我们使用张量奇异值分解（T-SVD）代数框架提出了一种新的快速流算法，用于抵抗缺失的低管级张量的缺失条目。我们展示T-SVD是三阶张量的研究型块术语分解的专业化，我们在该模型下呈现了一种算法，可以跟踪从不完全流2-D数据的可自由子模块。所提出的算法使用来自子空间的基层歧管的增量梯度下降的原理，以解决线性复杂度和时间样本的恒定存储器的张量完成问题。我们为我们的算法提供了局部预期的线性收敛结果。我们的经验结果在精确态度上具有竞争力，但在计算时间内比实际应用上的最先进的张量完成算法更快，以在有限的采样下恢复时间化疗和MRI数据。

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or N -way array. Decompositions of higher-order tensors (i.e., N -way arrays with N ≥ 3) have applications in psychometrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

低级张力完成已广泛用于计算机视觉和机器学习。本文开发了一种新型多模态核心张量分解（MCTF）方法，与张量低秩测量和该措施的更好的非凸弛豫形式（NC-MCTF）。所提出的模型编码由Tucker和T-SVD提供的一般张量的低秩见解，因此预计将在多个方向上同时模拟光谱低秩率，并准确地恢复基于几个观察到的条目的内在低秩结构的数据。此外，我们研究了MCTF和NC-MCTF正则化最小化问题，并设计了一个有效的块连续上限最小化（BSUM）算法来解决它们。该高效的求解器可以将MCTF扩展到各种任务，例如张量完成。一系列实验，包括高光谱图像（HSI），视频和MRI完成，确认了所提出的方法的卓越性能。

Tensor完成是矩阵完成的自然高阶泛化，其中目标是从其条目的稀疏观察中恢复低级张量。现有算法在没有可证明的担保的情况下是启发式，基于解决运行不切实际的大型半纤维程序，或者需要强大的假设，例如需要因素几乎正交。在本文中，我们介绍了交替最小化的新变型，其又通过了解如何对矩阵设置中的交替最小化的收敛性的进展措施来调整到张量设置的启发。我们展示了强大的可证明的保证，包括表明我们的算法即使当因素高度相关时，我们的算法也会在真正的张量线上会聚，并且可以在几乎线性的时间内实现。此外，我们的算法也非常实用，我们表明我们可以完成具有千维尺寸的三阶张量，从观察其条目的微小一部分。相比之下，有些令人惊讶的是，我们表明，如果没有我们的新扭曲，则表明交替最小化的标准版本可以在实践中以急剧速度收敛。

提供了一种强大而灵活的模型，可用于代表多属数据和多种方式相互作用，在科学和工程中的各个领域中发挥着现代数据科学中的不可或缺的作用。基本任务是忠实地以统计和计算的有效方式从高度不完整的测量中恢复张量。利用Tucker分解中的张量的低级别结构，本文开发了一个缩放的梯度下降（Scaledgd）算法，可以直接恢复具有定制频谱初始化的张量因子，并表明它以与条件号无关的线性速率收敛对于两个规范问题的地面真理张量 - 张量完成和张量回归 - 一旦样本大小高于$ n ^ {3/2} $忽略其他参数依赖项，$ n $是维度张量。这导致与现有技术相比的低秩张力估计的极其可扩展的方法，这些方法具有以下至少一个缺点：对记忆和计算方面的对不良，偏移成本高的极度敏感性，或差样本复杂性保证。据我们所知，Scaledgd是第一算法，它可以同时实现近最佳统计和计算复杂性，以便与Tucker分解进行低级张力完成。我们的算法突出了加速非耦合统计估计在加速非耦合统计估计中的适当预处理的功率，其中迭代改复的预处理器促进轨迹的所需的不变性属性相对于低级张量分解中的底层对称性。

We consider the nonlinear inverse problem of learning a transition operator $\mathbf{A}$ from partial observations at different times, in particular from sparse observations of entries of its powers $\mathbf{A},\mathbf{A}^2,\cdots,\mathbf{A}^{T}$. This Spatio-Temporal Transition Operator Recovery problem is motivated by the recent interest in learning time-varying graph signals that are driven by graph operators depending on the underlying graph topology. We address the nonlinearity of the problem by embedding it into a higher-dimensional space of suitable block-Hankel matrices, where it becomes a low-rank matrix completion problem, even if $\mathbf{A}$ is of full rank. For both a uniform and an adaptive random space-time sampling model, we quantify the recoverability of the transition operator via suitable measures of incoherence of these block-Hankel embedding matrices. For graph transition operators these measures of incoherence depend on the interplay between the dynamics and the graph topology. We develop a suitable non-convex iterative reweighted least squares (IRLS) algorithm, establish its quadratic local convergence, and show that, in optimal scenarios, no more than $\mathcal{O}(rn \log(nT))$ space-time samples are sufficient to ensure accurate recovery of a rank-$r$ operator $\mathbf{A}$ of size $n \times n$. This establishes that spatial samples can be substituted by a comparable number of space-time samples. We provide an efficient implementation of the proposed IRLS algorithm with space complexity of order $O(r n T)$ and per-iteration time complexity linear in $n$. Numerical experiments for transition operators based on several graph models confirm that the theoretical findings accurately track empirical phase transitions, and illustrate the applicability and scalability of the proposed algorithm.

