Search

This work proposes a variational inference (VI) framework for hyperspectral unmixing in the presence of endmember variability (HU-EV). An EV-accounted noisy linear mixture model (LMM) is considered, and the presence of outliers is also incorporated into the model. Following the marginalized maximum likelihood (MML) principle, a VI algorithmic structure is designed for probabilistic inference for HU-EV. Specifically, a patch-wise static endmember assumption is employed to exploit spatial smoothness and to try to overcome the ill-posed nature of the HU-EV problem. The design facilitates lightweight, continuous optimization-based updates under a variety of endmember priors. Some of the priors, such as the Beta prior, were previously used under computationally heavy, sampling-based probabilistic HU-EV methods. The effectiveness of the proposed framework is demonstrated through synthetic, semi-real, and real-data experiments.

As communication systems advance towards the future 6G era, the incorporation of large-scale antenna arrays in base stations (BSs) presents challenges such as increased hardware costs and energy consumption. To address these issues, the use of one-bit analog-to-digital converters (ADCs)/digital-to-analog converters (DACs) has gained significant attentions. This paper focuses on one-bit multiple-input multiple-output (MIMO) detection in an uplink multiuser transmission scenario where the BS employs one-bit ADCs. One-bit quantization retains only the sign information and loses the amplitude information, which poses a unique challenge in the corresponding detection problem. The maximum-likelihood (ML) formulation of one-bit MIMO detection has a challenging likelihood function that hinders the application of many high-performance detectors developed for classic MIMO detection (under high-resolution ADCs). While many approximate methods for the ML detection problem have been studied, it lacks an efficient global algorithm. This paper fills this gap by proposing an efficient branch-and-bound algorithm, which is guaranteed to find the global solution of the one-bit ML MIMO detection problem. Additionally, a new amplitude retrieval (AR) detection approach is developed, incorporating explicit amplitude variables into the problem formulation. The AR approach yields simpler objective functions that enable the development of efficient algorithms offering both global and approximate solutions. The paper also contributes to the computational complexity analysis of both ML and AR detection problems. Extensive simulations are conducted to demonstrate the effectiveness and efficiency of the proposed formulations and algorithms.

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.

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.

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 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.

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.

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.

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.

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.

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 an approximation scheme from a probabilistic simplex component analysis framework, which is statistical and is principally 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.

In this article the author reviews Jos\'e Bioucas-Dias' key contributions to hyperspectral unmixing (HU), in memory of him as an influential scholar and for his many beautiful ideas introduced to the hyperspectral community. Our story will start with vertex component analysis (VCA) -- one of the most celebrated HU algorithms, with more than 2,000 Google Scholar citations. VCA was pioneering, invented at a time when HU research just began to emerge, and it shows sharp insights on a then less-understood subject. Then we will turn to SISAL, another widely-used algorithm. SISAL is not only a highly successful algorithm, it is also a demonstration of its inventor's ingenuity on applied optimization and on smart formulation for practical noisy cases. Our tour will end with dependent component analysis (DECA), perhaps a less well-known contribution. DECA adopts a statistical inference framework, and the author's latest research indicates that such framework has great potential for further development, e.g., there are hidden connections between SISAL and DECA. The development of DECA shows foresight years ahead, in that regard., Comment: to appear in IGARSS 2021, Special Session on "The Contributions of Jos\'e Manuel Bioucas-Dias to Remote Sensing Data Processing"

Symbol-level precoding (SLP) has recently emerged as a new paradigm for physical-layer transmit precoding in multiuser multi-input-multi-output (MIMO) channels. It exploits the underlying symbol constellation structure, which the conventional paradigm of linear precoding does not, to enhance symbol-level performance such as symbol error probability (SEP). It also allows the precoder to take a more general form than linear precoding. This paper aims to better understand the relationships between SLP and linear precoding, subsequent design implications, and further connections beyond the existing SLP scope. Focused on the quadrature amplitude modulation (QAM) constellations, our study is built on a basic signal observation, namely, that SLP can be equivalently represented by a zero-forcing (ZF) linear precoding scheme augmented with some appropriately chosen symbol-dependent perturbation terms, and that some extended form of SLP is equivalent to a vector perturbation (VP) nonlinear precoding scheme augmented with the above-noted perturbation terms. We examine how insights arising from this perturbed ZF and VP interpretations can be leveraged to i) substantially simplify the optimization of certain SLP design criteria, namely, total or peak power minimization subject to SEP quality guarantees; and ii) draw connections with some existing SLP designs. We also touch on the analysis side by showing that, under the total power minimization criterion, the basic ZF scheme is a near-optimal SLP scheme when the QAM order is very high -- which gives a vital implication that SLP is more useful for lower-order QAM cases. Numerical results further indicate the merits and limitations of the different SLP designs derived from the perturbed ZF and VP interpretations.

