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Scalable Solvers of Random Quadratic Equations via Stochastic Truncated Amplitude Flow

IEEE TRANSACTIONS ON SIGNAL PROCESSING 2017年第8期65卷 1961-1974页

作者： Wang, Gang Giannakis, Georgios B. Chen, Jie Univ Minnesota Digital Technol Ctr Minneapolis MN 55455 USA Univ Minnesota Dept Elect & Comp Engn Minneapolis MN 55455 USA Beijing Inst Technol Sch Automat Beijing 100081 Peoples R China Beijing Inst Technol State Key Lab Intelligent Control & Decis Complex Beijing 100081 Peoples R China

A novel approach termed stochastic truncated amplitude flow (STAF) is developed to reconstruct an unknown n-dimensional real-/ complex-valued signal x from m "phaseless" quadratic equations of the form psi(i) = |< a(i), x >|center dot This problem, also known as phase retrieval from magnitude-only information, is NP-hard in general. Adopting an amplitude-based nonconvex formulation, STAF leads to an iterative solver comprising two stages: s1) Orthogonality-promoting initialization through a stochastic variance reduced gradient algorithm;and, s2) a series of iterative refinements of the initialization using stochastic truncated gradient iterations. Both stages involve a single equation per iteration, thus rendering STAF a simple, scalable, and fast approach amenable to large-scale implementations that are useful when n is large. When {a(i)}(i=1)(m) are independent Gaussian, STAF provably recovers exactly any x is an element of R-n exponentially fast based on order of n quadratic equations. STAF is also robust in the presence of additive noise of bounded support. Simulated tests involving real Gaussian {a(i)} vectors demonstrate that STAF empirically reconstructs any x is an element of R-n exactly from about 2.3n magnitude-only measurements, outperforming state-of-the-art approaches and narrowing the gap from the information-theoretic number of equations m = 2n-1. Extensive experiments using synthetic data and real images corroborate markedly improved performance of STAF over existing alternatives.

关键词： Nonconvex optimization phase retrieval variance reduction kaczmarz algorithm

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Divide-and-Iterate Approach to Big Data Systems

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IEEE TRANSACTIONS ON SERVICES COMPUTING 2022年第4期15卷 1967-1979页

作者： Chang, Shih Yu Wu, Hsiao-Chun San Jose State Univ Dept Comp Engn San Jose CA 95192 USA Louisiana State Univ Sch Elect Engn & Comp Sci Baton Rouge LA 70803 USA

Matrix calculations are often required for the analysis of any big-data cloud computing system. It is quite common to process big-data associated matrices possessing the sparsity and low-rank properties. In order to efficiently deal with big-data matrices, we propose a new divide-and-iterate framework, which can be invoked to solve an enormously large linear system of equations by taking advantage of factored matrices. The kaczmarz algorithm (KA) is utilized here to design the parallel iterative algorithms which are capable of solving a large system of equations by iteratively updating the solution through the reduction into the factorized subsystems in parallel. The convergences of our proposed new iterative algorithms are justified by the rigorous proofs. Besides, the time- and memory-complexities are studied to demonstrate the resource efficiency of the proposed algorithms. Numerical experiments are also presented to illustrate the effectiveness of this proposed new framework.

关键词： Big Data Sparse matrices Linear regression Linear systems Iterative methods Mathematical model Matrix decomposition kaczmarz algorithm random iterations matrix factorization tremendous linear systems big data

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Randomized Subspace Actions and Fusion Frames

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CONSTRUCTIVE APPROXIMATION 2016年第1期43卷 103-134页

作者： Chen, Xuemei Powell, Alexander M. Univ Missouri Dept Math Columbia MO 65211 USA Vanderbilt Univ Dept Math Nashville TN 37240 USA

A randomized subspace action algorithm is investigated for fusion frame signal recovery problems and for the problem of recovering a signal from projections onto random subspaces. It is noted that kaczmarz bounds provide upper bounds on the algorithm's error moments. The main question of which probability distributions on a random subspace lead to provably fast convergence is addressed. In particular, it is proved which distributions give minimal kaczmarz bounds, and hence give best control on error moment upper bounds arising from kaczmarz bounds. Uniqueness of the optimal distributions is also addressed.

关键词： Extremal probability measures Frame potential Fusion frames kaczmarz algorithm Potential function Randomization Subspace action methods

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Model subspaces techniques to study Fourier expansions in L² spaces associated to singular measures

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JOURNAL OF FUNCTIONAL ANALYSIS 2020年第10期279卷

作者： Antezana, Jorge Guadalupe Garcia, Maria IAM CONICET Inst Argentino Matemat Alberto P Calderon Buenos Aires DF Argentina Univ Nacl La Plata Dept Matemat Fac Ciencias Exactas La Plata Argentina

