$ and the average number of active users $k_n$ may grow with the blocklength $n$. For this channel, it studies the maximum number of bits that can be transmitted reliably per unit-energy as a function of $\ell_n$ and $k_n$. When all users are active with probability one, i.e., $\ell_n = k_n$, it is demonstrated that if $k_n$ is of an order strictly below $n/\log n$, then each user can achieve the single-user capacity per unit-energy $(\log e)/N_0$ (where $N_0/ 2$ is the nois…
Reducing The Sub-packetization Level of Optimal-Access Cooperative MSR Codes
Ministry of Education Key Laboratory of Mathematics and Information Networks, School of Mathematical Sciences, Beijing University of Posts and Telecommunications（北京邮电大学数学与信息技术教育部重点实验室）；School of Information Science and Engineering, Shandong Agricultural University（山东农业大学信息科学与工程学院）
Let $T_ε$, $0 \le ε\le 1/2$, be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$ and let $q \ge 1$. In arXiv:1809.09696 the $\ell_q$ norm of $T_ε f$ was upperbounded by the average $\ell_q$ norm of conditional expectations of $f$, given sets whose elements are chosen at random with probability $λ$, depending on $q$ and on $ε$. In this note we prove this inequality for integer $q \ge 2$ with a better (smaller) parameter $λ$. The new inequality is tight for characteristic functions of subcubes. As an application, following a…
RSP-Based Analysis for Sparsest and Least $\ell_1$-Norm Solutions to Underdetermined Linear Systems
近年来，最坏情况分析、概率分析和经验证明等方法被用来解决这样一个基本问题：当$\ell_1$ -极小化方法找到欠定线性系统的稀疏解时，它在什么情况下是最优的？在本文中，确定性分析，植根于经典的线性规划理论，进行进一步解决这个问题。首先给出了线性方程组最小$\ell_1$ -范数解唯一的一个充分必要条件。从这个条件出发，我们推出了一个稀疏解与线性方程组的唯一最小$\ell_1$ -范数解一致的充要条件是所谓的\emph{range space property}（RSP）在该解处成立.这产生了广泛的理解之间的关系$\ell_0$ -和$\ell_1$ -最小化问题。我们的分析显示，可吸入颗粒物确实是这两个问题之间关系的核心。通过基于RSP的分析，可以在很大程度上解决这一领域的几个重要问题。例如，如何通过确定性分析有效地解释当前理论与$\ell_1$ -最小化的实际数值性能之间的差距，以及如果线性系统有多个最稀疏解，那么什么时候$\ell_1$ -最小化能保证找到其中之一？此外，本文还引入了新的矩阵性质（如\emph{RSP of order $K$}和\emph{Weak-RSP of order $K$}），建立了基于$K$阶RSP的稀疏信号恢复理论。
Wireless communication systems that rely on orthogonal frequency-division multiplexing (OFDM) suffer from a high peak-to-average (power) ratio (PAR), which necessitates power-inefficient radio-frequency (RF) chains to avoid an increase in error-vector magnitude (EVM) and out-of-band (OOB) emissions. The situation is further aggravated in massive multiuser (MU) multiple-input multiple-output (MIMO) systems that would require hundreds of linear RF chains. In this paper, we present a novel approach to joint precoding and PAR reduction that builds upon a novel $\ell^p\!-\!\ell^q$-norm formulation,…
ShanghaiTech University（上海科技大学）；University of Washington（华盛顿大学）；Beihang University（北京航空航天大学）
This paper investigates the optimality conditions for characterizing the local minimizers of the constrained optimization problems involving an $\ell_p$ norm ($0<p<1$) of the variables, which may appear in either the objective or the constraint. This kind of problems have strong applicability to a wide range of areas since usually the $\ell_p$ norm can promote sparse solutions. However, the nonsmooth and non-Lipschtiz nature of the $\ell_p$ norm often cause these problems difficult to analyze and solve. We provide the calculation of the subgradients of the $\ell_p$ norm and the normal cones of…
Carnegie Mellon University（卡内基梅隆大学）；University of Pennsylvania（宾夕法尼亚大学）
An evolving line of machine learning works observe empirical evidence that suggests interpolating estimators -- the ones that achieve zero training error -- may not necessarily be harmful. This paper pursues theoretical understanding for an important type of interpolators: the minimum $\ell_{1}$-norm interpolator, which is motivated by the observation that several learning algorithms favor low $\ell_1$-norm solutions in the over-parameterized regime. Concretely, we consider the noisy sparse regression model under Gaussian design, focusing on linear sparsity and high-dimensional asymptotics (so…
