基于图像的异常检测系统在各种制造应用中至关重要。在图像传感技术的快速开发下，近年来，此类系统的分辨率和采集率正在显着提高。这可以实时检测微小的缺陷。但是，如此高的分辨率和获取率不仅会降低图像处理算法的速度，而且还增加了数据存储和传输成本。为了解决此问题，我们提出了一种具有理论性能保证的快速和数据效率方法，该方法适用于具有光滑背景的图像中的稀疏异常检测（光滑加上稀疏信号）。所提出的方法，命名为压缩平滑稀疏分解（CSSD），是一种单步方法，它统一了压缩图像采集和基于分解的图像处理技术。为了在高维情况下进一步提高其性能，提出了Kronecker压缩平滑的稀疏分解（KRONCSSD）方法。与传统的平滑和稀疏分解算法相比，通过可忽略不计的性能损失可以实现显着的传输成本和计算速度提升。在各种应用中的模拟示例和几个案例研究说明了提出的框架的有效性。

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.

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.

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.

Unsupervised pixel-level defective region segmentation is an important task in image-based anomaly detection for various industrial applications. The state-of-the-art methods have their own advantages and limitations: matrix-decomposition-based methods are robust to noise but lack complex background image modeling capability; representation-based methods are good at defective region localization but lack accuracy in defective region shape contour extraction; reconstruction-based methods detected defective region match well with the ground truth defective region shape contour but are noisy. To combine the best of both worlds, we present an unsupervised patch autoencoder based deep image decomposition (PAEDID) method for defective region segmentation. In the training stage, we learn the common background as a deep image prior by a patch autoencoder (PAE) network. In the inference stage, we formulate anomaly detection as an image decomposition problem with the deep image prior and domain-specific regularizations. By adopting the proposed approach, the defective regions in the image can be accurately extracted in an unsupervised fashion. We demonstrate the effectiveness of the PAEDID method in simulation studies and an industrial dataset in the case study.

库存记录不正确，经常发生，某些措施的年销售额约为4％。手动检测库存不准确性的成本较高，现有算法解决方案几乎完全依赖于从纵向数据中学习，这在现代零售操作引起的动态环境中不足。取而代之的是，我们提出了基于商店和SKU上的横截面数据的解决方案，观察到检测库存不准确性可以被视为识别（低级别）泊松矩阵中异常的问题。在低级别矩阵中检测到的最先进的方法显然不足。具体而言，从理论的角度来看，这些方法的恢复保证要求需要观察到无反对的条目，而噪音消失了（在我们的问题中，在许多应用中都不是这种情况）。如此有动力，我们提出了一种在概念上简单的入门方法，以在低级别的泊松矩阵中进行异常检测。我们的方法适合一类概率异常模型。我们表明，我们的算法所产生的成本以最低最佳最佳速率接近最佳算法。使用来自消费品零售商的合成数据和真实数据，我们表明我们的方法可提供超过现有检测方法的10倍成本降低。在此过程中，我们建立了最新的工作，该工作寻求矩阵完成的入门错误保证，并为次指定矩阵确定此类保证，这是独立利益的结果。

在本文中，我们提出了一种均匀抖动的一位量化方案，以进行高维统计估计。该方案包含截断，抖动和量化，作为典型步骤。作为规范示例，量化方案应用于三个估计问题：稀疏协方差矩阵估计，稀疏线性回归和矩阵完成。我们研究了高斯和重尾政权，假定重尾数据的基本分布具有有限的第二或第四刻。对于每个模型，我们根据一位量化的数据提出新的估计器。在高斯次级政权中，我们的估计器达到了对数因素的最佳最小速率，这表明我们的量化方案几乎没有额外的成本。在重尾状态下，虽然我们的估计量基本上变慢，但这些结果是在这种单位量化和重型尾部设置中的第一个结果，或者比现有可比结果表现出显着改善。此外，我们为一位压缩传感和一位矩阵完成的问题做出了巨大贡献。具体而言，我们通过凸面编程将一位压缩感传感扩展到次高斯甚至是重尾传感向量。对于一位矩阵完成，我们的方法与标准似然方法基本不同，并且可以处理具有未知分布的预量化随机噪声。提出了有关合成数据的实验结果，以支持我们的理论分析。

We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M . Can we complete the matrix and recover the entries that we have not seen?We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys m ≥ C n 1.2 r log n for some positive numerical constant C, then with very high probability, most n × n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.

