本文从压缩感测的角度研究时间序列预测（TSF）的问题。首先，我们将TSF转换为具有任意采样（TCAS）的更加包容性问题，称为TCOR完成，该问题是从其条目的子集中以任意方式恢复张量。虽然已知在Tucker低级别的框架中，但理论上是不可能根据一些任意选择的条目识别目标张量，在这项工作中，我们将表明TCAS根据称为新概念的光明粘附卷积低秩，这是众所周知的傅立叶稀疏性的概括。然后我们介绍了一个凸面的卷积核规范最小化（CNNM），我们证明CNNM在求解TCA时，只要采样条件取决于目标张量的卷积等级 - 遵守。该理论为制作给定数量预测所需的最小采样大小提供了有意义的答案。单变量时间序列，图像和视频的实验显示令人鼓舞的结果。

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。

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.

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.

我们介绍和分析了多元奇异频谱分析（MSSA）的变体，这是一种流行的时间序列方法，用于启用和预测多元时间序列。在我们介绍的时空因素模型下，给定$ n $时间序列和$ t $观测时间序列，我们为插补和样本外预测均有效地扩展为$ 1 / \ sqrt，为预测和样本预测有效地缩放均值{\ min（n，t）t} $。这是一个改进：（i）$ 1 /\ sqrt {t} $ SSA的错误缩放，MSSA限制对单变量时间序列； （ii）$ 1/\ min（n，t）$对于不利用数据中时间结构的矩阵估计方法的错误缩放。我们引入的时空模型包括：谐波，多项式，可区分的周期函数和持有人连续函数的任何有限总和和产物。在时空因素模型下，我们的样本外预测结果可能对在线学习具有独立的兴趣。从经验上讲，在基准数据集上，我们的MSSA变体通过最先进的神经网络时间序列方法（例如，DEEPAR，LSTM）竞争性能，并且明显优于诸如矢量自动化（VAR）之类的经典方法。最后，我们提出了MSSA的扩展：（i）估计时间序列的时变差异的变体； （ii）一种张量变体，对于$ n $和$ t $的某些制度具有更好的样本复杂性。

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。

在本文中，我们提供了有关Hankel低级近似和完成工作的综述和书目，特别强调了如何将这种方法用于时间序列分析和预测。我们首先描述问题的可能表述，并就获得全球最佳解决方案的相关主题和挑战提供评论。提供了关键定理，并且纸张以一些说明性示例关闭。

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。

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.

This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discrete-time signal f ∈ C N and a randomly chosen set of frequencies Ω of mean size τ N . Is it possible to reconstruct f from the partial knowledge of its Fourier coefficients on the set Ω?A typical result of this paper is as follows: for each M > 0, suppose that f obeysthen with probability at least 1 − O(N −M ), f can be reconstructed exactly as the solution to the ℓ 1 minimization problem min g N −1 t=0 |g(t)|, s.t. ĝ(ω) = f (ω) for all ω ∈ Ω.In short, exact recovery may be obtained by solving a convex optimization problem.We give numerical values for α which depends on the desired probability of success; except for the logarithmic factor, the condition on the size of the support is sharp.The methodology extends to a variety of other setups and higher dimensions. For example, we show how one can reconstruct a piecewise constant (one or two-dimensional) object from incomplete frequency samples-provided that the number of jumps (discontinuities) obeys the condition above-by minimizing other convex functionals such as the total-variation of f .

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。

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。

本文提出了弗兰克 - 沃尔夫（FW）的新变种​​，称为$ k $ fw。标准FW遭受缓慢的收敛性：迭代通常是Zig-zag作为更新方向振荡约束集的极端点。新变种，$ k $ fw，通过在每次迭代中使用两个更强的子问题oracelles克服了这个问题。第一个是$ k $线性优化Oracle（$ k $ loo），计算$ k $最新的更新方向（而不是一个）。第二个是$ k $方向搜索（$ k $ ds），最大限度地减少由$ k $最新更新方向和之前迭代表示的约束组的目标。当问题解决方案承认稀疏表示时，奥克斯都易于计算，而且$ k $ FW会迅速收敛，以便平滑凸起目标和几个有趣的约束集：$ k $ fw实现有限$ \ frac {4l_f ^ 3d ^} { \ Gamma \ Delta ^ 2} $融合在多台和集团规范球上，以及光谱和核规范球上的线性收敛。数值实验验证了$ k $ fw的有效性，并展示了现有方法的数量级加速。