在这项工作中，我们估计具有高概率的张量的随机选择元素的数量，保证了黎曼梯度下降的局部收敛性，以便张力列车完成。基于展开奇异值的谐波平均值，我们从正交投影的正交投影推导出一个新的界限，并引入张力列车的核心相干概念。我们还将结果扩展到张力列车完成与侧面信息，并获得相应的本地收敛保证。

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets.This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed-either explicitly or implicitly-to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis.The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an m × n matrix. (i) For a dense input matrix, randomized algorithms require O(mn log(k)) floating-point operations (flops) in contrast with O(mnk) for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multi-processor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to O(k) passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data.

kronecker回归是一个高度结构的最小二乘问题$ \ min _ {\ mathbf {x}}} \ lvert \ mathbf {k} \ mathbf {x} - \ mathbf {b} \ rvert_ \ rvert_ {2}^2 $矩阵$ \ mathbf {k} = \ mathbf {a}^{（1）} \ otimes \ cdots \ cdots \ otimes \ mathbf {a}^{（n）} $是因子矩阵的Kronecker产品。这种回归问题是在广泛使用的最小二乘（ALS）算法的每个步骤中都出现的，用于计算张量的塔克分解。我们介绍了第一个用于求解Kronecker回归的子次数算法，以避免在运行时间中避免指数项$ o（\ varepsilon^{ - n}）$的$（1+ \ varepsilon）$。我们的技术结合了利用分数抽样和迭代方法。通过扩展我们对一个块是Kronecker产品的块设计矩阵的方法，我们还实现了（1）Kronecker Ridge回归的亚次级时间算法，并且（2）更新ALS中Tucker分解的因子矩阵，这不是一个不是一个纯Kronecker回归问题，从而改善了Tucker ALS的所有步骤的运行时间。我们证明了该Kronecker回归算法在合成数据和现实世界图像张量上的速度和准确性。

We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by a constrained non-convex optimization problem involving the Grassmann manifold. We propose two sets of algorithms, one arising from Riemannian optimization and the other as an alternating minimization scheme, both of which include first- and second-order variants. Both sets of algorithms have theoretical guarantees. In particular, for the alternating minimization, we establish global convergence and worst-case complexity bounds. Additionally, using the Kurdyka-Lojasiewicz property, we show that the alternating minimization converges to a unique limit point. We provide extensive numerical results for the recovery of union of subspaces and clustering under entry sampling and dense Gaussian sampling. Our methods are competitive with existing approaches and, in particular, high accuracy is achieved in the recovery using Riemannian second-order methods.

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 consider the estimation of a low Tucker rank tensor from a number of noisy linear measurements. The general problem covers many specific examples arising from applications, including tensor regression, tensor completion, and tensor PCA/SVD. We consider an efficient Riemannian Gauss-Newton (RGN) method for low Tucker rank tensor estimation. Different from the generic (super)linear convergence guarantee of RGN in the literature, we prove the first local quadratic convergence guarantee of RGN for low-rank tensor estimation in the noisy setting under some regularity conditions and provide the corresponding estimation error upper bounds. A deterministic estimation error lower bound, which matches the upper bound, is provided that demonstrates the statistical optimality of RGN. The merit of RGN is illustrated through two machine learning applications: tensor regression and tensor SVD. Finally, we provide the simulation results to corroborate our theoretical findings.

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。

网络数据通常在各种应用程序中收集，代表感兴趣的功能之间直接测量或统计上推断的连接。在越来越多的域中，这些网络会随着时间的流逝而收集，例如不同日子或多个主题之间的社交媒体平台用户之间的交互，例如在大脑连接性的多主体研究中。在分析多个大型网络时，降低降低技术通常用于将网络嵌入更易于处理的低维空间中。为此，我们通过专门的张量分解来开发用于网络集合的主组件分析（PCA）的框架，我们将半对称性张量PCA或SS-TPCA术语。我们得出计算有效的算法来计算我们提出的SS-TPCA分解，并在标准的低级别信号加噪声模型下建立方法的统计效率。值得注意的是，我们表明SS-TPCA具有与经典矩阵PCA相同的估计精度，并且与网络中顶点数的平方根成正比，而不是预期的边缘数。我们的框架继承了古典PCA的许多优势，适用于广泛的无监督学习任务，包括识别主要网络，隔离有意义的更改点或外出观察，以及表征最不同边缘的“可变性网络”。最后，我们证明了我们的提案对模拟数据的有效性以及经验法律研究的示例。用于建立我们主要一致性结果的技术令人惊讶地简单明了，可能会在其他各种网络分析问题中找到使用。

