素描和项目是一个框架，它统一了许多已知的迭代方法来求解线性系统及其变体，并进一步扩展了非线性优化问题。它包括流行的方法，例如随机kaczmarz，坐标下降，凸优化的牛顿方法的变体等。在本文中，我们通过新的紧密频谱边界为预期的草图投影矩阵获得了素描和项目的收敛速率的敏锐保证。我们的估计值揭示了素描和项目的收敛率与另一个众所周知但看似无关的算法家族的近似误差之间的联系，这些算法使用草图加速了流行的矩阵因子化，例如QR和SVD。这种连接使我们更接近准确量化草图和项目求解器的性能如何取决于其草图大小。我们的分析不仅涵盖了高斯和次高斯的素描矩阵，还涵盖了一个有效的稀疏素描方法，称为较少的嵌入方法。我们的实验备份了理论，并证明即使极稀疏的草图在实践中也显示出相同的收敛属性。

通常希望通过将其投影到低维子空间来降低大数据集的维度。矩阵草图已成为一种非常有效地执行这种维度降低的强大技术。尽管有关于草图最差的表现的广泛文献，但现有的保证通常与实践中观察到的差异截然不同。我们利用随机矩阵的光谱分析中的最新发展来开发新技术，这些技术为通过素描获得的随机投影矩阵的期望值提供了准确的表达。这些表达式可以用来表征各种常见的机器学习任务中尺寸降低的性能，从低级别近似到迭代随机优化。我们的结果适用于几种流行的草图方法，包括高斯和拉德马赫草图，它们可以根据数据的光谱特性对这些方法进行精确的分析。经验结果表明，我们得出的表达式反映了这些草图方法的实际性能，直到低阶效应甚至不变因素。

算法高斯化是一种现象，当使用随机素描或采样方法生成较小的大数据集的较小表示时，可能会出现的现象：对于某些任务，已经观察到这些草图表示表现出许多可靠的性能特征，这些性能是在数据样本中出现的，这些性能来自次高斯随机设计，是一个强大的数据分布统计模型。但是，这种现象仅研究了特定的任务和指标，或依靠计算昂贵的方法。我们通过为平均值提供用于高斯数据分布的算法框架来解决这一问题，并证明可以有效构建几乎无法区分的数据草图（与亚高斯随机设计有关的总变化距离）。特别是，依靠最近引入的素描技术称为杠杆得分稀疏（少）嵌入，我们表明一个人可以构造$ n \ times d $矩阵$ a $的$ n \ times d $ sketch of $ n \ times d $ n \ ll n $，几乎与次高斯设计几乎没有区别$ a $中的非零条目的数量。结果，可以直接适用于我们的草图框架，可直接适用于我们的草图框架。我们通过对草图最小二乘正方形的新近似保证进行了说明。

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.

在本文中，我们提出{\ it \下划线{r} ecursive} {\ it \ usef \ undesline {i} mortance} {\ it \ it \ usew supsline {s} ketching} algorithM squares {\ it \下划线{o} ptimization}（risro）。 Risro的关键步骤是递归重要性草图，这是一个基于确定性设计的递归投影的新素描框架，它与文献中的随机素描\ Citep {Mahoney2011 randomized，Woodruff2014sketching}有很大不同。在这个新的素描框架下，可以重新解释文献中的几种现有算法，而Risro比它们具有明显的优势。 Risro易于实现，并在计算上有效，其中每次迭代中的核心过程是解决降低尺寸最小二乘问题的问题。我们在某些轻度条件下建立了Risro的局部二次线性和二次收敛速率。我们还发现了Risro与Riemannian Gauss-Newton算法在固定等级矩阵上的联系。在机器学习和统计数据中的两种应用中，RISRO的有效性得到了证明：低级别矩阵痕量回归和相位检索。仿真研究证明了Risro的出色数值性能。

