Structured Random Ensembles Learning for Signal Reconstruction: from Theory to Application


Student thesis: Doctoral Thesis

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Award date7 Sep 2021


Random matrix ensembles have found wide applications in fields of wireless communication, signal processing, and machine learning. Even though the most studied Gaussian measurement ensembles offer tractable analyses and appealing results, they are of somewhat limited use in practical applications because the design of measurement matrices is usually subject to physical or other constraints provided by a specific system architecture. Such constraints can be tractably captured by introducing structures to random ensembles. It is thus desired to explore structured random ensembles from a computational and an application-oriented point of view. In this thesis, we study structured random ensembles for signal reconstruction and their signal processing algorithms from two different perspectives: i) compressive sensing and ii) machine learning. Furthermore, the applications of such random ensemble to millimeter-wave (mmWave) communications are investigated. The thesis is organized as follows.

In Chapter 1, the background of the thesis is introduced. We consider, in Chapter 2, two structured random ensembles: i) the isotropic and ii) random phase-rotated (RPR) measurement matrices, both of which maintain a unit-norm and/or constant modulus property. These measurement ensembles have important practical applications in advanced beamforming and precoding in communication systems and have been popularly studied in radar signal processing. We analyze the coherence statistics of the two random measurement ensembles and apply them to acquire probabilistic performance guarantees of the orthogonal matching pursuit for support detection (SD). Despite their different coherence statistics, we show for both cases that the required number of measurements ensuring a bounded SD error scales with the number of supports and the signal dimension with a similar trend. The analysis strengthens the scaling laws by eliminating any nuisance constant or optimizing a free variable to tighten the bound. This is in contrast with the existing results for Gaussian random measurements, posing uncertainties due to the dependency on ambiguous constants. It is shown that the SD bounds for the isotropic and RPR measurements are closely related, while both show substantial tightness, especially when the signal is sparse. Besides, the asymptotic successful SD performance is discussed in the large system limits by considering a fixed or varying number of supports. We provide theoretical justifications as well as numerical demonstrations for the above combinations.

In Chapter 3, we focus on a machine learning-based data representation technique dubbed “Kolmogorov model (KM)” that models the outcome of a binary random variable as an inner product between two structured vectors, one on the unit simplex probability space and another on the binary elementary space. Leveraging the predictive and interpretable power, the KM plays an essential role in learning elementary representation of random variables by utilizing a subset of sampled statistics. Motivated by the lack of computational scalability of the existing KM learning algorithm that relies on the semi-definite relaxation with randomization (SDRwR), we propose two new approaches to the KM learning problem: i) discrete monotonic optimization (DMO) and ii) dual optimization. To address the computational bottleneck introduced by the costly eigenvalue decomposition (EVD) of the proposed dual optimization algorithm that is based on iterative gradient descent (GD), two acceleration schemes including the enhanced GD with EVD elimination and a proximal EVD algorithm are presented. A thresholding technique is devised to determine the number of iterations of the proximal EVD algorithm by exploiting the approximation error analysis and leveraging the normalized Minkowski ℓ1-norm. When applied to big data applications, it is demonstrated that the proposed methods can achieve comparable training and prediction performance with significantly reduced computational cost compared to the existing KM learning algorithm. Moreover, the interpretability of the proposed KM learning is validated via logical relation mining.

In Chapter 4, we propose a beam alignment and tracking framework for mmWave multiple-input multiple-output (MIMO) systems by incorporating the predictability and interpretability of KM. In particular, the proposed scheme enables a novel interpretable beam tracking that extracts insights on relations between distinctive beam pairs from sounded observations to reduce the beam training overhead after the initial beam alignment. Two developed solvers, i.e., the proposed DMO and dual optimization with GD, are adopted. Numerical simulations demonstrate that the proposed KM learning methods can reduce the computational cost by up to three orders of magnitude compared with the existing KM learning with SDRwR while maintaining an appreciable beam alignment performance. The superiority of the joint predictive beam alignment and interpretable beam tracking is corroborated in comparison with state-of-the-art techniques in terms of the beam tracking overhead, spectral efficiency, and robustness in the low signal-to-noise (SNR) regime.

A mmWave MIMO hybrid analog-digital precoding and combining system that facilitates multiple data streams is considered in Chapter 5. We focus on the angle of departure (AoD) and angle of arrival (AoA) tracking problem for temporally correlated sparse mmWave MIMO channels. A two-sided channel sounding scheme with low complexity is proposed. With the adoption of the beam space channel representation, the AoD and AoA estimation is converted to a sparse signal recovery problem, which can be efficiently resolved by employing an iterative approximate message passing algorithm. Motivated by the fact that the omni-directional beam suffers from performance deterioration at low SNR, several directional sounding beam designs adapted to the temporal AoD and AoA evolution statistics are provided. We create simulations to demonstrate the effectiveness of the proposed AoD and AoA tracking scheme employed with the directional sounding beams, especially in the low SNR regime.

Finally, Chapter 6 concludes the thesis and outlines possible future research directions.

    Research areas

  • Signal Processing, Millimeter wave communication systems, Machine Learning