我们通过基于压缩感测和多输出（MIMO）无线雷达来解决材料缺陷的检测，这些材料缺陷在层状材料结构内部。这里，由于层状结构的表面的反射导致的强杂波通常经常使缺陷挑战的缺陷。因此，需要改进的缺陷检测所需的复杂信号分离方法。在许多情况下，我们感兴趣的缺陷的数量是有限的，并且分层结构的信令响应可以被建模为低秩结构。因此，我们提出了对缺陷检测的关节等级和稀疏最小化。特别是，我们提出了一种基于迭代重量的核和$ \ ell_1- $规范（一种双重重量方法）的非凸法方法，与传统的核规范和$ \ ell_1- $常态最小化相比获得更高的准确性。为此，迭代算法旨在估计低级别和稀疏贡献。此外，我们建议深入学习来学习算法（即，算法展开）的参数，以提高算法的准确性和汇聚速度。我们的数值结果表明，该方法在恢复的低级别和稀疏组分的均方误差和收敛速度方面优于常规方法。

在许多工程应用中，例如雷达/声纳/超声成像等许多工程应用中，稀疏多通道盲卷（S-MBD）的问题经常出现。为了降低其计算和实施成本，我们提出了一种压缩方法，该方法可以及时从更少的测量值中进行盲目恢复。提出的压缩通过过滤器随后进行亚采样来测量信号，从而大大降低了实施成本。我们得出理论保证，可从压缩测量中识别和回收稀疏过滤器。我们的结果允许设计广泛的压缩过滤器。然后，我们提出了一个由数据驱动的展开的学习框架，以学习压缩过滤器并解决S-MBD问题。编码器是一个经常性的推理网络，该网络将压缩测量结果映射到稀疏过滤器的估计值中。我们证明，与基于优化的方法相比，我们展开的学习方法对源形状的选择更为强大，并且具有更好的恢复性能。最后，在具有有限数据的应用程序（少数图）的应用中，我们强调了与传统深度学习相比，展开学习的卓越概括能力。

Tomographic SAR technique has attracted remarkable interest for its ability of three-dimensional resolving along the elevation direction via a stack of SAR images collected from different cross-track angles. The emerged compressed sensing (CS)-based algorithms have been introduced into TomoSAR considering its super-resolution ability with limited samples. However, the conventional CS-based methods suffer from several drawbacks, including weak noise resistance, high computational complexity, and complex parameter fine-tuning. Aiming at efficient TomoSAR imaging, this paper proposes a novel efficient sparse unfolding network based on the analytic learned iterative shrinkage thresholding algorithm (ALISTA) architecture with adaptive threshold, named Adaptive Threshold ALISTA-based Sparse Imaging Network (ATASI-Net). The weight matrix in each layer of ATASI-Net is pre-computed as the solution of an off-line optimization problem, leaving only two scalar parameters to be learned from data, which significantly simplifies the training stage. In addition, adaptive threshold is introduced for each azimuth-range pixel, enabling the threshold shrinkage to be not only layer-varied but also element-wise. Moreover, the final learned thresholds can be visualized and combined with the SAR image semantics for mutual feedback. Finally, extensive experiments on simulated and real data are carried out to demonstrate the effectiveness and efficiency of the proposed method.

Discriminative features extracted from the sparse coding model have been shown to perform well for classification. Recent deep learning architectures have further improved reconstruction in inverse problems by considering new dense priors learned from data. We propose a novel dense and sparse coding model that integrates both representation capability and discriminative features. The model studies the problem of recovering a dense vector $\mathbf{x}$ and a sparse vector $\mathbf{u}$ given measurements of the form $\mathbf{y} = \mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}$. Our first analysis proposes a geometric condition based on the minimal angle between spanning subspaces corresponding to the matrices $\mathbf{A}$ and $\mathbf{B}$ that guarantees unique solution to the model. The second analysis shows that, under mild assumptions, a convex program recovers the dense and sparse components. We validate the effectiveness of the model on simulated data and propose a dense and sparse autoencoder (DenSaE) tailored to learning the dictionaries from the dense and sparse model. We demonstrate that (i) DenSaE denoises natural images better than architectures derived from the sparse coding model ($\mathbf{B}\mathbf{u}$), (ii) in the presence of noise, training the biases in the latter amounts to implicitly learning the $\mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}$ model, (iii) $\mathbf{A}$ and $\mathbf{B}$ capture low- and high-frequency contents, respectively, and (iv) compared to the sparse coding model, DenSaE offers a balance between discriminative power and representation.