在线张量分解（OTF）是一种从流媒体多模态数据学习低维解释特征的基本工具。虽然最近已经调查了OTF的各种算法和理论方面，但仍然甚至缺乏任何不连贯或稀疏假设的客观函数的静止点的一般会聚保证仍然缺乏仍然缺乏缺乏。案件。在这项工作中，我们介绍了一种新颖的算法，该算法从一般约束下的给定的张力值数据流中学习了CANDECOMP / PARAFAC（CP），包括诱导学习CP的解释性的非承诺约束。我们证明我们的算法几乎肯定会收敛到目标函数的一组静止点，在该假设下，数据张集的序列由底层马尔可夫链产生。我们的环境涵盖了古典的i.i.d.案例以及广泛的应用程序上下文，包括由独立或MCMC采样生成的数据流。我们的结果缩小了OTF和在线矩阵分解在全局融合分析中的OTF和在线矩阵分解之间的差距\ Commhl {对于CP - 分解}。实验，我们表明我们的算法比合成和实际数据的非负张量分解任务的标准算法更快地收敛得多。此外，我们通过图像，视频和时间序列数据展示了我们算法对来自图像，视频和时间序列数据的多样化示例的实用性，示出了通过以多种方式利用张量结构来利用张量结构，如何从相同的张量数据中学习定性不同的CP字典。 。

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.

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.

许多现代数据集，从神经影像和地统计数据等领域都以张量数据的随机样本的形式来说，这可以被理解为对光滑的多维随机功能的嘈杂观察。来自功能数据分析的大多数传统技术被维度的诅咒困扰，并且随着域的尺寸增加而迅速变得棘手。在本文中，我们提出了一种学习从多维功能数据样本的持续陈述的框架，这些功能是免受诅咒的几种表现形式的。这些表示由一组可分离的基函数构造，该函数被定义为最佳地适应数据。我们表明，通过仔细定义的数据的仔细定义的减少转换的张测仪分解可以有效地解决所得到的估计问题。使用基于差分运算符的惩罚，并入粗糙的正则化。也建立了相关的理论性质。在模拟研究中证明了我们对竞争方法的方法的优点。我们在神经影像动物中得出真正的数据应用。

In this paper, we study the problem of a batch of linearly correlated image alignment, where the observed images are deformed by some unknown domain transformations, and corrupted by additive Gaussian noise and sparse noise simultaneously. By stacking these images as the frontal slices of a third-order tensor, we propose to utilize the tensor factorization method via transformed tensor-tensor product to explore the low-rankness of the underlying tensor, which is factorized into the product of two smaller tensors via transformed tensor-tensor product under any unitary transformation. The main advantage of transformed tensor-tensor product is that its computational complexity is lower compared with the existing literature based on transformed tensor nuclear norm. Moreover, the tensor $\ell_p$ $(0<p<1)$ norm is employed to characterize the sparsity of sparse noise and the tensor Frobenius norm is adopted to model additive Gaussian noise. A generalized Gauss-Newton algorithm is designed to solve the resulting model by linearizing the domain transformations and a proximal Gauss-Seidel algorithm is developed to solve the corresponding subproblem. Furthermore, the convergence of the proximal Gauss-Seidel algorithm is established, whose convergence rate is also analyzed based on the Kurdyka-$\L$ojasiewicz property. Extensive numerical experiments on real-world image datasets are carried out to demonstrate the superior performance of the proposed method as compared to several state-of-the-art methods in both accuracy and computational time.

非凸松弛方法已被广泛用于张量恢复问题，并且与凸松弛方法相比，可以实现更好的恢复结果。在本文中，提出了一种新的非凸函数，最小值对数凹点（MLCP）函数，并分析了其某些固有属性，其中有趣的是发现对数函数是MLCP的上限功能。所提出的功能概括为张量病例，得出张量MLCP和加权张量$ l \ gamma $ -norm。考虑到将其直接应用于张量恢复问题时无法获得其明确解决方案。因此，给出了解决此类问题的相应等效定理，即张量等效的MLCP定理和等效加权张量$ l \ gamma $ -norm定理。此外，我们提出了两个基于EMLCP的经典张量恢复问题的模型，即低秩量张量完成（LRTC）和张量稳健的主组件分析（TRPCA）以及设计近端替代线性化最小化（棕榈）算法以单独解决它们。此外，基于Kurdyka - {\ l} ojasiwicz属性，证明所提出算法的溶液序列具有有限的长度并在全球范围内收敛到临界点。最后，广泛的实验表明，提出的算法取得了良好的结果，并证实MLCP函数确实比最小化问题中的对数函数更好，这与理论特性的分析一致。