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Register a new userGeneralized Expectation Consistent Signal Recovery for Nonlinear Measurements
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Added by
Hengtao He
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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In this paper, we propose a generalized expectation consistent signal recovery algorithm to estimate the signal $mathbf{x}$ from the nonlinear measurements of a linear transform output $mathbf{z}=mathbf{A}mathbf{x}$. This estimation problem has been encountered in many applications, such as communications with front-end impairments, compressed sensing, and phase retrieval. The proposed algorithm extends the prior art called generalized turbo signal recovery from a partial discrete Fourier transform matrix $mathbf{A}$ to a class of general matrices. Numerical results show the excellent agreement of the proposed algorithm with the theoretical Bayesian-optimal estimator derived using the replica method.
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The reconstruction of sparse signal is an active area of research. Different from a typical i.i.d. assumption, this paper considers a non-independent prior of group structure. For this more practical setup, we propose EM-aided HyGEC, a new algorithm to address the stability issue and the hyper-parameter issue facing the other algorithms. The instability problem results from the ill condition of the transform matrix, while the unavailability of the hyper-parameters is a ground truth that their values are not known beforehand. The proposed algorithm is built on the paradigm of HyGAMP (proposed by Rangan et al.) but we replace its inner engine, the GAMP, by a matrix-insensitive alternative, the GEC, so that the first issue is solved. For the second issue, we take expectation-maximization as an outer loop, and together with the inner engine HyGEC, we learn the value of the hyper-parameters. Effectiveness of the proposed algorithm is also verified by means of numerical simulations.
Compressive sensing relies on the sparse prior imposed on the signal of interest to solve the ill-posed recovery problem in an under-determined linear system. The objective function used to enforce the sparse prior information should be both effective and easily optimizable. Motivated by the entropy concept from information theory, in this paper we propose the generalized Shannon entropy function and R{e}nyi entropy function of the signal as the sparsity promoting regularizers. Both entropy functions are nonconvex, non-separable. Their local minimums only occur on the boundaries of the orthants in the Euclidean space. Compared to other popular objective functions, minimizing the generalized entropy functions adaptively promotes multiple high-energy coefficients while suppressing the rest low-energy coefficients. The corresponding optimization problems can be recasted into a series of reweighted $l_1$-norm minimization problems and then solved efficiently by adapting the FISTA. Sparse signal recovery experiments on both the simulated and real data show the proposed entropy functions minimization approaches perform better than other popular approaches and achieve state-of-the-art performances.
The generalized approximate message passing (GAMP) algorithm under the Bayesian setting shows advantage in recovering under-sampled sparse signals from corrupted observations. Compared to conventional convex optimization methods, it has a much lower complexity and is computationally tractable. In the GAMP framework, the sparse signal and the observation are viewed to be generated according to some pre-specified probability distributions in the input and output channels. However, the parameters of the distributions are usually unknown in practice. In this paper, we propose an extended GAMP algorithm with built-in parameter estimation (PE-GAMP) and present its empirical convergence analysis. PE-GAMP treats the parameters as unknown random variables with simple priors and jointly estimates them with the sparse signals. Compared with Expectation Maximization (EM) based parameter estimation methods, the proposed PE-GAMP could draw information from the prior distributions of the parameters to perform parameter estimation. It is also more robust and much simpler, which enables us to consider more complex signal distributions apart from the usual Bernoulli-Gaussian (BGm) mixture distribution. Specifically, the formulations of Bernoulli-Exponential mixture (BEm) distribution and Laplace distribution are given in this paper. Simulated noiseless sparse signal recovery experiments demonstrate that the performance of the proposed PE-GAMP matches the oracle GAMP algorithm. When noise is present, both the simulated experiments and the real image recovery experiments show that PE-GAMP is still able to maintain its robustness and outperform EM based parameter estimation method when the sampling ratio is small. Additionally, using the BEm formulation of the PE-GAMP, we can successfully perform non-negative sparse coding of local image patches and provide useful features for the image classification task.
Phase retrieval (PR) is an important component in modern computational imaging systems. Many algorithms have been developed over the past half century. Recent advances in deep learning have opened up a new possibility for robust and fast PR. An emerging technique, called deep unfolding, provides a systematic connection between conventional model-based iterative algorithms and modern data-based deep learning. Unfolded algorithms, powered by data learning, have shown remarkable performance and convergence speed improvement over the original algorithms. Despite their potential, most existing unfolded algorithms are strictly confined to a fixed number of iterations when employing layer-dependent parameters. In this study, we develop a novel framework for deep unfolding to overcome the existing limitations. Even if our framework can be widely applied to general inverse problems, we take PR as an example in the paper. Our development is based on an unfolded generalized expectation consistent signal recovery (GEC-SR) algorithm, wherein damping factors are left for data-driven learning. In particular, we introduce a hypernetwork to generate the damping factors for GEC-SR. Instead of directly learning a set of optimal damping factors, the hypernetwork learns how to generate the optimal damping factors according to the clinical settings, thus ensuring its adaptivity to different scenarios. To make the hypernetwork work adapt to varying layer numbers, we use a recurrent architecture to develop a dynamic hypernetwork, which generates a damping factor that can vary online across layers. We also exploit a self-attention mechanism to enhance the robustness of the hypernetwork. Extensive experiments show that the proposed algorithm outperforms existing ones in convergence speed and accuracy, and still works well under very harsh settings, that many classical PR algorithms unstable or even fail.
Compressive sensing has shown significant promise in biomedical fields. It reconstructs a signal from sub-Nyquist random linear measurements. Classical methods only exploit the sparsity in one domain. A lot of biomedical signals have additional structures, such as multi-sparsity in different domains, piecewise smoothness, low rank, etc. We propose a framework to exploit all the available structure information. A new convex programming problem is generated with multiple convex structure-inducing constraints and the linear measurement fitting constraint. With additional a priori information for solving the underdetermined system, the signal recovery performance can be improved. In numerical experiments, we compare the proposed method with classical methods. Both simulated data and real-life biomedical data are used. Results show that the newly proposed method achieves better reconstruction accuracy performance in term of both L1 and L2 errors.
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