This study presents PRISM, a probabilistic simplex component analysis approach to identifying the vertices of a data-circumscribing simplex from data. The problem has a rich variety of applications, the most notable being hyperspectral unmixing in remote sensing and non-negative matrix factorization in machine learning. PRISM uses a simple probabilistic model, namely, uniform simplex data distribution and additive Gaussian noise, and it carries out inference by maximum likelihood. The inference model is sound in the sense that the vertices are provably identifiable under some assumptions, and it suggests that PRISM can be effective in combating noise when the number of data points is large. PRISM has strong, but hidden, relationships with simplex volume minimization, a powerful geometric approach for the same problem. We study these fundamental aspects, and we also consider algorithmic schemes based on importance sampling and variational inference. In particular, the variational inference scheme is shown to resemble a matrix factorization problem with a special regularizer, which draws an interesting connection to the matrix factorization approach. Numerical results are provided to demonstrate the potential of PRISM.

In federated learning, models are learned from users' data that are held private in their edge devices, by aggregating them in the service provider's "cloud" to obtain a global model. Such global model is of great commercial value in, e.g., improving the customers' experience. In this paper we focus on two possible areas of improvement of the state of the art. First, we take the difference between user habits into account and propose a quadratic penalty-based formulation, for efficient learning of the global model that allows to personalize local models. Second, we address the latency issue associated with the heterogeneous training time on edge devices, by exploiting a hierarchical structure modeling communication not only between the cloud and edge devices, but also within the cloud. Specifically, we devise a tailored block coordinate descent-based computation scheme, accompanied with communication protocols for both the synchronous and asynchronous cloud settings. We characterize the theoretical convergence rate of the algorithm, and provide a variant that performs empirically better. We also prove that the asynchronous protocol, inspired by multi-agent consensus technique, has the potential for large gains in latency compared to a synchronous setting when the edge-device updates are intermittent. Finally, experimental results are provided that corroborate not only the theory, but also show that the system leads to faster convergence for personalized models on the edge devices, compared to the state of the art., Comment: 31 pages, 5 figures. Codes available at this url {https://github.com/REIYANG/FedBCD}. To appear in AAAI 2021

Consider a robust multiple-input single-output downlink beamforming optimization problem in a frequency division duplexing system. The base station (BS) sends training signals to the users, and every user estimates the channel coefficients, quantizes the gain and the direction of the estimated channel and sends them back to the BS. Suppose that the channel state information at the transmitter is imperfectly known mainly due to the channel direction quantization errors, channel estimation errors and outdated channel effects. The actual channel is modeled as in an uncertainty set composed of two inequality homogeneous and one equality inhomogeneous quadratic constraints, in order to account for the aforementioned errors and effects. Then the transmit power minimization problem is formulated subject to robust signal-to-noise-plus-interference ratio constraints. Each robust constraint is transformed equivalently into a quadratic matrix inequality (QMI) constraint with respect to the beamforming vectors. The transformation is accomplished by an equivalent phase rotation process and the strong duality result for a quadratically constrained quadratic program. The minimization problem is accordingly turned into a QMI problem, and the problem is solved by a restricted linear matrix inequality relaxation with additional valid convex constraints. Simulation results are presented to demonstrate the performance of the proposed method, and show the efficiency of the restricted relaxation.

Adopting one-bit analog-to-digital convertors (ADCs) for massive multiple-input multiple-output (MIMO) implementations has great potential in reducing the hardware cost and power consumption. However, distortions caused by quantization raise great challenges. In MIMO orthogonal frequency-division modulation (OFDM) detection, coarse quantization renders the orthogonal separation among subcarriers inapplicable, forcing us to deal with a problem that has a very large problem size. In this paper we study the expectation-maximization (EM) approach for one-bit MIMO-OFDM detection. The idea is to iteratively decouple the MIMO-OFDM detection problem among subcarriers. Using the perspective of block coordinate descent, we describe inexact variants of the classical EM method for providing more flexible and computationally efficient designs. Simulation results are provided to illustrate the potential of the divide-and-conquer strategy enabled by EM.