约束的张量和矩阵分子化模型允许从多道数据中提取可解释模式。因此，对于受约束的低秩近似度的可识别性特性和有效算法是如此重要的研究主题。这项工作涉及低秩近似的因子矩阵的列，以众所周知的和可能的过度顺序稀疏，该模型包括基于字典的低秩近似（DLRA）。虽然早期的贡献集中在候选列字典内的发现因子列，即一稀疏的近似值，这项工作是第一个以大于1的稀疏性解决DLRA。我建议专注于稀疏编码的子问题，在解决DLRA时出现的混合稀疏编码（MSC）以交替的优化策略在解决DLRA时出现。提供了基于稀疏编码启发式的几种算法（贪婪方法，凸起放松）以解决MSC。在模拟数据上评估这些启发式的性能。然后，我展示了如何基于套索来调整一个有效的MSC求解器，以计算高光谱图像处理和化学测量学的背景下的基于词典的基于矩阵分解和规范的多adic分解。这些实验表明，DLRA扩展了低秩近似的建模能力，有助于降低估计方差并提高估计因子的可识别性和可解释性。

Quantum state tomography aims to estimate the state of a quantum mechanical system which is described by a trace one, Hermitian positive semidefinite complex matrix, given a set of measurements of the state. Existing works focus on estimating the density matrix that represents the state, using a compressive sensing approach, with only fewer measurements than that required for a tomographically complete set, with the assumption that the true state has a low rank. One very popular method to estimate the state is the use of the Singular Value Thresholding (SVT) algorithm. In this work, we present a machine learning approach to estimate the quantum state of n-qubit systems by unrolling the iterations of SVT which we call Learned Quantum State Tomography (LQST). As merely unrolling SVT may not ensure that the output of the network meets the constraints required for a quantum state, we design and train a custom neural network whose architecture is inspired from the iterations of SVT with additional layers to meet the required constraints. We show that our proposed LQST with very few layers reconstructs the density matrix with much better fidelity than the SVT algorithm which takes many hundreds of iterations to converge. We also demonstrate the reconstruction of the quantum Bell state from an informationally incomplete set of noisy measurements.

近年来，在诸如denoing，压缩感应，介入和超分辨率等反问题中使用深度学习方法的使用取得了重大进展。尽管这种作品主要是由实践算法和实验驱动的，但它也引起了各种有趣的理论问题。在本文中，我们调查了这一作品中一些突出的理论发展，尤其是生成先验，未经训练的神经网络先验和展开算法。除了总结这些主题中的现有结果外，我们还强调了一些持续的挑战和开放问题。

In this paper, we investigate the joint device activity and data detection in massive machine-type communications (mMTC) with a one-phase non-coherent scheme, where data bits are embedded in the pilot sequences and the base station simultaneously detects active devices and their embedded data bits without explicit channel estimation. Due to the correlated sparsity pattern introduced by the non-coherent transmission scheme, the traditional approximate message passing (AMP) algorithm cannot achieve satisfactory performance. Therefore, we propose a deep learning (DL) modified AMP network (DL-mAMPnet) that enhances the detection performance by effectively exploiting the pilot activity correlation. The DL-mAMPnet is constructed by unfolding the AMP algorithm into a feedforward neural network, which combines the principled mathematical model of the AMP algorithm with the powerful learning capability, thereby benefiting from the advantages of both techniques. Trainable parameters are introduced in the DL-mAMPnet to approximate the correlated sparsity pattern and the large-scale fading coefficient. Moreover, a refinement module is designed to further advance the performance by utilizing the spatial feature caused by the correlated sparsity pattern. Simulation results demonstrate that the proposed DL-mAMPnet can significantly outperform traditional algorithms in terms of the symbol error rate performance.