Let mu be a probability measure on T that is singular with respect to the Haar measure. In this paper we study Fourier expansions in L-2(T, mu) using techniques from the theory of model subspaces of the Hardy space. Since the sequence of monomials {z(n)}(n is an element of N) is effective in L-2(T, mu), it has a Parseval frame associated via the kaczmarz algorithm. Our first main goal is to identify the aforementioned frame with boundary values of the frame P-phi(z(n)) for the model subspace H(phi) = H-2 circle minus phi H-2, where P-phi, is the orthogonal projection from the Hardy space H-2 onto H(phi). The study of Fourier expansions in L-2(T, mu) also leads to consider positive kernels in the Hardy space. Our second main goal is to study the set of measures it which reproduce a kernel contained in a model subspace. We completely characterize this set when the kernel is the reproducing kernel of a model subspace, and we study the consequences of this characterization. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Model subspaces kaczmarz algorithm Fourier expansions Parseval frames

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Low-complexity kaczmarz precoding in DL massive MIMO with partial CSI and correlation

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PHYSICAL COMMUNICATION 2019年 37卷

作者： Rosa, Karina Bernardin Marinello, Jose Carlos Abrao, Taufik Londrina State Univ DEEL UEL Elect Engn Dept Rod Celso Garcia Cid-PR445POB 10-011 BR-86057970 Londrina Parana Brazil

The capacity achieved by three low-complexity precoders is compared considering a single-cell massive MIMO (M-MIMO) broadcast channel;these linear precoders include the conventional zero-forcing (ZF) beamforming, regularized channel inversion (RCI) precoding, and a precoding version based on the iterative randomized kaczmarz algorithm (rKA). The capacity vs complexity tradeoff is analyzed in the downlink (DL) with a base-station (BS) deploying massive uniform linear array (ULA) antennas, while mobile terminals (MT) are equipped with single-antenna. Based on the precoding matrix associated with each scheme, we first derive the interference matrix and, consequently, the capacity achieved by each analyzed precoder. Numerical results demonstrate that the rKA's performance-complexity tradeoff is superior compared to conventional ZF and RCI precoding when operating in M-MIMO scenario, since it holds a relative robustness against system loading increasing, appears to be equally efficient under a simple equal power assignment (EPA) policy and implies in a considerably reduced number of complex operations. (c) 2019 Elsevier B.V. All rights reserved.

关键词： Channel inversion kaczmarz algorithm Massive MIMO Precoding Capacity mMTC URLLC Massive MIMO 5G Activity detection

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Low-Complexity Statistically Robust Precoder/Detector Computation for Massive MIMO Systems

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 2018年第10期17卷 6516-6530页

作者： Boroujerdi, Mahdi Nouri Haghighatshoar, Saeid Caire, Giuseppe Univ Tehran Sch Elect & Comp Engn Tehran *** Iran Tech Univ Berlin Commun & Informat Theory Grp D-10623 Berlin Germany

Massive MIMO is a variant of multi-user MIMO in which the number of antennas at the base station (BS) M is very large and typically much larger than the number of served users (data streams) K. Recent research has widely investigated the system-level advantages of the massive MIMO, and in particular, the beneficial effect of increasing the number of antennas M. These benefits, however, come at the cost of a dramatic increase in hardware and computational complexity. This is partly due to the fact that the BS needs to compute precoding/receiving vectors in order to coherently transmit/detect data to/from each user, where the resulting complexity grows proportionally to the number of antennas M and the number of served users K. Recently, different algorithms based on tools from asymptotic random matrix theory and/or approximated message passing have been proposed to reduce such complexity. The underlying assumption in all these techniques, however, is that the exact statistics (covariance matrix) of the channel vectors of the users is a priori known. This is far from being realistic, especially taking into account that, in the high-dim regime of M >> 1, estimating the channel covariance matrices of the users is also challenging in terms of both computation and storage requirements. In this paper, we propose a novel technique for computing the precoder/detector in a massive MIMO system. Our method is based on the randomized kaczmarz algorithm and does not require a priori knowledge of the statistics of users' channel vectors. We analyze the performance of our proposed algorithm theoretically and compare its performance with that of other techniques based on random matrix theory and approximate message passing via numerical simulations. Our results indicate that our proposed technique is computationally very competitive and yields quite a comparable performance while it does not require the knowledge of the statistics of users' channel vectors.

关键词： Massive MIMO kaczmarz algorithm detection precoding

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Global One-Bit Phase Retrieval via Sample Abundance-Including an Application to STFT Measurements 57

Global One-Bit Phase Retrieval via Sample Abundance-Includin...

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57th Asilomar Conference on Signals, Systems and Computers

作者： Eamaz, Arian Yeganegi, Farhang Soltanalian, Mojtaba Univ Illinois Chicago IL 60607 USA

ISBN: (纸本)9798350325744

We introduce a novel technique for reconstructing signals from phaseless measurements obtained through one-bit ADCs. Our approach adopts the Sampling kaczmarz-Motzkin algorithm to efficiently and globally optimize the phase retrieval objective. In particular, it utilizes the sample abundance provided in one-bit sensing with time-varying thresholds to overcome the limitations of traditional convex phase retrieval methods, such as costly semi-definiteness and rank constraints. Additionally, our approach eliminates the need for lifting up the unknown signal, leading to further computationally efficiency. We demonstrate the effectiveness of our technique by successfully recovering the signal of interest from one-bit sampled short-time Fourier transform magnitudes-an interesting problem that has not been adequately addressed prior to this work. The results not only demonstrate the success of our approach in the context of phase retrieval but also the potential of one-bit sampling and the associated sample abundance gains in information processing at large.