Many contemporary applications in signal processing and machine learning give rise to structured non-convex non-smooth optimization problems that can often be tackled by simple iterative methods quite effectively. One of the keys to understanding such a phenomenon---and, in fact, one of the very difficult conundrums even for experts---lie in the study of "stationary points" of the problem in question. Unlike smooth optimization, for which the definition of a stationary point is rather standard, there is a myriad of definitions of stationarity in non-smooth optimization. In this article, we give an introduction to different stationarity concepts for several important classes of non-convex non-smooth functions and discuss the geometric interpretations and further clarify the relationship among these different concepts. We then demonstrate the relevance of these constructions in some representative applications and how they could affect the performance of iterative methods for tackling these applications., Comment: Accepted for publication in IEEE Signal Processing Magazine, 2020

In this paper we consider maximum-likelihood (ML) MIMO detection under one-bit quantized observations and binary symbol constellations. This problem is motivated by the recent interest in adopting coarse quantization in massive MIMO systems--as an effective way to scale down the hardware complexity and energy consumption. Classical MIMO detection techniques consider unquantized observations, and many of them are not applicable to the one-bit MIMO case. We develop a new non-convex optimization algorithm for the one-bit ML MIMO detection problem, using a strategy called homotopy optimization. The idea is to transform the ML problem into a sequence of approximate problems, from easy (convex) to hard (close to ML), and with each problem being a gradual modification of its previous. Then, our attempt is to iteratively trace the solution path of these approximate problems. This homotopy algorithm is well suited to the application of deep unfolding, a recently popular approach for turning certain model-based algorithms into data-driven, and performance enhanced, ones. While our initial focus is on one-bit MIMO detection, the proposed technique also applies naturally to the classical unquantized MIMO detection. We performed extensive simulations and show that the proposed homotopy algorithms, both non-deep and deep, have satisfactory bit-error probability performance compared to many state-of-the-art algorithms. Also, the deep homotopy algorithm has attractively low computational complexity.

Recently, the use of intelligent reflecting surface (IRS) has gained considerable attention in wireless communications. By intelligently adjusting the passive reflection angle, IRS is able to assist the base station (BS) to extend the coverage and improve spectral efficiency. This paper considers a joint symbol-level precoding (SLP) and IRS reflecting design to minimize the symbol-error probability (SEP) of the intended users in an IRS-aided multiuser MISO downlink. We formulate the SEP minimization problems to pursue uniformly good performance for all users for both QAM and PSK constellations. The resulting problem is non-convex and we resort to alternating minimization to obtain a stationary solution. Simulation results demonstrate that under the aid of IRS our proposed design indeed enhances the bit-error rate performance. In particular, the performance improvement is significant when the number of IRS elements is large., Comment: Accepted by IEEE Wireless Communications Letters

This paper develops a first-order optimization method for coupled structured matrix factorization (CoSMF) problems that arise in the context of hyperspectral super-resolution (HSR) in remote sensing. To best leverage the problem structures for computational efficiency, we introduce a hybrid inexact block coordinate descent (HiBCD) scheme wherein one coordinate is updated via the fast proximal gradient (FPG) method, while another via the Frank-Wolfe (FW) method. The FPG-type methods are known to take less number of iterations to converge, by numerical experience, while the FW-type methods can offer lower per-iteration complexity in certain cases; and we wish to take the best of both. We show that the limit points of this HiBCD scheme are stationary. Our proof treats HiBCD as an optimization framework for a class of multi-block structured optimization problems, and our stationarity claim is applicable not only to CoSMF but also to many other problems. Previous optimization research showed the same stationarity result for inexact block coordinate descent with either FPG or FW updates only. Numerical results indicate that the proposed HiBCD scheme is computationally much more efficient than the state-of-the-art CoSMF schemes in HSR., Comment: 32 pages, 3 figures, 5 tables. Codes available at https://github.com/REIYANG/HiBCD. To appear in IEEE Transactions on Signal Processing

Coupled structured matrix factorization (CoSMF) for hyperspectral super-resolution (HSR) has recently drawn significant interest in hyperspectral imaging for remote sensing. Presently there is very few work that studies the theoretical recovery guarantees of CoSMF. This paper makes one such endeavor by considering the CoSMF formulation by Wei et al., which, simply speaking, is similar to coupled non-negative matrix factorization. Assuming no noise, we show sufficient conditions under which the globably optimal solution to the CoSMF problem is guaranteed to deliver certain recovery accuracies. Our analysis suggests that sparsity and the pure-pixel (or separability) condition play a hidden role in enabling CoSMF to achieve some good recovery characteristics., Comment: submitted to CAMSAP 2019, extended version