It is known that the decomposition in low-rank and sparse matrices (\textbf{L+S} for short) can be achieved by several Robust PCA techniques. Besides the low rankness, the local smoothness (\textbf{LSS}) is a vitally essential prior for many real-world matrix data such as hyperspectral images and surveillance videos, which makes such matrices have low-rankness and local smoothness properties at the same time. This poses an interesting question: Can we make a matrix decomposition in terms of \textbf{L\&LSS +S } form exactly? To address this issue, we propose in this paper a new RPCA model based on three-dimensional correlated total variation regularization (3DCTV-RPCA for short) by fully exploiting and encoding the prior expression underlying such joint low-rank and local smoothness matrices. Specifically, using a modification of Golfing scheme, we prove that under some mild assumptions, the proposed 3DCTV-RPCA model can decompose both components exactly, which should be the first theoretical guarantee among all such related methods combining low rankness and local smoothness. In addition, by utilizing Fast Fourier Transform (FFT), we propose an efficient ADMM algorithm with a solid convergence guarantee for solving the resulting optimization problem. Finally, a series of experiments on both simulations and real applications are carried out to demonstrate the general validity of the proposed 3DCTV-RPCA model.

本文研究了聚类基质值观测值的计算和统计限制。我们提出了一个低级别的混合模型（LRMM），该模型适用于经典的高斯混合模型（GMM）来处理基质值观测值，该观测值假设人口中心矩阵的低级别。通过集成Lloyd算法和低级近似值设计了一种计算有效的聚类方法。一旦定位良好，该算法将快速收敛并达到最小值最佳的指数型聚类错误率。同时，我们表明一种基于张量的光谱方法可提供良好的初始聚类。与GMM相当，最小值最佳聚类错误率是由分离强度（即种群中心矩阵之间的最小距离）决定的。通过利用低级度，提出的算法对分离强度的要求较弱。但是，与GMM不同，LRMM的统计难度和计算难度的特征是信号强度，即最小的人口中心矩阵的非零奇异值。提供了证据表明，即使信号强度不够强，即使分离强度很强，也没有多项式时间算法是一致的。在高斯以下噪声下进一步证明了我们低级劳埃德算法的性能。讨论了LRMM下估计和聚类之间的有趣差异。通过全面的仿真实验证实了低级劳埃德算法的优点。最后，我们的方法在现实世界数据集的文献中优于其他方法。

在这项工作中，我们研究了缺少数据（ST-MISS）和离群值（强大的ST-MISS）的子空间跟踪问题。我们提出了一种新颖的算法，并为这两个问题提供了保证。与过去在该主题上的工作不同，当前的工作并不强加分段恒定的子空间变更假设。此外，所提出的算法比我们以前的工作要简单得多（使用较少的参数）。其次，我们将方法及其分析扩展到当数据联合到数据时，以及在$ k $对等点点和中心之间的信息交换时，可以证明解决这些问题。我们通过广泛的数值实验来验证理论主张。

