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Register a new userApproximate Message Passing with a Colored Aliasing Model for Variable Density Fourier Sampled Images
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The Approximate Message Passing (AMP) algorithm efficiently reconstructs signals which have been sampled with large i.i.d. sub-Gaussian sensing matrices. Central to AMP is its state evolution, which guarantees that the difference between the current estimate and ground truth (the aliasing) at every iteration obeys a Gaussian distribution that can be fully characterized by a scalar. However, when Fourier coefficients of a signal with non-uniform spectral density are sampled, such as in Magnetic Resonance Imaging (MRI), the aliasing is intrinsically colored, AMPs scalar state evolution is no longer accurate and the algorithm encounters convergence problems. In response, we propose the Variable Density Approximate Message Passing (VDAMP) algorithm, which uses the wavelet domain to model the colored aliasing. We present empirical evidence that VDAMP obeys a colored state evolution, where the aliasing obeys a Gaussian distribution that can be fully characterized with one scalar per wavelet subband. A benefit of state evolution is that Steins Unbiased Risk Estimate (SURE) can be effectively implemented, yielding an algorithm with subband-dependent thresholding that has no free parameters. We empirically evaluate the effectiveness of VDAMP on three variations of Fast Iterative Shrinkage-Thresholding (FISTA) and find that it converges in around 10 times fewer iterations on average than the next-fastest method, and to a comparable mean-squared-error.
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For certain sensing matrices, the Approximate Message Passing (AMP) algorithm efficiently reconstructs undersampled signals. However, in Magnetic Resonance Imaging (MRI), where Fourier coefficients of a natural image are sampled with variable density, AMP encounters convergence problems. In response we present an algorithm based on Orthogonal AMP constructed specifically for variable density partial Fourier sensing matrices. For the first time in this setting a state evolution has been observed. A practical advantage of state evolution is that Steins Unbiased Risk Estimate (SURE) can be effectively implemented, yielding an algorithm with no free parameters. We empirically evaluate the effectiveness of the parameter-free algorithm on simulated data and find that it converges over 5x faster and to a lower mean-squared error solution than Fast Iterative Shrinkage-Thresholding (FISTA).
We consider a broad class of Approximate Message Passing (AMP) algorithms defined as a Lipschitzian functional iteration in terms of an $ntimes n$ random symmetric matrix $A$. We establish universality in noise for this AMP in the $n$-limit and validate this behavior in a number of AMPs popularly adapted in compressed sensing, statistical inferences, and optimizations in spin glasses.
Compressed sensing (CS) deals with the problem of reconstructing a sparse vector from an under-determined set of observations. Approximate message passing (AMP) is a technique used in CS based on iterative thresholding and inspired by belief propagation in graphical models. Due to the high transmission rate and a high molecular absorption, spreading loss and reflection loss, the discrete-time channel impulse response (CIR) of a typical indoor THz channel is very long and exhibits an approximately sparse characteristic. In this paper, we develop AMP based channel estimation algorithms for indoor THz communications. The performance of these algorithms is compared to the state of the art. We apply AMP with soft- and hard-thresholding. Unlike the common applications in which AMP with hard-thresholding diverges, the properties of the THz channel favor this approach. It is shown that THz channel estimation via hard-thresholding AMP outperforms all previously proposed methods and approaches the oracle based performance closely.
Sparse Bayesian learning (SBL) can be implemented with low complexity based on the approximate message passing (AMP) algorithm. However, it does not work well for a generic measurement matrix, which may cause AMP to diverge. Damped AMP has been used for SBL to alleviate the problem at the cost of reducing convergence speed. In this work, we propose a new SBL algorithm based on structured variational inference, leveraging AMP with a unitary transformation (UAMP). Both single measurement vector and multiple measurement vector problems are investigated. 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.
We consider the problem of estimating a signal from measurements obtained via a generalized linear model. We focus on estimators based on approximate message passing (AMP), a family of iterative algorithms with many appealing features: the performance of AMP in the high-dimensional limit can be succinctly characterized under suitable model assumptions; AMP can also be tailored to the empirical distribution of the signal entries, and for a wide class of estimation problems, AMP is conjectured to be optimal among all polynomial-time algorithms. However, a major issue of AMP is that in many models (such as phase retrieval), it requires an initialization correlated with the ground-truth signal and independent from the measurement matrix. Assuming that such an initialization is available is typically not realistic. In this paper, we solve this problem by proposing an AMP algorithm initialized with a spectral estimator. With such an initialization, the standard AMP analysis fails since the spectral estimator depends in a complicated way on the design matrix. Our main contribution is a rigorous characterization of the performance of AMP with spectral initialization in the high-dimensional limit. The key technical idea is to define and analyze a two-phase artificial AMP algorithm that first produces the spectral estimator, and then closely approximates the iterates of the true AMP. We also provide numerical results that demonstrate the validity of the proposed approach.
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