Hyperspectral super-resolution (HSR) is a problem that aims to estimate an image of high spectral and spatial resolutions from a pair of co-registered multispectral (MS) and hyperspectral (HS) images, which have coarser spectral and spatial resolutions, respectively. In this paper we pursue a low-rank matrix estimation approach for HSR. We assume that the spectral-spatial matrices associated with the whole image and the local areas of the image have low-rank structures. The local low-rank assumption, in particular, has the aim of providing a more flexible model for accounting for local variation effects due to endmember variability. We formulate the HSR problem as a global-local rank-regularized least-squares problem. By leveraging on the recent advances in non-convex large-scale optimization, namely, the smooth Schatten-p approximation and the accelerated majorization-minimization method, we develop an efficient algorithm for the global-local low-rank problem. Numerical experiments on synthetic, semi-real and real data show that the proposed algorithm outperforms a number of benchmark algorithms in terms of recovery performance., Comment: 30 pages, 12 figures, 5 tables. Codes available at https://github.com/REIYANG/GLORIA. To appear in IEEE Transactions on Geoscience and Remote Sensing

Hyperspectral image unmixing has proven to be a useful technique to interpret hyperspectral data, and is a prolific research topic in the community. Most of the approaches used to perform linear unmixing are based on convex geometry concepts, because of the strong geometrical structure of the linear mixing model. However, two main phenomena lead to question this model, namely nonlinearities and the spectral variability of the materials. Many algorithms based on convex geometry are still used when considering these two limitations of the linear model. A natural question is to wonder to what extent these concepts and tools (Intrinsic Dimensionality estimation, endmember extraction algorithms, pixel purity) can be safely used in these different scenarios. In this paper, we analyze them with a focus on endmember variability, assuming that the linear model holds. In the light of this analysis, we propose an integrated unmixing chain which tries to adress the shortcomings of the classical tools used in the linear case, based on our previously proposed extended linear mixing model. We show the interest of the proposed approach on simulated and real datasets.

Coarsely quantized MIMO signalling methods have gained popularity in the recent developments of massive MIMO as they open up opportunities for massive MIMO implementation using cheap and power-efficient radio-frequency front-ends. This paper presents a new one-bit MIMO precoding approach using spatial Sigma-Delta ($\Sigma\Delta$) modulation. In previous one-bit MIMO precoding research, one mainly focuses on using optimization to tackle the difficult binary signal optimization problem that arises from the precoding design. Our approach attempts a different route. Assuming angular MIMO channels, we apply $\Sigma\Delta$ modulation---a classical concept in analog-to-digital conversion of temporal signals---in space. The resulting $\Sigma\Delta$ precoding approach has two main advantages: First, we no longer need to deal with binary optimization in $\Sigma\Delta$ precoding design. Particularly, the binary signal restriction is replaced by peak signal amplitude constraints. Second, the impact of the quantization error can be well controlled via modulator design and under appropriate operating conditions. Through symbol error probability analysis, we reveal that the very large number of antennas in massive MIMO provides favorable operating conditions for $\Sigma\Delta$ precoding. In addition, we develop a new $\Sigma\Delta$ modulation architecture that is capable of adapting the channel to achieve nearly zero quantization error for a targeted user. Furthermore, we consider multi-user $\Sigma\Delta$ precoding using the zero-forcing and symbol-level precoding schemes. These two $\Sigma\Delta$ precoding schemes perform considerably better than their direct one-bit quantized counterparts, as simulation results show.

Consider the following problem: A multi-antenna base station (BS) sends multiple symbol streams to multiple single-antenna users via precoding. However, unlike conventional multiuser precoding, the transmitted signals are subjected to binary, unit-modulus, or even discrete unit-modulus constraints. Such constraints arise in the one-bit and constant-envelope (CE) massive MIMO scenarios, wherein high-resolution digital-to-analog converters (DACs) are replaced by one-bit DACs and phase shifters, respectively, for cutting down hardware cost and energy consumption. Multiuser precoding under one-bit and CE restrictions poses significant design difficulty. In this paper we establish a framework for designing multiuser precoding under the one-bit, continuous CE and discrete CE scenarios---all within one theme. We first formulate a precoding design that focuses on minimizations of symbol-error probabilities (SEPs), assuming quadrature amplitude modulation (QAM) constellations. We then devise an algorithm for our SEP-based design. The algorithm combines i) a novel penalty method for handling binary, unit-modulus and discrete unit-modulus constraints; and ii) a first-order non-convex optimization recipe custom-built for the design. Specifically, the latter is an inexact majorization-minimization method via accelerated projected gradient, which, as shown by simulations, runs very fast and can handle a large number of decision variables. Simulation results indicate that the proposed design offers significantly better bit-error rate performance than the existing designs., Comment: To appear in IEEE Trans. Signal Processing