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。

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。

我们研究基于Krylov子空间的迭代方法，用于在任何Schatten $ p $ Norm中的低级别近似值。在这里，通过矩阵向量产品访问矩阵$ a $ $如此$ \ | a（i -zz^\ top）\ | _ {s_p} \ leq（1+ \ epsilon）\ min_ {u^\ top u = i_k} } $，其中$ \ | m \ | _ {s_p} $表示$ m $的单数值的$ \ ell_p $ norm。对于$ p = 2 $（frobenius norm）和$ p = \ infty $（频谱规范）的特殊情况，musco and Musco（Neurips 2015）获得了基于Krylov方法的算法，该方法使用$ \ tilde {o}（k）（k /\ sqrt {\ epsilon}）$ matrix-vector产品，改进na \“ ive $ \ tilde {o}（k/\ epsilon）$依赖性，可以通过功率方法获得，其中$ \ tilde {o} $抑制均可抑制poly $（\ log（dk/\ epsilon））$。我们的主要结果是仅使用$ \ tilde {o}（kp^{1/6}/\ epsilon^{1/3} {1/3}）$ matrix $ matrix的算法 - 矢量产品，并为所有$ p \ geq 1 $。为$ p = 2 $工作，我们的限制改进了先前的$ \ tilde {o}（k/\ epsilon^{1/2}）$绑定到$ \ tilde {o}（k/\ epsilon^{1/3}）$。由于schatten- $ p $和schatten-$ \ infty $ norms在$（1+ \ epsilon）$ pers $ p时相同\ geq（\ log d）/\ epsilon $，我们的界限恢复了Musco和Musco的结果，以$ p = \ infty $。此外，我们证明了矩阵矢量查询$ \ omega的下限（1/\ epsilon^ {1/3}）$对于任何固定常数$ p \ geq 1 $，表明令人惊讶的$ \ tilde {\ theta}（1/\ epsilon^{ 1/3}）$是常数〜$ k $的最佳复杂性。为了获得我们的结果，我们介绍了几种新技术，包括同时对多个Krylov子空间进行优化，以及针对分区操作员的不平等现象。我们在[1,2] $中以$ p \的限制使用了Araki-lieb-thirring Trace不平等，而对于$ p> 2 $，我们呼吁对安装分区操作员的规范压缩不平等。

求解线性系统的迭代方法的收敛速率$ \ mathbf {a} x = b $通常取决于矩阵$ \ mathbf {a} $的条件号。预处理是通过以计算廉价的方式减少该条件号来加速这些方法的常用方式。在本文中，我们通过左或右对角线重构重新审视如何最好地提高$ \ mathbf {a}条件号的数十年。我们在几个方向上取得了这个问题。首先，我们为缩放$ \ mathbf {a} $的经典启发式提供了新的界限（a.k.a.jacobi预处理）。我们证明了这种方法将$ \ MATHBF {a} $的条件号减少到最佳可能缩放的二次因素中。其次，我们为结构化混合包装和覆盖了Semidefinite程序（MPC SDP）提供了一个求解器，它计算$ \ mathbf {a} $ in $ \ widetilde {o}（\ text {nnz}（\ mathbf {a}）\ cdot \ text {poly}（\ kappa ^ \ star））$ time;这与在缩放到$ \ widetilde {o}（\ text {poly}（\ kappa ^ \ star））$ factors之后求解线性系统的成本匹配。第三，我们证明了足够一般的宽度无关的MPC SDP求解器将暗示我们考虑的缩放问题的近乎最佳的运行时间，以及与平均调理措施有关的自然变体。最后，我们突出了我们的预处理技术与半随机噪声模型的连接，以及在几种统计回归模型中降低风险的应用。

我们提供了新的基于梯度的方法，以便有效解决广泛的病态化优化问题。我们考虑最小化函数$ f：\ mathbb {r} ^ d \ lightarrow \ mathbb {r} $的问题，它是隐含的可分解的，作为$ m $未知的非交互方式的总和，强烈的凸起功能并提供方法这解决了这个问题，这些问题是缩放（最快的对数因子）作为组件的条件数量的平方根的乘积。这种复杂性绑定（我们证明几乎是最佳的）可以几乎指出的是加速梯度方法的几乎是指数的，这将作为$ F $的条件数量的平方根。此外，我们提供了求解该多尺度优化问题的随机异标变体的有效方法。而不是学习$ F $的分解（这将是过度昂贵的），而是我们的方法应用一个清洁递归“大步小步”交错标准方法。由此产生的算法使用$ \ tilde {\ mathcal {o}}（d m）$空间，在数字上稳定，并打开门以更细粒度的了解凸优化超出条件号的复杂性。