越来越多的数据科学和机器学习问题依赖于张量的计算，这些计算比矩阵更好地捕获数据的多路关系和相互作用。当利用这一关键优势时，一个关键的挑战是开发计算上有效的算法，以从张量数据中提取有用的信息，这些信息同时构成腐败和不良条件。本文解决了张量强大的主成分分析（RPCA），该分析旨在从塔克分解下的稀疏腐败污染的观察结果中回收低排名的张量。为了最大程度地减少计算和内存足迹，我们建议通过缩放梯度下降（scaledgd）直接恢复低维张量因子（从量身定制的光谱初始化开始），并与迭代变化的阈值操作相结合腐败。从理论上讲，我们确定所提出的算法以恒定的速率与真实的低级张量线性收敛，而恒定的速率与其条件编号无关，只要损坏的水平不大。从经验上讲，我们证明，通过合成实验和现实世界应用，提出的算法比最先进的矩阵和张量RPCA算法更好，更可扩展的性能。

在本文中，我们利用过度参数化来设计高维单索索引模型的无规矩算法，并为诱导的隐式正则化现象提供理论保证。具体而言，我们研究了链路功能是非线性且未知的矢量和矩阵单索引模型，信号参数是稀疏向量或低秩对称矩阵，并且响应变量可以是重尾的。为了更好地理解隐含正规化的角色而没有过度的技术性，我们假设协变量的分布是先验的。对于载体和矩阵设置，我们通过采用分数函数变换和专为重尾数据的强大截断步骤来构造过度参数化最小二乘损耗功能。我们建议通过将无规则化的梯度下降应用于损耗函数来估计真实参数。当初始化接近原点并且步骤中足够小时，我们证明了所获得的解决方案在载体和矩阵案件中实现了最小的收敛统计速率。此外，我们的实验结果支持我们的理论调查结果，并表明我们的方法在$ \ ell_2 $ -staticatisticated率和变量选择一致性方面具有明确的正则化的经验卓越。

异常值广泛发生在大数据应用中，可能严重影响统计估计和推理。在本文中，引入了抗强估计的框架，以强制任意给出的损耗函数。它与修剪方法密切连接，并且包括所有样本的显式外围参数，这反过来促进计算，理论和参数调整。为了解决非凸起和非体性的问题，我们开发可扩展的算法，以实现轻松和保证快速收敛。特别地，提出了一种新的技术来缓解对起始点的要求，使得在常规数据集上，可以大大减少数据重采样的数量。基于组合的统计和计算处理，我们能够超越M估计来执行非因思分析。所获得的抗性估算器虽然不一定全局甚至是局部最佳的，但在低维度和高维度中享有最小的速率最优性。回归，分类和神经网络的实验表明，在总异常值发生的情况下提出了拟议方法的优异性能。

我们的目标是在沿着张量模式的协变量信息存在中可获得稀疏和高度缺失的张量。我们的动机来自在线广告，在各种设备上的广告上的用户点击率（CTR）形成了大约96％缺失条目的CTR张量，并且在非缺失条目上有许多零，这使得独立的张量完井方法不满意。除了CTR张量旁边，额外的广告功能或用户特性通常可用。在本文中，我们提出了协助协助的稀疏张力完成（Costco），以合并复苏恢复稀疏张量的协变量信息。关键思想是共同提取来自张量和协变矩阵的潜伏组分以学习合成表示。从理论上讲，我们导出了恢复的张量组件的错误绑定，并明确地量化了由于协变量引起的显露概率条件和张量恢复精度的改进。最后，我们将Costco应用于由CTR张量和广告协变矩阵组成的广告数据集，从而通过基线的23％的准确性改进。重要的副产品是来自Costco的广告潜在组件显示有趣的广告集群，这对于更好的广告目标是有用的。

In this paper, we study the trace regression when a matrix of parameters B* is estimated via the convex relaxation of a rank-regularized regression or via regularized non-convex optimization. It is known that these estimators satisfy near-optimal error bounds under assumptions on the rank, coherence, and spikiness of B*. We start by introducing a general notion of spikiness for B* that provides a generic recipe to prove the restricted strong convexity of the sampling operator of the trace regression and obtain near-optimal and non-asymptotic error bounds for the estimation error. Similar to the existing literature, these results require the regularization parameter to be above a certain theory-inspired threshold that depends on observation noise that may be unknown in practice. Next, we extend the error bounds to cases where the regularization parameter is chosen via cross-validation. This result is significant in that existing theoretical results on cross-validated estimators (Kale et al., 2011; Kumar et al., 2013; Abou-Moustafa and Szepesvari, 2017) do not apply to our setting since the estimators we study are not known to satisfy their required notion of stability. Finally, using simulations on synthetic and real data, we show that the cross-validated estimator selects a near-optimal penalty parameter and outperforms the theory-inspired approach of selecting the parameter.