Most recently, there has been a flurry of research activities on studying how massive MIMO precoding should be designed when the digital-to-analog conversion at the transmitter side is operated by cheap one-bit digital-to-analog converters (DACs). Such research is motivated by the desire to substantially cut down the hardware cost and power consumption of the radio-frequency chain, which is unaffordable in massive MIMO if high-resolution DACs are still used. One-bit MIMO precoding design problems are much harder to solve than their high-resolution DAC counterparts. In our previous work, we developed a minimum symbol-error probability (SEP) design for one-bit precoding under the multiuser MISO downlink scenario and under quadrature amplitude modulation signaling. Leveraging on the previous work, this work shows how the minimum SEP design is applied to M-ary phase shift keying (MPSK) signaling. Simulation results show that our minimum SEP design delivers significantly better bit-error rate (BER) performance than the other designs for higher-order PSK such as 8-PSK and 16-PSK. As a minor, but useful, side contribution, we also tackle an MPSK SEP characterization problem which was only intuitively treated in the prior arts., Comment: IEEE GlobalSIP 2018

This paper considers the problem of constant envelope (CE) precoder designs for multiuser massive MISO downlink channels. The use of CE signals allows one to employ low-cost radio frequency chains and thereby facilitates the implementation of massive MIMO. However, the subsequent CE precoder designs are usually challenging owing to the non-convex CE constraints. The existing CE precoder designs consider minimization of some measures on the distortion levels of the received symbols, and they usually aim at improving the symbol-error rate (SER) performances. In this paper we formulate a minimum SER-based design for CE precoding. The design formulation is non-convex and we propose two low-complexity first-order algorithms using gradient projection. Curiously, our simulation results show that the proposed designs can achieve bit-error rate performance close to that of zero-forcing beamforming without CE signaling restrictions., Comment: To appear in IEEE Statistical Signal Processing Workshop 2018

Hyperspectral super-resolution refers to the problem of fusing a hyperspectral image (HSI) and a multispectral image (MSI) to produce a super-resolution image (SRI) that has fine spatial and spectral resolution. State-of-the-art methods approach the problem via low-rank matrix approximations to the matricized HSI and MSI. These methods are effective to some extent, but a number of challenges remain. First, HSIs and MSIs are naturally third-order tensors (data "cubes") and thus matricization is prone to loss of structural information--which could degrade performance. Second, it is unclear whether or not these low-rank matrix-based fusion strategies can guarantee identifiability or exact recovery of the SRI. However, identifiability plays a pivotal role in estimation problems and usually has a significant impact on performance in practice. Third, the majority of the existing methods assume that there are known (or easily estimated) degradation operators applied to the SRI to form the corresponding HSI and MSI--which is hardly the case in practice. In this work, we propose to tackle the super-resolution problem from a tensor perspective. Specifically, we utilize the multidimensional structure of the HSI and MSI to propose a coupled tensor factorization framework that can effectively overcome the aforementioned issues. The proposed approach guarantees the identifiability of the SRI under mild and realistic conditions. Furthermore, it works with little knowledge of the degradation operators, which is clearly an advantage over the existing methods. Semi-real numerical experiments are included to show the effectiveness of the proposed approach.

This paper considers symbol-level precoding (SLP) for multiuser multiple-input single-output (MISO) downlink. SLP is a nonlinear precoding scheme that utilizes symbol constellation structures. It has been shown that SLP can outperform the popular linear beamforming scheme. In this work we reveal a hidden connection between SLP and linear beamforming. We show that under an energy minimization design, SLP is equivalent to a zero-forcing (ZF) beamforming scheme with perturbations on symbols. This identity gives new insights and they are discussed in the paper. As a side contribution, this work also develops a symbol error probability (SEP)-constrained SLP design formulation under quadrature amplitude modulation (QAM) constellations., Comment: accepted by ICASSP 2018

Massive multiple-input multiple-output (MIMO) has the potential to substantially improve the spectral efficiency, robustness and coverage of mobile networks. However, such potential is limited by hardware cost and power consumption associated with a large number of RF chains. Recently, one-bit quantization is proposed to address this issue by replacing high-resolution digital-to-analog converters (DACs) with one-bit DACs, thereby simplifying the RF chains. Despite low system cost, advanced signal processing techniques are needed to compensate for quantization distortions caused by low resolution DACs. In this paper, a symbol-error-rate (SER)-based one-bit precoding scheme is proposed to minimize the detection error probability of all users under one-bit constraints. The problem is recast as a continuous optimization problem with a biconvex objective. By applying the block coordinate descent (BCD) method and the FISTA method, we develop an efficient iterative algorithm to obtain a one-bit precoding solution. Simulation results demonstrate its superiority over state-of-the-art algorithms in terms of bit error rate performance in high-order modulation cases., Comment: Accepted by ICASSP 2018

