The classical alternating minimization (or projection) algorithm has been successful in the context of solving optimization problems over two variables. The iterative nature and simplicity of the algorithm has led to its application to many areas such as signal processing, information theory, control, and finance. A general set of sufficient conditions for the convergence and correctness of the algorithm is quite well-known when the underlying problem parameters are fixed. In many practical situations, however, the underlying problem parameters are changing over time, and the use of an adaptive algorithm is more appropriate. In this paper, we study such an adaptive version of the alternating minimization algorithm. As a main result of this paper, we provide a general set of sufficient conditions for the convergence and correctness of the adaptive algorithm. Perhaps surprisingly, these conditions seem to be the minimal ones one would expect in such an adaptive setting. We present applications of our results to adaptive decomposition of mixtures, adaptive log-optimal portfolio selection, and adaptive filter design.
We consider the total variation (TV) minimization problem used for compressive sensing and solve it using the generalized alternating projection (GAP) algorithm. Extensive results demonstrate the high performance of proposed algorithm on compressive sensing, including two dimensional images, hyperspectral images and videos. We further derive the Alternating Direction Method of Multipliers (ADMM) framework with TV minimization for video and hyperspectral image compressive sensing under the CACTI and CASSI framework, respectively. Connections between GAP and ADMM are also provided.
In this paper, we propose an algorithm based on the Alternating Minimization technique to solve the uplink massive MIMO detection problem. The proposed algorithm provides a lower complexity compared to the conventional MMSE detection technique, especially when the number of user equipment (UE) antennas is close to the number of base station (BS) antennas. This improvement is obtained without any matrix inversion. Moreover, the algorithm re-formulates the maximum likelihood (ML) detection problem as a sum of convex functions based on decomposing the received vector into multiple vectors. Each v…
Index codes reduce the number of bits broadcast by a wireless transmitter to a number of receivers with different demands and with side information. It is known that the problem of finding optimal linear index codes is NP-hard. We investigate the performance of different heuristics based on rank minimization and matrix completion methods, such as alternating projections and alternating minimization, for constructing linear index codes over the reals. As a summary of our results, the alternating projections method gives the best results in terms of minimizing the number of broadcast bits and co…
Hybrid precoding is a cost-effective approach to support directional transmissions for millimeter wave (mmWave) communications. While existing works on hybrid precoding mainly focus on single-user single-carrier transmission, in practice multicarrier transmission is needed to combat the much increased bandwidth, and multiuser MIMO can provide additional spatial multiplexing gains. In this paper, we propose a new hybrid precoding structure for multiuser OFDM mmWave systems, which greatly simplifies the hybrid precoder design and is able to approach the performance of the fully digital precoder.…
Alternating Minimization Schemes for Computing Rate-Distortion-Perception Functions with $f$-Divergence Perception Constraints
Communication Systems Department at EURECOM（EURECOM通信系统部门）；Andalusian Research Institute in Data Science and Computational Intelligence (DaSCI)（数据科学与计算智能安达卢西亚研究机构）；University of Granada（Granada大学）
Microsoft Research India（微软印度研究院）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）
Phase retrieval problems involve solving linear equations, but with missing sign (or phase, for complex numbers) information. More than four decades after it was first proposed, the seminal error reduction algorithm of (Gerchberg and Saxton 1972) and (Fienup 1982) is still the popular choice for solving many variants of this problem. The algorithm is based on alternating minimization; i.e. it alternates between estimating the missing phase information, and the candidate solution. Despite its wide usage in practice, no global convergence guarantees for this algorithm are known. In this paper, w…
We consider the problem of solving mixed random linear equations with $k$ components. This is the noiseless setting of mixed linear regression. The goal is to estimate multiple linear models from mixed samples in the case where the labels (which sample corresponds to which model) are not observed. We give a tractable algorithm for the mixed linear equation problem, and show that under some technical conditions, our algorithm is guaranteed to solve the problem exactly with sample complexity linear in the dimension, and polynomial in $k$, the number of components. Previous approaches have requir…
