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Structured Sub-Nyquist Sampling with Applications in Compressive Toeplitz Covariance Estimation, Super-Resolution and Phase Retrieval
Sub-Nyquist sampling has received a huge amount of interest in the past decade. In classical compressed sensing theory, if the measurement procedure satisfies a particular condition known as Restricted Isometry Property (RIP), we can achieve stable recovery of signals of low-dimensional intrinsic structures with an order-wise optimal sample size. Such low-dimensional structures include sparse and low rank for both vector and matrix cases. The main drawback of conventional compressed sensing theory is that random measurements are required to ensure the RIP property. However, in many applications such as imaging and array signal processing, applying independent random measurements may not be practical as the systems are deterministic. Moreover, random measurements based compressed sensing always exploits convex programs for signal recovery even in the noiseless case, and solving those programs is computationally intensive if the ambient dimension is large, especially in the matrix case. The main contribution of this dissertation is that we propose a deterministic sub-Nyquist sampling framework for compressing the structured signal and come up with computationally efficient algorithms. Besides widely studied sparse and low-rank structures, we particularly focus on the cases that the signals of interest are stationary or the measurements are of Fourier type. The key difference between our work from classical compressed sensing theory is that we explicitly exploit the second-order statistics of the signals, and study the equivalent quadratic measurement model in the correlation domain. The essential observation made in this dissertation is that a difference/sum coarray structure will arise from the quadratic model if the measurements are of Fourier type. With these observations, we are able to achieve a better compression rate for covariance estimation, identify more sources in array signal processing or recover the signals of larger sparsity. In this dissertation, we will first study the problem of Toeplitz covariance estimation. In particular, we will show how to achieve an order-wise optimal compression rate using the idea of sparse arrays in both general and low-rank cases. Then, an analysis framework of super-resolution with positivity constraint is established. We will present fundamental robustness guarantees, efficient algorithms and applications in practices. Next, we will study the problem of phase-retrieval for which we successfully apply the sparse array ideas by fully exploiting the quadratic measurement model. We achieve near-optimal sample complexity for both sparse and general cases with practical Fourier measurements and provide efficient and deterministic recovery algorithms. In the end, we will further elaborate on the essential role of non-negative constraint in underdetermined inverse problems. In particular, we will analyze the nonlinear co-array interpolation problem and develop a universal upper bound of the interpolation error. Bilinear problem with non-negative constraint will be considered next and the exact characterization of the ambiguous solutions will be established for the first time in literature. At last, we will show how to apply the nested array idea to solve real problems such as Kriging. Using spatial correlation information, we are able to have a stable estimate of the field of interest with fewer sensors than classic methodologies. Extensive numerical experiments are implemented to demonstrate our theoretical claims
Three more Decades in Array Signal Processing Research: An Optimization and Structure Exploitation Perspective
The signal processing community currently witnesses the emergence of sensor
array processing and Direction-of-Arrival (DoA) estimation in various modern
applications, such as automotive radar, mobile user and millimeter wave indoor
localization, drone surveillance, as well as in new paradigms, such as joint
sensing and communication in future wireless systems. This trend is further
enhanced by technology leaps and availability of powerful and affordable
multi-antenna hardware platforms. The history of advances in super resolution
DoA estimation techniques is long, starting from the early parametric
multi-source methods such as the computationally expensive maximum likelihood
(ML) techniques to the early subspace-based techniques such as Pisarenko and
MUSIC. Inspired by the seminal review paper Two Decades of Array Signal
Processing Research: The Parametric Approach by Krim and Viberg published in
the IEEE Signal Processing Magazine, we are looking back at another three
decades in Array Signal Processing Research under the classical narrowband
array processing model based on second order statistics. We revisit major
trends in the field and retell the story of array signal processing from a
modern optimization and structure exploitation perspective. In our overview,
through prominent examples, we illustrate how different DoA estimation methods
can be cast as optimization problems with side constraints originating from
prior knowledge regarding the structure of the measurement system. Due to space
limitations, our review of the DoA estimation research in the past three
decades is by no means complete. For didactic reasons, we mainly focus on
developments in the field that easily relate the traditional multi-source
estimation criteria and choose simple illustrative examples.Comment: 16 pages, 8 figures. This work has been submitted to the IEEE for
possible publication. Copyright may be transferred without notice, after
which this version may no longer be accessibl
Theory and Algorithms for Reliable Multimodal Data Analysis, Machine Learning, and Signal Processing
Modern engineering systems collect large volumes of data measurements across diverse sensing modalities. These measurements can naturally be arranged in higher-order arrays of scalars which are commonly referred to as tensors. Tucker decomposition (TD) is a standard method for tensor analysis with applications in diverse fields of science and engineering. Despite its success, TD exhibits severe sensitivity against outliers —i.e., heavily corrupted entries that appear sporadically in modern datasets. We study L1-norm TD (L1-TD), a reformulation of TD that promotes robustness. For 3-way tensors, we show, for the first time, that L1-TD admits an exact solution via combinatorial optimization and present algorithms for its solution. We propose two novel algorithmic frameworks for approximating the exact solution to L1-TD, for general N-way tensors. We propose a novel algorithm for dynamic L1-TD —i.e., efficient and joint analysis of streaming tensors. Principal-Component Analysis (PCA) (a special case of TD) is also outlier responsive. We consider Lp-quasinorm PCA (Lp-PCA) for
Covariance Estimation from Compressive Data Partitions using a Projected Gradient-based Algorithm
Covariance matrix estimation techniques require high acquisition costs that
challenge the sampling systems' storing and transmission capabilities. For this
reason, various acquisition approaches have been developed to simultaneously
sense and compress the relevant information of the signal using random
projections. However, estimating the covariance matrix from the random
projections is an ill-posed problem that requires further information about the
data, such as sparsity, low rank, or stationary behavior. Furthermore, this
approach fails using high compression ratios. Therefore, this paper proposes an
algorithm based on the projected gradient method to recover a low-rank or
Toeplitz approximation of the covariance matrix. The proposed algorithm divides
the data into subsets projected onto different subspaces, assuming that each
subset contains an approximation of the signal statistics, improving the
inverse problem's condition. The error induced by this assumption is
analytically derived along with the convergence guarantees of the proposed
method. Extensive simulations show that the proposed algorithm can effectively
recover the covariance matrix of hyperspectral images with high compression
ratios (8-15% approx) in noisy scenarios. Additionally, simulations and
theoretical results show that filtering the gradient reduces the estimator's
error recovering up to twice the number of eigenvectors.Comment: submitted to IEEE Transactions on Image Processin
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