Nonnegative matrix factorization (NMF) has become a workhorse for signal and data analytics, triggered by its model parsimony and interpretability. Perhaps a bit surprisingly, the understanding to its model identifiability---the major reason behind the interpretability in many applications such as topic mining and hyperspectral imaging---had been rather limited until recent years. Beginning from the 2010s, the identifiability research of NMF has progressed considerably: Many interesting and important results have been discovered by the signal processing (SP) and machine learning (ML) communities. NMF identifiability has a great impact on many aspects in practice, such as ill-posed formulation avoidance and performance-guaranteed algorithm design. On the other hand, there is no tutorial paper that introduces NMF from an identifiability viewpoint. In this paper, we aim at filling this gap by offering a comprehensive and deep tutorial on model identifiability of NMF as well as the connections to algorithms and applications. This tutorial will help researchers and graduate students grasp the essence and insights of NMF, thereby avoiding typical `pitfalls' that are often times due to unidentifiable NMF formulations. This paper will also help practitioners pick/design suitable factorization tools for their own problems., Comment: accepted version, IEEE Signal Processing Magazine; supplementary materials added. Some minor revisions implemented

Consider a structured matrix factorization model where one factor is restricted to have its columns lying in the unit simplex. This simplex-structured matrix factorization (SSMF) model and the associated factorization techniques have spurred much interest in research topics over different areas, such as hyperspectral unmixing in remote sensing, topic discovery in machine learning, to name a few. In this paper we develop a new theoretical SSMF framework whose idea is to study a maximum volume ellipsoid inscribed in the convex hull of the data points. This maximum volume inscribed ellipsoid (MVIE) idea has not been attempted in prior literature, and we show a sufficient condition under which the MVIE framework guarantees exact recovery of the factors. The sufficient recovery condition we show for MVIE is much more relaxed than that of separable non-negative matrix factorization (or pure-pixel search); coincidentally it is also identical to that of minimum volume enclosing simplex, which is known to be a powerful SSMF framework for non-separable problem instances. We also show that MVIE can be practically implemented by performing facet enumeration and then by solving a convex optimization problem. The potential of the MVIE framework is illustrated by numerical results.

Spectrally efficient multi-antenna wireless communication systems are a key challenge as service demands continue to increase. At the same time, powering up radio access networks is facing environmental and regulation limitations. In order to achieve more power efficiency, we design a directional modulation precoder by considering an $M$-QAM constellation, particularly with $M=4,8,16,32$. First, extended detection regions are defined for desired constellations using analytical geometry. Then, constellation points are placed in the optimal positions of these regions while the minimum Euclidean distance to adjacent constellation points and detection region boundaries is kept as in the conventional $M$-QAM modulation. For further power efficiency and symbol error rate similar to that of fixed design in high SNR, relaxed detection regions are modeled for inner points of $M=16,32$ constellations. The modeled extended and relaxed detection regions as well as the modulation characteristics are utilized to formulate symbol-level precoder design problems for directional modulation to minimize the transmission power while preserving the minimum required SNR at the destination. In addition, the extended and relaxed detection regions are used for precoder design to minimize the output of each power amplifier. We transform the design problems into convex ones and devise an interior point path-following iterative algorithm to solve the mentioned problems and provide details on finding the initial values of the parameters and the starting point. Results show that compared to the benchmark schemes, the proposed method performs better in terms of power and peak power reduction as well as symbol error rate reduction for a wide range of SNRs.

Consider a two-way communication scenario where two single-antenna nodes, operating under full-duplex mode, exchange information to one another through the aid of a (full-duplex) multi-antenna relay, and there is another single-antenna node who intends to eavesdrop. The relay employs artificial noise (AN) to interfere the eavesdropper's channel, and amplify-forward (AF) Alamouti-based rank-two beamforming to establish the two-way communication links of the legitimate nodes. Our problem is to optimize the rank-two beamformer and AN covariance for sum secrecy rate maximization (SSRM). This SSRM problem is nonconvex, and we develop an efficient solution approach using semidefinite relaxation (SDR) and minorization-maximization (MM). We prove that SDR is tight for the SSRM problem and thus introduces no loss. Also, we consider an inexact MM method where an approximately but computationally cheap MM solution update is used in place of the exact update in conventional MM. We show that this inexact MM method guarantees convergence to a stationary solution to the SSRM problem. The effectiveness of our proposed approach is further demonstrated by an energy-harvesting scenario extension, and by extensive simulation results., Comment: 15 pages, 7 figures, accepted by IEEE Journal of Selected Topics in Signal Processing