我们研究了用于线性回归的主动采样算法，该算法仅旨在查询目标向量$ b \ in \ mathbb {r} ^ n $的少量条目，并将近最低限度输出到$ \ min_ {x \ In \ mathbb {r} ^ d} \ | ax-b \ | $，其中$ a \ in \ mathbb {r} ^ {n \ times d} $是一个设计矩阵和$ \ | \ cdot \ | $是一些损失函数。对于$ \ ell_p $ norm回归的任何$ 0 <p <\ idty $，我们提供了一种基于Lewis权重采样的算法，其使用只需$ \ tilde {o}输出$（1+ \ epsilon）$近似解决方案（d ^ {\ max（1，{p / 2}）} / \ mathrm {poly}（\ epsilon））$查询到$ b $。我们表明，这一依赖于$ D $是最佳的，直到对数因素。我们的结果解决了陈和Derezi的最近开放问题，陈和Derezi \'{n} Ski，他们为$ \ ell_1 $ norm提供了附近的最佳界限，以及$ p \中的$ \ ell_p $回归的次优界限（1,2） $。我们还提供了$ O的第一个总灵敏度上限（D ^ {\ max \ {1，p / 2 \} \ log ^ 2 n）$以满足最多的$ p $多项式增长。这改善了Tukan，Maalouf和Feldman的最新结果。通过将此与我们的技术组合起来的$ \ ell_p $回归结果，我们获得了一个使$ \ tilde o的活动回归算法（d ^ {1+ \ max \ {1，p / 2 \}} / \ mathrm {poly}。 （\ epsilon））$疑问，回答陈和德里兹的另一个打开问题{n}滑雪。对于Huber损失的重要特殊情况，我们进一步改善了我们对$ \ tilde o的主动样本复杂性的绑定（d ^ {（1+ \ sqrt2）/ 2} / \ epsilon ^ c）$和非活跃$ \ tilde o的样本复杂性（d ^ {4-2 \ sqrt 2} / \ epsilon ^ c）$，由于克拉克森和伍德拉夫而改善了Huber回归的以前的D ^ 4 $。我们的敏感性界限具有进一步的影响，使用灵敏度采样改善了各种先前的结果，包括orlicz规范子空间嵌入和鲁棒子空间近似。最后，我们的主动采样结果为每种$ \ ell_p $ norm提供的第一个Sublinear时间算法。

We consider minimizing a smooth and strongly convex objective function using a stochastic Newton method. At each iteration, the algorithm is given an oracle access to a stochastic estimate of the Hessian matrix. The oracle model includes popular algorithms such as Subsampled Newton and Newton Sketch. Despite using second-order information, these existing methods do not exhibit superlinear convergence, unless the stochastic noise is gradually reduced to zero during the iteration, which would lead to a computational blow-up in the per-iteration cost. We propose to address this limitation with Hessian averaging: instead of using the most recent Hessian estimate, our algorithm maintains an average of all the past estimates. This reduces the stochastic noise while avoiding the computational blow-up. We show that this scheme exhibits local $Q$-superlinear convergence with a non-asymptotic rate of $(\Upsilon\sqrt{\log (t)/t}\,)^{t}$, where $\Upsilon$ is proportional to the level of stochastic noise in the Hessian oracle. A potential drawback of this (uniform averaging) approach is that the averaged estimates contain Hessian information from the global phase of the method, i.e., before the iterates converge to a local neighborhood. This leads to a distortion that may substantially delay the superlinear convergence until long after the local neighborhood is reached. To address this drawback, we study a number of weighted averaging schemes that assign larger weights to recent Hessians, so that the superlinear convergence arises sooner, albeit with a slightly slower rate. Remarkably, we show that there exists a universal weighted averaging scheme that transitions to local convergence at an optimal stage, and still exhibits a superlinear convergence rate nearly (up to a logarithmic factor) matching that of uniform Hessian averaging.

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.

在线性回归中，我们希望根据少量样本估算超过$ d $维的输入点和实价响应的最佳最小二乘预测。根据标准随机设计分析，其中绘制样品i.i.d。从输入分布中，该样品的最小二乘解决方案可以看作是最佳的自然估计器。不幸的是，该估计器几乎总是产生来自输入点的随机性的不良偏置，这在模型平均中是一个重要的瓶颈。在本文中，我们表明可以绘制非i.i.d。输入点的样本，无论响应模型如何，最小二乘解决方案都是最佳的无偏估计器。此外，可以通过增强先前绘制的I.I.D。可以有效地生产该样本。带有额外的$ d $点的样品，根据点由点跨越的平方量重新缩放的输入分布构建的一定确定点过程，共同绘制。在此激励的基础上，我们开发了一个理论框架来研究体积响应的采样，并在此过程中证明了许多新的矩阵期望身份。我们使用它们来表明，对于任何输入分布和$ \ epsilon> 0 $，有一个随机设计由$ o（d \ log d+ d+ d+ d/\ epsilon）$点，从中可以从中构造出无偏见的估计器，其预期的是正方形损耗在整个发行版中，$ 1+\ epsilon $ times最佳损失。我们提供有效的算法来在许多实际设置中生成这种无偏估计量，并在实验中支持我们的主张。

