Extreme Point Pursuit -- Part II: Further Error Bound Analysis and Applications

In the first part of this study, a convex-constrained penalized formulation was studied for a class of constant modulus (CM) problems. In particular, the error bound techniques were shown to play a vital role in providing exact penalization results. In this second part of the study, we continue our error bound analysis for the cases of partial permutation matrices, size-constrained assignment matrices and non-negative semi-orthogonal matrices. We develop new error bounds and penalized formulations for these three cases, and the new formulations possess good structures for building computationally efficient algorithms. Moreover, we provide numerical results to demonstrate our framework in a variety of applications such as the densest k-subgraph problem, graph matching, size-constrained clustering, non-negative orthogonal matrix factorization and sparse fair principal component analysis.

Extreme Point Pursuit -- Part I: A Framework for Constant Modulus Optimization

This study develops a framework for a class of constant modulus (CM) optimization problems, which covers binary constraints, discrete phase constraints, semi-orthogonal matrix constraints, non-negative semi-orthogonal matrix constraints, and several types of binary assignment constraints. Capitalizing on the basic principles of concave minimization and error bounds, we study a convex-constrained penalized formulation for general CM problems. The advantage of such formulation is that it allows us to leverage non-convex optimization techniques, such as the simple projected gradient method, to build algorithms. As the first part of this study, we explore the theory of this framework. We study conditions under which the formulation provides exact penalization results. We also examine computational aspects relating to the use of the projected gradient method for each type of CM constraint. Our study suggests that the proposed framework has a broad scope of applicability.

Multilayer Simplex-structured Matrix Factorization for Hyperspectral Unmixing with Endmember Variability

Given a hyperspectral image, the problem of hyperspectral unmixing (HU) is to identify the endmembers (or materials) and the abundance (or endmembers' contributions on pixels) that underlie the image. HU can be seen as a matrix factorization problem with a simplex structure in the abundance matrix factor. In practice, hyperspectral images may exhibit endmember variability (EV) effects -- the endmember matrix factor varies from one pixel to another. In this paper we consider a multilayer simplex-structured matrix factorization model to account for the EV effects. Our multilayer model is based on the postulate that if we arrange the varied endmembers as an expanded endmember matrix, that matrix exhibits a low-rank structure. A variational inference-based maximum-likelihood estimation method is employed to tackle the multilayer factorization problem. Simulation results are provided to demonstrate the performance of our multilayer factorization method.

Transmitting Data Through Reconfigurable Intelligent Surface: A Spatial Sigma-Delta Modulation Approach

Transmitting data using the phases on reconfigurable intelligent surfaces (RIS) is a promising solution for future energy-efficient communication systems. Recent work showed that a virtual phased massive multiuser multiple-input-multiple-out (MIMO) transmitter can be formed using only one active antenna and a large passive RIS. In this paper, we are interested in using such a system to perform MIMO downlink precoding. In this context, we may not be able to apply conventional MIMO precoding schemes, such as the simple zero-forcing (ZF) scheme, and we typically need to design the phase signals by solving optimization problems with constant modulus constraints or with discrete phase constraints, which pose challenges with high computational complexities. In this work, we propose an alternative approach based on Sigma-Delta ($\Sigma\Delta$) modulation, which is classically famous for its noise-shaping ability. Specifically, first-order $\Sigma\Delta$ modulation is applied in the spatial domain to handle phase quantization in generating constant envelope signals. Under some mild assumptions, the proposed phased $\Sigma\Delta$ modulator allows us to use the ZF scheme to synthesize the RIS reflection phases with negligible complexity. The proposed approach is empirically shown to achieve comparable bit error rate performance to the unquantized ZF scheme.

Spatial Sigma-Delta Modulation for Coarsely Quantized Massive MIMO Downlink: Flexible Designs by Convex Optimization

This paper considers the context of multiuser massive MIMO downlink precoding with low-resolution digital-to-analog converters (DACs) at the transmitter. This subject is motivated by the consideration that it is expensive to employ high-resolution DACs for practical massive MIMO implementations. The challenge with using low-resolution DACs is to overcome the detrimental quantization error effects. Recently, spatial Sigma-Delta modulation has arisen as a viable way to put quantization errors under control. This approach takes insight from temporal Sigma-Delta modulation in classical DAC studies. Assuming a 1D uniform linear transmit antenna array, the principle is to shape the quantization errors in space such that the shaped quantization errors are pushed away from the user-serving angle sector. In the previous studies, spatial Sigma-Delta modulation was performed by direct application of the basic first- and second-order modulators from the Sigma-Delta literature. In this paper, we develop a general Sigma-Delta modulator design framework for any given order, for any given number of quantization levels, and for any given angle sector. We formulate our design as a problem of maximizing the signal-to-quantization-and-noise ratios experienced by the users. The formulated problem is convex and can be efficiently solved by available solvers. Our proposed framework offers the alternative option of focused quantization error suppression in accordance with channel state information. Our framework can also be extended to 2D planar transmit antenna arrays. We perform numerical study under different operating conditions, and the numerical results suggest that, given a moderate number of quantization levels, say, 5 to 7 levels, our optimization-based Sigma-Delta modulation schemes can lead to bit error rate performance close to that of the unquantized counterpart.