This paper considers \emph{volume minimization} (VolMin)-based structured matrix factorization (SMF). VolMin is a factorization criterion that decomposes a given data matrix into a basis matrix times a structured coefficient matrix via finding the minimum-volume simplex that encloses all the columns of the data matrix. Recent work showed that VolMin guarantees the identifiability of the factor matrices under mild conditions that are realistic in a wide variety of applications. This paper focuses on both theoretical and practical aspects of VolMin. On the theory side, exact equivalence of two independently developed sufficient conditions for VolMin identifiability is proven here, thereby providing a more comprehensive understanding of this aspect of VolMin. On the algorithm side, computational complexity and sensitivity to outliers are two key challenges associated with real-world applications of VolMin. These are addressed here via a new VolMin algorithm that handles volume regularization in a computationally simple way, and automatically detects and {iteratively downweights} outliers, simultaneously. Simulations and real-data experiments using a remotely sensed hyperspectral image and the Reuters document corpus are employed to showcase the effectiveness of the proposed algorithm.

In this paper, we study the amplify-and-forward (AF) schemes in two-hop one-way relay networks. In particular, we consider the multigroup multicast transmission between long-distance users. Given that perfect channel state information is perceived, our goal is to design the AF process so that the max-min-fair (MMF) signal-to-interference-plus-noise ratio (SINR) is optimized subject to generalized power constraints. We propose a rank-two beamformed Alamouti (BFA) AF scheme and formulate the corresponding AF design problem as a \emph{two-variable} fractional quadratically-constrained quadratic program (QCQP), which is further tackled by the semidefinite relaxation (SDR) technique. We analyze the approximation quality of two-variable fractional SDRs under the Gaussian randomization algorithm. These results are fundamentally new and reveal that the proposed BFA AF scheme can outperform the traditional BF AF scheme, especially when there are many users in the system or many generalized power constraints in the problem formulation. From a practical perspective, the BFA AF scheme offers two degrees of freedom (DoFs) in beamformer design, as opposed to the one DoF offered by the BF AF scheme, to improve the receivers' SINR. In the latter part of this paper, we demonstrate how this extra DoF leads to provable performance gains by considering two special cases of multicasting, where the AF process is shown to employ a special structure. The numerical simulations further validate that the proposed BFA AF scheme outperforms the BF AF scheme and works well for large-scale relay systems.

This is a companion technical report of the main manuscript "Semidefinite Relaxation and Approximation Analysis of a Beamformed Alamouti Scheme for Relay Beamforming Networks". The report serves to give detailed derivations of Lemma 1-2 in the main manuscript, which are too long to be included in the latter. In addition, more simulation results are presented to verify the viability of the BF Alamouti AF schemes developed in the main manuscript.

This paper concentrates on a robust transmit optimization problem for the multiuser multi-input single-output (MISO) downlink scenario and under inaccurate channel state information (CSI). This robust problem deals with a general-rank transmit covariance design, and it follows a safe rate-constrained formulation under spherically bounded CSI uncertainties. Curiously, simulation results in previous works suggested that the robust problem admits rank-one optimal transmit covariances in most cases. Such a numerical finding is appealing because transmission with rank-one covariances can be easily realized by single-stream transmit beamforming. This gives rise to a fundamentally important question, namely, whether we can theoretically identify conditions under which the robust problem admits a rank-one solution. In this paper, we identify one such condition. Simply speaking, we show that the robust problem is guaranteed to admit a rank-one solution if the CSI uncertainties are not too large and the multiuser channel is not too poorly conditioned. To establish the aforementioned condition, we develop a novel duality framework, through which an intimate relationship between the robust problem and a related maximin problem is revealed. Our condition involves only a simple expression with respect to the multiuser channel and other system parameters. In particular, unlike other sufficient rank-one conditions that have appeared in the literature, ours is verifiable. The application of our analysis framework to several other CSI uncertainty models is also discussed., Comment: To appear in IEEE Trans. Signal Process. This version combines the main manuscript and its accompanied supplementary report into one single article