关键词： kaczmarz algorithm one-bit sampling phase retrieval sample abundance time-varying thresholds

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Unlimited Sampling of FRI Signals with Dithered One-Bit Quantization 57

Unlimited Sampling of FRI Signals with Dithered One-Bit Quan...

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57th Asilomar Conference on Signals, Systems and Computers

作者： Eamaz, Arian Yeganegi, Farhang Mishra, Kumar Vijay Soltanalian, Mojtaba Univ Illinois ECE Dept Chicago IL 60607 USA US Army US DEVCOM Res Lab Adelphi MD 20783 USA

ISBN: (纸本)9798350325744

Recent research in sampling theory suggests utilizing unlimited sampling with one-bit quantization time-varying threshold (UNO) for bandlimited signals. The UNO framework addresses the problems associated with the finite dynamic range and limited quantization levels in conventional ADCs. However, the input signal in many engineering applications is not bandlimited. Of particular interest are the finite rate of innovation (FRI) signals that are characterized by finitely many samples per unit of time and need not be bandlimited. In this work, leveraging upon the generalized FRI sampling theory, we introduce a novel sampling approach for FRI signals that utilizes low-cost one-bit ADCs. We extend this one-bit generalized FRI (OBi-GeFRI) sampling to include unlimited sampling for FRI signals in the form of a new unlimited one-bit FRI (UNOFRI) framework. We rely on the highly efficient random kaczmarz algorithm, a linear feasibility problem solver, for signal reconstruction in both OBi-GeFRI and UNOFRI. Our numerical examples illustrate the viability of these sampling techniques to tackle the shortcomings of the state-of-the-art.

关键词： FRI signals kaczmarz algorithm one-bit quantization time-varying sampling threshold unlimited sampling

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Wideband Integrated Sensing and Communication on Synthetic Aperture Radar Platforms 57

Wideband Integrated Sensing and Communication on Synthetic A...

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57th Asilomar Conference on Signals, Systems and Computers

作者： Vouras, Peter US Dept Def Washington DC 20375 USA

ISBN: (纸本)9798350325744

Integrated sensing and communication (ISAC) is a critical emerging technology that will be an important feature in sixth generation (6G) wireless networks. Combining the detection and information transmission functions of a radar and communication system into a single waveform is a challenging problem even in the simplest case with a single shared aperture. In the case of synthetic aperture radar (SAR), the motion of an aircraft is used to synthesize an aperture larger than the physical antenna. This technique is only possible however if the received waveform is phase coherent across separated spatial samples. Maintaining spatial coherence with a temporally modulated waveform useful for sending information requires accurate knowledge of the phase of the transmitted waveform at every point along the length of the synthetic aperture. Systems that remember the phase of previous transmissions are known as coherent-on-receive and are the focus of this paper. Additionally, if the waveform is wideband, the SAR image focusing process will require greater geolocation accuracy. This paper proposes to embed communication information into the sweep rate of a nonlinear chirp that is suitable for detection purposes after matched filtering. An iterative autofocusing technique for coherent-on-receive ISAC systems is presented that leverages the kaczmarz algorithm and reduces the memory demands on a processor.

关键词： synthetic aperture radar kaczmarz algorithm integrated sensing and communication wideband autofocusing

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Persistence Homology with Applications in Analysis and Data Science

Persistence Homology with Applications in Analysis and Data ...

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作者： Przybylski, Lee Iowa State University

学位级别：Ph.D., Doctor of Philosophy

This thesis is a compilation of three research projects that I developed as a graduate student at Iowa State. The main results of these projects are distributed into the final three chapters. Each of those chapters begin with their own overview to introduce the ideas unique to their respective projects. The first chapter provides an overview to the three projects and introduces some of the motivating concepts and notation leading to the three projects. Chapter 2 provides background on topological data analysis that is necessary to understand chapters 3 and 4. We now provide a brief description of the major contributions in each of the three projects. Topological data analysis (TDA) is often considered as the way to characterize the shape of data. The way we do this is by taking a set of data points, computing its Cech complex across a range of resolutions, and recording how the homology groups change in what is called a persistence landscape. This can be done for any collection of points in a metric space, and we apply this to fractals which are the invariant sets of an iterated function system (IFS). Using the self-similarity properties of these fractals and some assumptions about the distance between the images of the similitudes in our IFS, we prove several isomorphisms that allow us to establish a formula for the persistence landscape of the invariant set of an IFS. Another tool from TDA is the persistence landscape. In this project we develop a novel algorithm for anomaly detection in time-series data using persistence diagrams measured by the bottleneck distance. Specifically, we generate anomaly scores by randomly collecting the data into reference bags, and then for each data point, we assign an anomaly score by replacing a randomly chosen point in each reference bag with the data point of interest to create a modified bag. The anomaly score is then computed using the bottleneck distances between the persistence diagrams generated by the reference bags and t

关键词： Anomaly detection Fractals Hardy space kaczmarz algorithm Persistence homology Topological data analysis

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