Deep neural network (DNN) has a wide range of applications in various fields, including solving sparse inverse problems. In this paper, we propose a novel network called the Stein's unbiased risk estimate based-tr...
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ISBN:
(纸本)9781479981311
Deep neural network (DNN) has a wide range of applications in various fields, including solving sparse inverse problems. In this paper, we propose a novel network called the Stein's unbiased risk estimate based-trainable iterative thresholding algorithm (SURE-TISTA) for sparse signal recovery problems. Without prior information, SURE-TISTA outperforms TISTA, an algorithm based on the minimum mean squared error (MMSE) estimator. SURE-TISTA also shows a great robustness in many cases including large-scale and large-variance problems. Meanwhile, SURE-TISTA uses fewer learnable variables to achieve similar performance as learned approximate message passing (LAMP), which has more learnable parameters. Without any error measure estimator, SURE-TISTA achieves a near MMSE-based performance. Our numerical results indicate that SURE-TISTA is superior to TISTA and other traditional algorithms in many aspects, which can be promising in image denoising.
State-of the art audio codecs use time-frequency transforms derived from cosine bases, followed by a quantification stage. The quantization steps are set according to perceptual considerations. In the last decade, sev...
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State-of the art audio codecs use time-frequency transforms derived from cosine bases, followed by a quantification stage. The quantization steps are set according to perceptual considerations. In the last decade, several studies applied adaptive sparse time-frequency transforms to audio coding, e.g. on unions of cosine bases using a Matching-Pursuit-derived algorithm [1]. This was shown to significantly improve the coding efficiency. We propose another approach based on a variational algorithm, i.e. the optimization of a cost function taking into account both a perceptual distortion measure derived form a hearing model and a sparsity constraint, which favors the coding efficiency. In this early version, we show that, using a coding scheme without perceptual control of quantization, our method outperforms a codec from the literature with the same quantization scheme [1]. In future work, a more sophisticated quantization scheme would probably allow our method to challenge standard codecs e.g. AAC.
In this paper, we first establish a weak unique continuation property for time-fractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for el...
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In this paper, we first establish a weak unique continuation property for time-fractional diffusion-advection equations. The proof is mainly based on the Laplace transform and the unique continuation properties for elliptic and parabolic equations. The result is weaker than its parabolic counterpart in the sense that we additionally impose the homogeneous boundary condition. As a direct application, we prove the uniqueness for an inverse problem on determining the spatial component in the source term by interior measurements. Numerically, we reformulate our inverse source problem as an optimization problem, and propose an iterative thresholding algorithm. Finally, several numerical experiments are presented to show the accuracy and efficiency of the algorithm.
The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biom...
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The goal of compressed sensing is to reconstruct a sparse signal under a few linear measurements far less than the dimension of the ambient space of the signal. However, many real-life applications in physics and biomedical sciences carry some strongly nonlinear structures, and the linear model is no longer suitable. Compared with the compressed sensing under the linear circumstance, this nonlinear compressed sensing is much more difficult, in fact also NP-hard, combinatorial problem, because of the discrete and discontinuous nature of the L-0-norm and the nonlinearity. In order to get a convenience for sparse signal recovery, we set the nonlinear models have a smooth quasi-linear nature in this paper, and study a non-convex fraction function rho(a) in this quasi-linear compressed sensing. We propose an iterative fraction thresholdingalgorithm to solve the regularization problem (QP(a)(lambda)) for all a > 0. With the change of parameter a > 0, our algorithm could get a promising result, which is one of the advantages for our algorithm compared with some state-of-art algorithms. Numerical experiments show that our method performs much better than some state-of-the-art methods.
In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabi...
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In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabilized nonlinear minimization system by an appropriately selected Tikhonov regularization. The existence and the stability of the optimization system are demonstrated. The nonlinear optimization problem is approximated by a fully discrete scheme, whose convergence is established under a novel result verified in this study that the H-1-norm of the solution to the discrete forward system is uniformly bounded. The iterative thresholding algorithm is proposed to solve the discrete minimization, and several numerical experiments are presented to show the efficiency and the accuracy of the algorithm.
In this paper, we investigate a group sparse optimization problem via lp,q regularization in three aspects: theory, algorithm and application. In the theoretical aspect, by introducing a notion of group restricted eig...
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In this paper, we investigate a group sparse optimization problem via lp,q regularization in three aspects: theory, algorithm and application. In the theoretical aspect, by introducing a notion of group restricted eigenvalue condition, we establish an oracle property and a global recovery bound of order O(λ2/2-q) for any point in a level set of the lp,q regularization problem, and by virtue of modern variational analysis techniques, we also provide a local analysis of recovery bound of order O(λ2) for a path of local minima. In the algorithmic aspect, we apply the well-known proximal gradient method to solve the p,q regularization problems, either by analytically solving some specific lp,q regularization subproblems, or by using the Newton method to solve general lp,q regularization subproblems. In particular, we establish a local linear convergence rate of the proximal gradient method for solving the l1,q regularization problem under some mild conditions and by first proving a second-order growth condition. As a consequence, the local linear convergence rate of proximal gradient method for solving the usual lq regularization problem (0 < q < 1) is obtained. Finally in the aspect of application, we present some numerical results on both the simulated data and the real data in gene transcriptional regulation.
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