In this paper, we consider a two-hop one-way relay network for multigroup multicast transmission between long-distance users, in which the relay is equipped with multiple antennas, while the transmitters and receivers are all with a single antenna. Assuming that perfect channel state information is available, we study amplify-and-forward (AF) schemes that aim at optimizing the max-min-fair (MMF) rate. We begin by considering the classic beamformed AF (BF-AF) scheme, whose corresponding MMF design problem can be formulated as a rank-constrained fractional semidefinite program (SDP). We show that the gap between the BF-AF rate and the SDR rate associated with an optimal SDP solution is sensitive to the number of users as well as the number of power constraints in the relay system. This reveals that the BF-AF scheme may not be well suited for large-scale systems. We therefore propose the stochastic beamformed AF (SBF-AF) schemes, which differ from the BF-AF scheme in that time-varying AF weights are used. We prove that the MMF rates of the proposed SBF-AF schemes are at most $0.8317$ bits/s/Hz less than the SDR rate, irrespective of the number of users or power constraints. Thus, SBF-AF can outperform BF-AF especially in large-scale systems. Finally, we present numerical results to demonstrate the viability of our proposed schemes., Comment: This paper was first submitted for possible journal publication on 24-May-2015. A preliminary version of this paper has been published in IEEE SPAWC'15

We consider factoring low-rank tensors in the presence of outlying slabs. This problem is important in practice, because data collected in many real-world applications, such as speech, fluorescence, and some social network data, fit this paradigm. Prior work tackles this problem by iteratively selecting a fixed number of slabs and fitting, a procedure which may not converge. We formulate this problem from a group-sparsity promoting point of view, and propose an alternating optimization framework to handle the corresponding $\ell_p$ ($0

The dictionary-aided sparse regression (SR) approach has recently emerged as a promising alternative to hyperspectral unmixing (HU) in remote sensing. By using an available spectral library as a dictionary, the SR approach identifies the underlying materials in a given hyperspectral image by selecting a small subset of spectral samples in the dictionary to represent the whole image. A drawback with the current SR developments is that an actual spectral signature in the scene is often assumed to have zero mismatch with its corresponding dictionary sample, and such an assumption is considered too ideal in practice. In this paper, we tackle the spectral signature mismatch problem by proposing a dictionary-adjusted nonconvex sparsity-encouraging regression (DANSER) framework. The main idea is to incorporate dictionary correcting variables in an SR formulation. A simple and low per-iteration complexity algorithm is tailor-designed for practical realization of DANSER. Using the same dictionary correcting idea, we also propose a robust subspace solution for dictionary pruning. Extensive simulations and real-data experiments show that the proposed method is effective in mitigating the undesirable spectral signature mismatch effects.

This paper considers a recently emerged hyperspectral unmixing formulation based on sparse regression of a self-dictionary multiple measurement vector (SD-MMV) model, wherein the measured hyperspectral pixels are used as the dictionary. Operating under the pure pixel assumption, this SD-MMV formalism is special in that it allows simultaneous identification of the endmember spectral signatures and the number of endmembers. Previous SD-MMV studies mainly focus on convex relaxations. In this study, we explore the alternative of greedy pursuit, which generally provides efficient and simple algorithms. In particular, we design a greedy SD-MMV algorithm using simultaneous orthogonal matching pursuit. Intriguingly, the proposed greedy algorithm is shown to be closely related to some existing pure pixel search algorithms, especially, the successive projection algorithm (SPA). Thus, a link between SD-MMV and pure pixel search is revealed. We then perform exact recovery analyses, and prove that the proposed greedy algorithm is robust to noise---including its identification of the (unknown) number of endmembers---under a sufficiently low noise level. The identification performance of the proposed greedy algorithm is demonstrated through both synthetic and real-data experiments.

In this paper, we analyze different preconditionings designed to enhance robustness of pure-pixel search algorithms, which are used for blind hyperspectral unmixing and which are equivalent to near-separable nonnegative matrix factorization algorithms. Our analysis focuses on the successive projection algorithm (SPA), a simple, efficient and provably robust algorithm in the pure-pixel algorithm class. Recently, a provably robust preconditioning was proposed by Gillis and Vavasis (arXiv:1310.2273) which requires the resolution of a semidefinite program (SDP) to find a data points-enclosing minimum volume ellipsoid. Since solving the SDP in high precisions can be time consuming, we generalize the robustness analysis to approximate solutions of the SDP, that is, solutions whose objective function values are some multiplicative factors away from the optimal value. It is shown that a high accuracy solution is not crucial for robustness, which paves the way for faster preconditionings (e.g., based on first-order optimization methods). This first contribution also allows us to provide a robustness analysis for two other preconditionings. The first one is pre-whitening, which can be interpreted as an optimal solution of the same SDP with additional constraints. We analyze robustness of pre-whitening which allows us to characterize situations in which it performs competitively with the SDP-based preconditioning. The second one is based on SPA itself and can be interpreted as an optimal solution of a relaxation of the SDP. It is extremely fast while competing with the SDP-based preconditioning on several synthetic data sets., Comment: 25 pages, 3 figures