Newton-Type Optimal Thresholding Algorithms for Sparse Optimization Problems

Sparse signals can be possibly reconstructed by an algorithm which merges a traditional nonlinear optimization method and a certain thresholding technique. Different from existing thresholding methods, a novel thresholding technique referred to as the optimal k-thresholding was recently proposed by Zhao (SIAM J Optim 30(1):31–55, 2020). This technique simultaneously performs the minimization of an error metric for the problem and thresholding of the iterates generated by the classic gradient method. In this paper, we propose the so-called Newton-type optimal k-thresholding (NTOT) algorithm which is motivated by the appreciable performance of both Newton-type methods and the optimal k-thresholding technique for signal recovery. The guaranteed performance (including convergence) of the proposed algorithms is shown in terms of suitable choices of the algorithmic parameters and the restricted isometry property (RIP) of the sensing matrix which has been widely used in the analysis of compressive sensing algorithms. The simulation results based on synthetic signals indicate that the proposed algorithms are stable and efficient for signal recovery.

An inertial parallel algorithm for a finite family of $ G $-nonexpansive mappings applied to signal recovery

<abstract><p>This study investigates the weak convergence of the sequences generated by the inertial technique combining the parallel monotone hybrid method for finding a common fixed point of a finite family of $ G $-nonexpansive mappings under suitable conditions in Hilbert spaces endowed with graphs. Some numerical examples are also presented, providing applications to signal recovery under situations without knowing the type of noises. Besides, numerical experiments of the proposed algorithms, defined by different types of blurred matrices and noises on the algorithm, are able to show the efficiency and the implementation for LASSO problem in signal recovery.</p></abstract>

Matrix methods for perfect signal recovery underlying range space of operators

The purpose of this work is to investigate perfect reconstruction underlying range space of operators in finite dimensional Hilbert spaces by a new matrix method. To this end, first we obtain more structures of the canonical $K$-dual. % and survey optimal $K$-dual problem under probabilistic erasures. Then, we survey the problem of recovering and robustness of signals when the erasure set satisfies the minimal redundancy condition or the $K$-frame is maximal robust. Furthermore, we show that the error rate is reduced under erasures if the $K$-frame is of uniform excess. Toward the protection of encoding frame (K-dual) against erasures, we introduce a new concept so called $(r,k)$-matrix to recover lost data and solve the perfect recovery problem via matrix equations. Moreover, we discuss the existence of such matrices by using minimal redundancy condition on decoding frames for operators. We exhibit several examples that illustrate the advantage of using the new matrix method with respect to the previous approaches in existence construction. And finally, we provide the numerical results to confirm the main results in the case noise-free and test sensitivity of the method with respect to noise.

The Dantzig selector: recovery of signal via ℓ 1 − αℓ 2 minimization

In the paper, we proposed the Dantzig selector based on the ℓ 1 − αℓ 2 (0 < α ⩽ 1) minimization for the signal recovery. In the Dantzig selector, the constraint ‖ A ⊤ ( b − Ax )‖∞ ⩽ η for some small constant η > 0 means the columns of A has very weakly correlated with the error vector e = Ax − b . First, recovery guarantees based on the restricted isometry property are established for signals. Next, we propose the effective algorithm to solve the proposed Dantzig selector. Last, we illustrate the proposed model and algorithm by extensive numerical experiments for the recovery of signals in the cases of Gaussian, impulsive and uniform noises. And the performance of the proposed Dantzig selector is better than that of the existing methods.

Matrix inference and estimation in multi-layer models*

We consider the problem of estimating the input and hidden variables of a stochastic multi-layer neural network (NN) from an observation of the output. The hidden variables in each layer are represented as matrices with statistical interactions along both rows as well as columns. This problem applies to matrix imputation, signal recovery via deep generative prior models, multi-task and mixed regression, and learning certain classes of two-layer NNs. We extend a recently-developed algorithm—multi-layer vector approximate message passing, for this matrix-valued inference problem. It is shown that the performance of the proposed multi-layer matrix vector approximate message passing algorithm can be exactly predicted in a certain random large-system limit, where the dimensions N × d of the unknown quantities grow as N → ∞ with d fixed. In the two-layer neural-network learning problem, this scaling corresponds to the case where the number of input features as well as training samples grow to infinity but the number of hidden nodes stays fixed. The analysis enables a precise prediction of the parameter and test error of the learning.

Numerical simulation of the frequency response and phase response of combined UWB antennas in the receiving-transmitting mode

Results of a simulation and experimental study of a combined ultra-wideband antenna, which is a combination of electric and magnetic dipoles, are presented. The input signal recovery of the transceiver path of combined antennas made by the MathCad code and the experimental one were compared. The reconstruction took place according to the frequency and phase responses of the transceiver path of these antennas.