A Spatial Sigma-Delta Approach to Mitigation of Power Amplifier Distortions in Massive MIMO Downlink

In massive multiple-input multiple-output (MIMO) downlink systems, the physical implementation of the base stations (BSs) requires the use of cheap and power-efficient power amplifiers (PAs) to avoid high hardware cost and high power consumption. However, such PAs usually have limited linear amplification ranges. Nonlinear distortions arising from operation beyond the linear amplification ranges can significantly degrade system performance. Existing approaches to handle the nonlinear distortions, such as digital predistortion (DPD), typically require accurate knowledge, or acquisition, of the PA transfer function. In this paper, we present a new concept for mitigation of the PA distortions. Assuming a uniform linear array (ULA) at the BS, the idea is to apply a Sigma-Delta ($\Sigma \Delta$) modulator to spatially shape the PA distortions to the high-angle region. By having the system operating in the low-angle region, the received signals are less affected by the PA distortions. To demonstrate the potential of this spatial $\Sigma \Delta$ approach, we study the application of our approach to the multi-user MIMO-orthogonal frequency division modulation (OFDM) downlink scenario. A symbol-level precoding (SLP) scheme and a zero-forcing (ZF) precoding scheme, with the new design requirement by the spatial $\Sigma \Delta$ approach being taken into account, are developed. Numerical simulations are performed to show the effectiveness of the developed $\Sigma \Delta$ precoding schemes.

An Efficient Global Algorithm for One-Bit Maximum-Likelihood MIMO Detection

There has been growing interest in implementing massive MIMO systems by one-bit analog-to-digital converters (ADCs), which have the benefit of reducing the power consumption and hardware complexity. One-bit MIMO detection arises in such a scenario. It aims to detect the multiuser signals from the one-bit quantized received signals in an uplink channel. In this paper, we consider one-bit maximum-likelihood (ML) MIMO detection in massive MIMO systems, which amounts to solving a large-scale nonlinear integer programming problem. We propose an efficient global algorithm for solving the one-bit ML MIMO detection problem. We first reformulate the problem as a mixed integer linear programming (MILP) problem that has a massive number of linear constraints. The massive number of linear constraints raises computational challenges. To solve the MILP problem efficiently, we custom build a light-weight branch-and-bound tree search algorithm, where the linear constraints are incrementally added during the tree search procedure and only small-size linear programming subproblems need to be solved at each iteration. We provide simulation results to demonstrate the efficiency of the proposed method.

Accelerated and Deep Expectation Maximization for One-Bit MIMO-OFDM Detection

In this paper we study the expectation maximization (EM) technique for one-bit MIMO-OFDM detection (OMOD). Arising from the recent interest in massive MIMO with one-bit analog-to-digital converters, OMOD is a massive-scale problem. EM is an iterative method that can exploit the OFDM structure to process the problem in a per-iteration efficient fashion. In this study we analyze the convergence rate of EM for a class of approximate maximum-likelihood OMOD formulations, or, in a broader sense, a class of problems involving regression from quantized data. We show how the SNR and channel conditions can have an impact on the convergence rate. We do so by making a connection between the EM and the proximal gradient methods in the context of OMOD. This connection also gives us insight to build new accelerated and/or inexact EM schemes. The accelerated scheme has faster convergence in theory, and the inexact scheme provides us with the flexibility to implement EM more efficiently, with convergence guarantee. Furthermore we develop a deep EM algorithm, wherein we take the structure of our inexact EM algorithm and apply deep unfolding to train an efficient structured deep net. Simulation results show that our accelerated exact/inexact EM algorithms run much faster than their standard EM counterparts, and that the deep EM algorithm gives promising detection and runtime performances.

SISAL Revisited

Simplex identification via split augmented Lagrangian (SISAL) is a popularly-used algorithm in blind unmixing of hyperspectral images. Developed by Jos\'{e} M. Bioucas-Dias in 2009, the algorithm is fundamentally relevant to tackling simplex-structured matrix factorization, and by extension, non-negative matrix factorization, which have many applications under their umbrellas. In this article, we revisit SISAL and provide new meanings to this quintessential algorithm. The formulation of SISAL was motivated from a geometric perspective, with no noise. We show that SISAL can be explained as a heuristic from a probabilistic simplex component analysis framework, which is statistical and is, by principle, more powerful in accommodating the presence of noise. The algorithm for SISAL was designed based on a successive convex approximation method, with a focus on practical utility. It was not known, by analyses, whether the SISAL algorithm has any kind of guarantee of convergence to a stationary point. By establishing associations between the SISAL algorithm and a line-search-based proximal gradient method, we confirm that SISAL can indeed guarantee convergence to a stationary point. Our re-explanation of SISAL also reveals new formulations and algorithms. The performance of these new possibilities is demonstrated by numerical experiments.