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。

我们提出了一个算法框架，用于近距离矩阵上的量子启发的经典算法，概括了Tang的突破性量子启发算法开始的一系列结果，用于推荐系统[STOC'19]。由量子线性代数算法和gily \'en，su，low和wiebe [stoc'19]的量子奇异值转换（SVT）框架[SVT）的动机[STOC'19]，我们开发了SVT的经典算法合适的量子启发的采样假设。我们的结果提供了令人信服的证据，表明在相应的QRAM数据结构输入模型中，量子SVT不会产生指数量子加速。由于量子SVT框架基本上概括了量子线性代数的所有已知技术，因此我们的结果与先前工作的采样引理相结合，足以概括所有有关取消量子机器学习算法的最新结果。特别是，我们的经典SVT框架恢复并经常改善推荐系统，主成分分析，监督聚类，支持向量机器，低秩回归和半决赛程序解决方案的取消结果。我们还为汉密尔顿低级模拟和判别分析提供了其他取消化结果。我们的改进来自识别量子启发的输入模型的关键功能，该模型是所有先前量子启发的结果的核心：$ \ ell^2 $ -Norm采样可以及时近似于其尺寸近似矩阵产品。我们将所有主要结果减少到这一事实，使我们的简洁，独立和直观。

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.

诸如压缩感测，图像恢复，矩阵/张恢复和非负矩阵分子等信号处理和机器学习中的许多近期问题可以作为约束优化。预计的梯度下降是一种解决如此约束优化问题的简单且有效的方法。本地收敛分析将我们对解决方案附近的渐近行为的理解，与全球收敛分析相比，收敛率的较小界限提供了较小的界限。然而，本地保证通常出现在机器学习和信号处理的特定问题领域。此稿件在约束最小二乘范围内，对投影梯度下降的局部收敛性分析提供了统一的框架。该建议的分析提供了枢转局部收敛性的见解，例如线性收敛的条件，收敛区域，精确的渐近收敛速率，以及达到一定程度的准确度所需的迭代次数的界限。为了证明所提出的方法的适用性，我们介绍了PGD的收敛分析的配方，并通过在四个基本问题上的配方的开始延迟应用来证明它，即线性约束最小二乘，稀疏恢复，最小二乘法使用单位规范约束和矩阵完成。

Tensor robust principal component analysis (TRPCA) is a promising way for low-rank tensor recovery, which minimizes the convex surrogate of tensor rank by shrinking each tensor singular values equally. However, for real-world visual data, large singular values represent more signifiant information than small singular values. In this paper, we propose a nonconvex TRPCA (N-TRPCA) model based on the tensor adjustable logarithmic norm. Unlike TRPCA, our N-TRPCA can adaptively shrink small singular values more and shrink large singular values less. In addition, TRPCA assumes that the whole data tensor is of low rank. This assumption is hardly satisfied in practice for natural visual data, restricting the capability of TRPCA to recover the edges and texture details from noisy images and videos. To this end, we integrate nonlocal self-similarity into N-TRPCA, and further develop a nonconvex and nonlocal TRPCA (NN-TRPCA) model. Specifically, similar nonlocal patches are grouped as a tensor and then each group tensor is recovered by our N-TRPCA. Since the patches in one group are highly correlated, all group tensors have strong low-rank property, leading to an improvement of recovery performance. Experimental results demonstrate that the proposed NN-TRPCA outperforms some existing TRPCA methods in visual data recovery. The demo code is available at https://github.com/qguo2010/NN-TRPCA.

我们考虑凸优化问题，这些问题被广泛用作低级基质恢复问题的凸松弛。特别是，在几个重要问题（例如相位检索和鲁棒PCA）中，在许多情况下的基本假设是最佳解决方案是排名一列。在本文中，我们考虑了目标上的简单自然的条件，以使这些放松的最佳解决方案确实是独特的，并且是一个排名。主要是，我们表明，在这种情况下，使用线路搜索的标准Frank-Wolfe方法（即，没有任何参数调整），该方法仅需要单个排名一级的SVD计算，可以找到$ \ epsilon $ - 仅在$ o（\ log {1/\ epsilon}）$迭代（而不是以前最著名的$ o（1/\ epsilon）$）中的近似解决方案，尽管目的不是强烈凸。我们考虑了基本方法的几种变体，具有改善的复杂性，以及由强大的PCA促进的扩展，最后是对非平滑问题的扩展。