Sparse reduced rank regression is an essential statistical learning method. In the contemporary literature, estimation is typically formulated as a nonconvex optimization that often yields to a local optimum in numerical computation. Yet, their theoretical analysis is always centered on the global optimum, resulting in a discrepancy between the statistical guarantee and the numerical computation. In this research, we offer a new algorithm to address the problem and establish an almost optimal rate for the algorithmic solution. We also demonstrate that the algorithm achieves the estimation with a polynomial number of iterations. In addition, we present a generalized information criterion to simultaneously ensure the consistency of support set recovery and rank estimation. Under the proposed criterion, we show that our algorithm can achieve the oracle reduced rank estimation with a significant probability. The numerical studies and an application in the ovarian cancer genetic data demonstrate the effectiveness and scalability of our approach.

稀疏数据的恢复是机器学习和信号处理中许多应用的核心。虽然可以使用$ \ ell_1 $ -regularization在套索估算器中使用此类问题，但在基础上，通常需要专用算法来解决大型实例的相应高维非平滑优化。迭代地重新重复的最小二乘（IRLS）是一种广泛使用的算法，其出于其优异的数值性能。然而，虽然现有理论能够保证该算法的收敛到最小化器，但它不提供全局收敛速度。在本文中，我们证明了IRLS的变型以全局线性速率收敛到稀疏解决方案，即，如果测量结果满足通常的空空间属性假设，则立即发生线性误差。我们通过数值实验支持我们的理论，表明我们的线性速率捕获了正确的维度依赖性。我们预计我们的理论调查结果将导致IRLS算法的许多其他用例的新见解，例如在低级矩阵恢复中。

随机奇异值分解（RSVD）是用于计算大型数据矩阵截断的SVD的一类计算算法。给定A $ n \ times n $对称矩阵$ \ mathbf {m} $，原型RSVD算法输出通过计算$ \ mathbf {m mathbf {m} $的$ k $引导singular vectors的近似m}^{g} \ mathbf {g} $;这里$ g \ geq 1 $是一个整数，$ \ mathbf {g} \ in \ mathbb {r}^{n \ times k} $是一个随机的高斯素描矩阵。在本文中，我们研究了一般的“信号加上噪声”框架下的RSVD的统计特性，即，观察到的矩阵$ \ hat {\ mathbf {m}} $被认为是某种真实但未知的加法扰动信号矩阵$ \ mathbf {m} $。我们首先得出$ \ ell_2 $（频谱规范）和$ \ ell_ {2 \ to \ infty} $（最大行行列$ \ ell_2 $ norm）$ \ hat {\ hat {\ Mathbf {M}} $和信号矩阵$ \ Mathbf {M} $的真实单数向量。这些上限取决于信噪比（SNR）和功率迭代$ g $的数量。观察到一个相变现象，其中较小的SNR需要较大的$ g $值以保证$ \ ell_2 $和$ \ ell_ {2 \ to \ fo \ infty} $ distances的收敛。我们还表明，每当噪声矩阵满足一定的痕量生长条件时，这些相变发生的$ g $的阈值都会很清晰。最后，我们得出了近似奇异向量的行波和近似矩阵的进入波动的正常近似。我们通过将RSVD的几乎最佳性能保证在应用于三个统计推断问题的情况下，即社区检测，矩阵完成和主要的组件分析，并使用缺失的数据来说明我们的理论结果。

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。