我们研究了张量张量的回归，其中的目标是将张量的响应与张量协变量与塔克等级参数张量/矩阵连接起来，而没有其内在等级的先验知识。我们提出了Riemannian梯度下降（RGD）和Riemannian Gauss-Newton（RGN）方法，并通过研究等级过度参数化的影响来应对未知等级的挑战。我们通过表明RGD和RGN分别线性地和四边形地收敛到两个等级的统计最佳估计值，从而为一般的张量调节回归提供了第一个收敛保证。我们的理论揭示了一种有趣的现象：Riemannian优化方法自然地适应了过度参数化，而无需修改其实施。我们还为低度多项式框架下的标量调整回归中的统计计算差距提供了第一个严格的证据。我们的理论证明了``统计计算差距的祝福''现象：在张张量的张量回归中，对于三个或更高的张紧器，在张张量的张量回归中，计算所需的样本量与中等级别相匹配的计算量相匹配。在考虑计算可行的估计器时，虽然矩阵设置没有此类好处。这表明中等等级的过度参数化本质上是``在张量调整的样本量三分或更高的样本大小上，三分或更高的样本量。最后，我们进行仿真研究以显示我们提出的方法的优势并证实我们的理论发现。

In a mixed generalized linear model, the objective is to learn multiple signals from unlabeled observations: each sample comes from exactly one signal, but it is not known which one. We consider the prototypical problem of estimating two statistically independent signals in a mixed generalized linear model with Gaussian covariates. Spectral methods are a popular class of estimators which output the top two eigenvectors of a suitable data-dependent matrix. However, despite the wide applicability, their design is still obtained via heuristic considerations, and the number of samples $n$ needed to guarantee recovery is super-linear in the signal dimension $d$. In this paper, we develop exact asymptotics on spectral methods in the challenging proportional regime in which $n, d$ grow large and their ratio converges to a finite constant. By doing so, we are able to optimize the design of the spectral method, and combine it with a simple linear estimator, in order to minimize the estimation error. Our characterization exploits a mix of tools from random matrices, free probability and the theory of approximate message passing algorithms. Numerical simulations for mixed linear regression and phase retrieval display the advantage enabled by our analysis over existing designs of spectral methods.

Autoencoders are a popular model in many branches of machine learning and lossy data compression. However, their fundamental limits, the performance of gradient methods and the features learnt during optimization remain poorly understood, even in the two-layer setting. In fact, earlier work has considered either linear autoencoders or specific training regimes (leading to vanishing or diverging compression rates). Our paper addresses this gap by focusing on non-linear two-layer autoencoders trained in the challenging proportional regime in which the input dimension scales linearly with the size of the representation. Our results characterize the minimizers of the population risk, and show that such minimizers are achieved by gradient methods; their structure is also unveiled, thus leading to a concise description of the features obtained via training. For the special case of a sign activation function, our analysis establishes the fundamental limits for the lossy compression of Gaussian sources via (shallow) autoencoders. Finally, while the results are proved for Gaussian data, numerical simulations on standard datasets display the universality of the theoretical predictions.

In many modern applications of deep learning the neural network has many more parameters than the data points used for its training. Motivated by those practices, a large body of recent theoretical research has been devoted to studying overparameterized models. One of the central phenomena in this regime is the ability of the model to interpolate noisy data, but still have test error lower than the amount of noise in that data. arXiv:1906.11300 characterized for which covariance structure of the data such a phenomenon can happen in linear regression if one considers the interpolating solution with minimum $\ell_2$-norm and the data has independent components: they gave a sharp bound on the variance term and showed that it can be small if and only if the data covariance has high effective rank in a subspace of small co-dimension. We strengthen and complete their results by eliminating the independence assumption and providing sharp bounds for the bias term. Thus, our results apply in a much more general setting than those of arXiv:1906.11300, e.g., kernel regression, and not only characterize how the noise is damped but also which part of the true signal is learned. Moreover, we extend the result to the setting of ridge regression, which allows us to explain another interesting phenomenon: we give general sufficient conditions under which the optimal regularization is negative.

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.

We study inductive matrix completion (matrix completion with side information) under an i.i.d. subgaussian noise assumption at a low noise regime, with uniform sampling of the entries. We obtain for the first time generalization bounds with the following three properties: (1) they scale like the standard deviation of the noise and in particular approach zero in the exact recovery case; (2) even in the presence of noise, they converge to zero when the sample size approaches infinity; and (3) for a fixed dimension of the side information, they only have a logarithmic dependence on the size of the matrix. Differently from many works in approximate recovery, we present results both for bounded Lipschitz losses and for the absolute loss, with the latter relying on Talagrand-type inequalities. The proofs create a bridge between two approaches to the theoretical analysis of matrix completion, since they consist in a combination of techniques from both the exact recovery literature and the approximate recovery literature.