This paper is about a curious phenomenon. Suppose we have a data matrix, which is the superposition of a low-rank component and a sparse component. Can we recover each component individually? We prove that under some suitable assumptions, it is possible to recover both the low-rank and the sparse components exactly by solving a very convenient convex program called Principal Component Pursuit; among all feasible decompositions, simply minimize a weighted combination of the nuclear norm and of the 1 norm. This suggests the possibility of a principled approach to robust principal component analysis since our methodology and results assert that one can recover the principal components of a data matrix even though a positive fraction of its entries are arbitrarily corrupted. This extends to the situation where a fraction of the entries are missing as well. We discuss an algorithm for solving this optimization problem, and present applications in the area of video surveillance, where our methodology allows for the detection of objects in a cluttered background, and in the area of face recognition, where it offers a principled way of removing shadows and specularities in images of faces.

一位压缩感知与从其一位噪声测量值中准确恢复了潜在的稀疏信号。该问题的常规信号恢复方法主要是基于以下假设开发的：感应矩阵的确切知识。但是，在这项工作中，我们提出了一种新颖的数据驱动和基于模型的方法，可以实现盲目恢复。即，信号恢复而无需了解传感矩阵。为此，我们利用了深层发展技术，并开发了用于该特定任务的模型驱动的深神经架构。拟议的深度体系结构能够通过利用基础展开的算法来学习替代感测矩阵 - 位噪声测量。此外，由于将域知识和系统的数学模型合并到拟议的深度体系结构中，因此由增强的可解释性带来的网络受益，具有少量的可训练参数，并且需要少量的培训样本，即与常用的黑盒深神经网络替代方案相比。

Countless signal processing applications include the reconstruction of signals from few indirect linear measurements. The design of effective measurement operators is typically constrained by the underlying hardware and physics, posing a challenging and often even discrete optimization task. While the potential of gradient-based learning via the unrolling of iterative recovery algorithms has been demonstrated, it has remained unclear how to leverage this technique when the set of admissible measurement operators is structured and discrete. We tackle this problem by combining unrolled optimization with Gumbel reparametrizations, which enable the computation of low-variance gradient estimates of categorical random variables. Our approach is formalized by GLODISMO (Gradient-based Learning of DIscrete Structured Measurement Operators). This novel method is easy-to-implement, computationally efficient, and extendable due to its compatibility with automatic differentiation. We empirically demonstrate the performance and flexibility of GLODISMO in several prototypical signal recovery applications, verifying that the learned measurement matrices outperform conventional designs based on randomization as well as discrete optimization baselines.

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard, because it contains vector cardinality minimization as a special case.In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is Ω(r(m + n) log mn), where m, n are the dimensions of the matrix, and r is its rank.The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to solving the norm minimization relaxations, and illustrate our results with numerical examples.

As a convex relaxation of the low rank matrix factorization problem, the nuclear norm minimization has been attracting significant research interest in recent years. The standard nuclear norm minimization regularizes each singular value equally to pursue the convexity of the objective function. However, this greatly restricts its capability and flexibility in dealing with many practical problems (e.g., denoising), where the singular values have clear physical meanings and should be treated differently. In this paper we study the weighted nuclear norm minimization (WNNM) problem, where the singular values are assigned different weights. The solutions of the WNNM problem are analyzed under different weighting conditions. We then apply the proposed WNNM algorithm to image denoising by exploiting the image nonlocal self-similarity. Experimental results clearly show that the proposed WNNM algorithm outperforms many state-of-the-art denoising algorithms such as BM3D in terms of both quantitative measure and visual perception quality.