Generalized Linear Models (GLMs), where a random vector $\mathbf{x}$ is observed through a noisy, possibly nonlinear, function of a linear transform $\mathbf{z}=\mathbf{Ax}$ arise in a range of applications in nonlinear filtering and regression. Approximate Message Passing (AMP) methods, based on loopy belief propagation, are a promising class of approaches for approximate inference in these models. AMP methods are computationally simple, general, and admit precise analyses with testable conditions for optimality for large i.i.d. transforms $\mathbf{A}$. However, the algorithms can easily dive…
Millimeter wave (mmWave) communications has been regarded as a key enabling technology for 5G networks. In contrast to conventional multiple-input-multiple-output (MIMO) systems, precoding in mmWave MIMO cannot be performed entirely at baseband using digital precoders, as only a limited number of signal mixers and analog-to-digital converters (ADCs) can be supported considering their cost and power consumption. As a cost-effective alternative, a hybrid precoding transceiver architecture, combining a digital precoder and an analog precoder, has recently received considerable attention. However,…
Colorado School of Mines（科罗拉多矿业学院）；Johns Hopkins University（约翰斯·霍普金斯大学）
The (global) Lipschitz smoothness condition is crucial in establishing the convergence theory for most optimization methods. Unfortunately, most machine learning and signal processing problems are not Lipschitz smooth. This motivates us to generalize the concept of Lipschitz smoothness condition to the relative smoothness condition, which is satisfied by any finite-order polynomial objective function. Further, this work develops new Bregman-divergence based algorithms that are guaranteed to converge to a second-order stationary point for any relatively smooth problem. In addition, the proposed…
This paper presents a novel adaptive reduced-rank {multi-input multi-output} (MIMO) equalization scheme and algorithms based on alternating optimization design techniques for MIMO spatial multiplexing systems. The proposed reduced-rank equalization structure consists of a joint iterative optimization of two equalization stages, namely, a transformation matrix that performs dimensionality reduction and a reduced-rank estimator that retrieves the desired transmitted symbol. The proposed reduced-rank architecture is incorporated into an equalization structure that allows both decision feedback an…
This paper presents a novel distributed low-rank scheme and adaptive algorithms for distributed estimation over wireless networks. The proposed distributed scheme is based on a transformation that performs dimensionality reduction at each agent of the network followed by transmission of a reduced set of parameters to other agents and reduced-dimension parameter estimation. Distributed low-rank joint iterative estimation algorithms based on alternating optimization strategies are developed, which can achieve significantly reduced communication overhead and improved performance when compared wit…
The low-tubal-rank tensor model has been recently proposed for real-world multidimensional data. In this paper, we study the low-tubal-rank tensor completion problem, i.e., to recover a third-order tensor by observing a subset of its elements selected uniformly at random. We propose a fast iterative algorithm, called {\em Tubal-Alt-Min}, that is inspired by a similar approach for low-rank matrix completion. The unknown low-tubal-rank tensor is represented as the product of two much smaller tensors with the low-tubal-rank property being automatically incorporated, and Tubal-Alt-Min alternates b…
In this article, we present a new approach to optimizing the energy efficiency of the cost-efficiency of quantized hybrid pre-encoding (HP) design. We present effective alternating minimization algorithms (AMA) based on the zero gradient method to produce completely connected structures (CCSs) and partially connected structures (PCSs). Alternative minimization algorithms offer lower complexity by introducing orthogonal constraints on digital pre-codes to concurrently maximize computing complexity and communication power. As a result, by improving CCS through advanced phase extraction, the alte…
A sum inverse energy efficiency (SIEE) minimization problem is solved. Compared with conventional sum energy efficiency (EE) maximization problems, minimizing SIEE achieves a better fairness. The paper begins by proposing a framework for solving sum-fraction minimization (SFMin) problems, then uses a novel transform to solve the SIEE minimization problem in a multiple base station (BS) system. After the reformulation into a multi-convex problem, the alternating direction method of multipliers (ADMM) is used to further simplify the problem. Numerical results confirm the efficiency of the transf…
