Rectified Gaussian Scale Mixtures and the Sparse Non-Negative Least Squares Problem

In this paper, we develop a Bayesian evidence maximization framework to solve the sparse non-negative least squares (S-NNLS) problem. We introduce a family of probability densities referred to as the Rectified Gaussian Scale Mixture (R- GSM) to model the sparsity enforcing prior distribution for the solution. The R-GSM prior encompasses a variety of heavy-tailed densities such as the rectified Laplacian and rectified Student- t distributions with a proper choice of the mixing density. We utilize the hierarchical representation induced by the R-GSM prior and develop an evidence maximization framework based on the Expectation-Maximization (EM) algorithm. Using the EM based method, we estimate the hyper-parameters and obtain a point estimate for the solution. We refer to the proposed method as rectified sparse Bayesian learning (R-SBL). We provide four R- SBL variants that offer a range of options for computational complexity and the quality of the E-step computation. These methods include the Markov chain Monte Carlo EM, linear minimum mean-square-error estimation, approximate message passing and a diagonal approximation. Using numerical experiments, we show that the proposed R-SBL method outperforms existing S-NNLS solvers in terms of both signal and support recovery performance, and is also very robust against the structure of the design matrix.Comment: Under Review by IEEE Transactions on Signal Processin

Unitary Approximate Message Passing for Sparse Bayesian Learning and Bilinear Recovery

Over the past several years, the approximate message passing (AMP) algorithm has been applied to a broad range of problems, including compressed sensing (CS), robust regression, Bayesian estimation, etc. AMP was originally developed for compressed sensing based on the loopy belief propagation (BP). Compared to convex optimization based algorithms, AMP has low complexity and its performance can be rigorously characterized by a scalar state evolution (SE) in the case of a large independent and identically distributed (i.i.d.) (sub-) Gaussian matrix. AMP was then extended to solve general estimation problems with a generalized linear observation model. However, AMP performs poorly on a generic matrix such as non-zero mean, rank-deficient, correlated, or ill-conditioned matrix, resulting in divergence and degraded performance. It was discovered later that applying AMP to a unitary transform of the original model can remarkably enhance the robustness to difficult matrices. This variant is named unitary AMP (UAMP), or formally called UTAMP. In this thesis, leveraging UAMP, we propose UAMP-SBL for sparse signal recovery and Bi-UAMP for bilinear recovery, both of which inherit the low complexity and robustness of UAMP. Sparse Bayesian learning (SBL) is a powerful tool for recovering a sparse signal from noisy measurements, which finds numerous applications in various areas. As a traditional implementation of SBL, e.g., Tipping’s method, involves matrix inversion in each iteration, the computational complexity can be prohibitive for large scale problems. To circumvent this, AMP and its variants have been used as low-complexity solutions. Unfortunately, they will diverge for ‘difficult’ measurement matrices as previously mentioned. In this thesis, leveraging UAMP, a novel SBL algorithm called UAMP-SBL is proposed where UAMP is incorporated into the structured variational message passing (SVMP) to handle the most computationally intensive part of message computations. It is shown that, compared to state-of-the-art AMP based SBL algorithms, the proposed UAMP-SBL is more robust and efficient, leading to remarkably better performance. The bilinear recovery problem has many applications such as dictionary learning, selfcalibration, compressed sensing with matrix uncertainty, etc. Compared to existing nonmessage passing alternates, several AMP based algorithms have been developed to solve bilinear problems. By using UAMP, a more robust and faster approximate inference algorithm for bilinear recovery is proposed in this thesis, which is called Bi-UAMP. With the lifting approach, the original bilinear problem is reformulated as a linear one. Then, variational inference (VI), expectation propagation (EP) and BP are combined with UAMP to implement the approximate inference algorithm Bi-UAMP, where UAMP is adopted for the most computationally intensive part. It is shown that, compared to state-of-the-art bilinear recovery algorithms, the proposed Bi-UAMP is much more robust and faster, and delivers significantly better performance. Recently, UAMP has also been employed for many other applications such as inverse synthetic aperture radar (ISAR) imaging, low-complexity direction of arrival (DOA) estimation, iterative detection for orthogonal time frequency space modulation (OTFS), channel estimation for RIS-Aided MIMO communications, etc. Promising performance was achieved in all of the applications, and more applications of UAMP are expected in the future

Metodi Matriciali per l'Acquisizione Efficiente e la Crittografia di Segnali in Forma Compressa

The idea of balancing the resources spent in the acquisition and encoding of natural signals strictly to their intrinsic information content has interested nearly a decade of research under the name of compressed sensing. In this doctoral dissertation we develop some extensions and improvements upon this technique's foundations, by modifying the random sensing matrices on which the signals of interest are projected to achieve different objectives. Firstly, we propose two methods for the adaptation of sensing matrix ensembles to the second-order moments of natural signals. These techniques leverage the maximisation of different proxies for the quantity of information acquired by compressed sensing, and are efficiently applied in the encoding of electrocardiographic tracks with minimum-complexity digital hardware. Secondly, we focus on the possibility of using compressed sensing as a method to provide a partial, yet cryptanalysis-resistant form of encryption; in this context, we show how a random matrix generation strategy with a controlled amount of perturbations can be used to distinguish between multiple user classes with different quality of access to the encrypted information content. Finally, we explore the application of compressed sensing in the design of a multispectral imager, by implementing an optical scheme that entails a coded aperture array and Fabry-Pérot spectral filters. The signal recoveries obtained by processing real-world measurements show promising results, that leave room for an improvement of the sensing matrix calibration problem in the devised imager