Deep neural networks provide unprecedented performance gains in many real world problems in signal and image processing. Despite these gains, future development and practical deployment of deep networks is hindered by their blackbox nature, i.e., lack of interpretability, and by the need for very large training sets. An emerging technique called algorithm unrolling or unfolding offers promise in eliminating these issues by providing a concrete and systematic connection between iterative algorithms that are used widely in signal processing and deep neural networks. Unrolling methods were first proposed to develop fast neural network approximations for sparse coding. More recently, this direction has attracted enormous attention and is rapidly growing both in theoretic investigations and practical applications. The growing popularity of unrolled deep networks is due in part to their potential in developing efficient, high-performance and yet interpretable network architectures from reasonable size training sets. In this article, we review algorithm unrolling for signal and image processing. We extensively cover popular techniques for algorithm unrolling in various domains of signal and image processing including imaging, vision and recognition, and speech processing. By reviewing previous works, we reveal the connections between iterative algorithms and neural networks and present recent theoretical results. Finally, we provide a discussion on current limitations of unrolling and suggest possible future research directions.

Channel estimation is a critical task in multiple-input multiple-output (MIMO) digital communications that substantially effects end-to-end system performance. In this work, we introduce a novel approach for channel estimation using deep score-based generative models. A model is trained to estimate the gradient of the logarithm of a distribution and is used to iteratively refine estimates given measurements of a signal. We introduce a framework for training score-based generative models for wireless MIMO channels and performing channel estimation based on posterior sampling at test time. We derive theoretical robustness guarantees for channel estimation with posterior sampling in single-input single-output scenarios, and experimentally verify performance in the MIMO setting. Our results in simulated channels show competitive in-distribution performance, and robust out-of-distribution performance, with gains of up to $5$ dB in end-to-end coded communication performance compared to supervised deep learning methods. Simulations on the number of pilots show that high fidelity channel estimation with $25$% pilot density is possible for MIMO channel sizes of up to $64 \times 256$. Complexity analysis reveals that model size can efficiently trade performance for estimation latency, and that the proposed approach is competitive with compressed sensing in terms of floating-point operation (FLOP) count.

我们考虑具有某些约束的矩阵分解（MF），在各个领域找到广泛的应用。利用变异推理（VI）和单一近似消息传递（UAMP），我们通过有效的消息传递实现（称为UAMPMF）开发了MF的贝叶斯方法。通过对因子矩阵施加的适当先验，UAMPMF可用于解决许多可以表达为MF的问题，例如非负基质分解，词典学习，具有矩阵不确定性的压缩感，可靠的主成分分析和稀疏矩阵分解。提供了广泛的数值示例，以表明UAMPMF在恢复精度，鲁棒性和计算复杂性方面显着优于最先进的算法。

传统上，信号处理，通信和控制一直依赖经典的统计建模技术。这种基于模型的方法利用代表基本物理，先验信息和其他领域知识的数学公式。简单的经典模型有用，但对不准确性敏感，当真实系统显示复杂或动态行为时，可能会导致性能差。另一方面，随着数据集变得丰富，现代深度学习管道的力量增加，纯粹的数据驱动的方法越来越流行。深度神经网络（DNNS）使用通用体系结构，这些架构学会从数据中运行，并表现出出色的性能，尤其是针对受监督的问题。但是，DNN通常需要大量的数据和巨大的计算资源，从而限制了它们对某些信号处理方案的适用性。我们对将原则数学模型与数据驱动系统相结合的混合技术感兴趣，以从两种方法的优势中受益。这种基于模型的深度学习方法通​​过为特定问题设计的数学结构以及从有限的数据中学习来利用这两个部分领域知识。在本文中，我们调查了研究和设计基于模型的深度学习系统的领先方法。我们根据其推理机制将基于混合模型/数据驱动的系统分为类别。我们对以系统的方式将基于模型的算法与深度学习以及具体指南和详细的信号处理示例相结合的领先方法进行了全面综述。我们的目的是促进对未来系统的设计和研究信号处理和机器学习的交集，这些系统结合了两个领域的优势。

深度无形的神经网络（NNS）受到了极大的关注，因为它们的复杂性相对较低。通常，这些深度折​​叠的NN仅限于所有输入的固定深度。但是，收敛所需的最佳层随着不同的输入而变化。在本文中，我们首先开发了一个深层确定性策略梯度（DDPG）驱动的深度无折叠的框架，并针对不同输入进行自适应深度，在该框架中，DDPG学习了可训练的深度NN的可训练参数，而不是由随机梯度更新下降算法直接。具体而言，DDPG的状态，动作和状态过渡分别将优化变量，可训练的参数和架构分别设计为DDPG的状态，动作和状态过渡。然后，使用此框架来处理大量多输入多输出系统中的通道估计问题。具体而言，首先，我们通过离网基准制定了通道估计问题，并开发了稀疏的贝叶斯学习（SBL）基于基于的算法来解决它。其次，将基于SBL的算法展开为一组带有一组可训练参数的层结构。第三，采用了提出的DDPG驱动的深度解释框架来基于基于SBL的算法的展开结构来解决此通道估计问题。为了实现自适应深度，我们设计了停止分数以指示何时停止，这是通道重建误差的函数。此外，提出的框架被扩展到实现一般深度神经网络（DNNS）的适应性深度。仿真结果表明，所提出的算法的表现优于固定深度的常规优化算法和DNN，层数量大多。

最近，刘和张研究了从压缩传感的角度研究了时间序列预测的相当具有挑战性的问题。他们提出了一个没有学习的方法，名为卷积核规范最小化（CNNM），并证明了CNNM可以完全从其观察到的部分恢复一系列系列的部分，只要该系列是卷积的低级。虽然令人印象深刻，但是每当系列远离季节性时可能不满足卷积的低秩条件，并且实际上是脆弱的趋势和动态的存在。本文试图通过将学习，正常的转换集成到CNNM中，以便将一系列渐开线结构转换为卷积低等级的常规信号的目的。我们证明，由于系列的变换是卷积低级的转换，所以，所产生的模型是基于学习的基于学习的CNNM（LBCNM），严格成功地识别了一个系列的未来部分。为了学习可能符合所需成功条件的适当转换，我们设计了一种基于主成分追求（PCP）的可解释方法。配备了这种学习方法和一些精心设计的数据论证技巧，LBCNM不仅可以处理时间序列的主要组成部分（包括趋势，季节性和动态），还可以利用其他一些预测方法提供的预测;这意味着LBCNNM可以用作模型组合的一般工具。从时间序列数据库（TSDL）和M4竞争（M4）的100,452个现实世界时间序列的大量实验证明了LBCNNM的卓越性能。

从高度不足的数据中恢复颜色图像和视频是面部识别和计算机视觉中的一项基本且具有挑战性的任务。通过颜色图像和视频的多维性质，在本文中，我们提出了一种新颖的张量完成方法，该方法能够有效探索离散余弦变换（DCT）下张量数据的稀疏性。具体而言，我们介绍了两个``稀疏 +低升级''张量完成模型，以及两种可实现的算法来找到其解决方案。第一个是基于DCT的稀疏加权核标准诱导低级最小化模型。第二个是基于DCT的稀疏加上$ P $换图映射引起的低秩优化模型。此外，我们因此提出了两种可实施的增强拉格朗日算法，以解决基础优化模型。一系列数值实验在内，包括颜色图像介入和视频数据恢复表明，我们所提出的方法的性能要比许多现有的最新张量完成方法更好，尤其是对于缺少数据比率较高的情况。