In this paper we establish a mathematically rigorous connection between Causal inference (C-inf) and the low-rank recovery (LRR). Using Random Duality Theory (RDT) concepts developed in [46,48,50] and novel mathematical strategies related to free probability theory, we obtain the exact explicit typical (and achievable) worst case phase transitions (PT). These PT precisely separate scenarios where causal inference via LRR is possible from those where it is not. We supplement our mathematical analysis with numerical experiments that confirm the theoretical predictions of PT phenomena, and further show that the two closely match for fairly small sample sizes. We obtain simple closed form representations for the resulting PTs, which highlight direct relations between the low rankness of the target C-inf matrix and the time of the treatment. Hence, our results can be used to determine the range of C-inf's typical applicability.
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In this paper, we revisit and further explore a mathematically rigorous connection between Causal inference (C-inf) and the Low-rank recovery (LRR) established in [10]. Leveraging the Random duality - Free probability theory (RDT-FPT) connection, we obtain the exact explicit typical C-inf asymmetric phase transitions (PT). We uncover a doubling low-rankness phenomenon, which means that exactly two times larger low rankness is allowed in asymmetric scenarios compared to the symmetric worst case ones considered in [10]. Consequently, the final PT mathematical expressions are as elegant as those obtained in [10], and highlight direct relations between the targeted C-inf matrix low rankness and the time of treatment. Our results have strong implications for applications, where C-inf matrices are not necessarily symmetric.
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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.
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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.
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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.
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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.
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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。
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强大的机器学习模型的开发中的一个重要障碍是协变量的转变,当训练和测试集的输入分布时发生的分配换档形式在条件标签分布保持不变时发生。尽管现实世界应用的协变量转变普遍存在,但在现代机器学习背景下的理论理解仍然缺乏。在这项工作中,我们检查协变量的随机特征回归的精确高尺度渐近性,并在该设置中提出了限制测试误差,偏差和方差的精确表征。我们的结果激发了一种自然部分秩序,通过协变速转移,提供足够的条件来确定何时何时损害(甚至有助于)测试性能。我们发现,过度分辨率模型表现出增强的协会转变的鲁棒性,为这种有趣现象提供了第一个理论解释之一。此外,我们的分析揭示了分销和分发外概率性能之间的精确线性关系,为这一令人惊讶的近期实证观察提供了解释。
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近似消息传递(AMP)是解决高维统计问题的有效迭代范式。但是,当迭代次数超过$ o \ big(\ frac {\ log n} {\ log log \ log \ log n} \时big)$(带有$ n $问题维度)。为了解决这一不足,本文开发了一个非吸附框架,用于理解峰值矩阵估计中的AMP。基于AMP更新的新分解和可控的残差项,我们布置了一个分析配方,以表征在存在独立初始化的情况下AMP的有限样本行为,该过程被进一步概括以进行光谱初始化。作为提出的分析配方的两个具体后果:(i)求解$ \ mathbb {z} _2 $同步时,我们预测了频谱初始化AMP的行为,最高为$ o \ big(\ frac {n} {\ mathrm {\ mathrm { poly} \ log n} \ big)$迭代,表明该算法成功而无需随后的细化阶段(如最近由\ citet {celentano2021local}推测); (ii)我们表征了稀疏PCA中AMP的非反应性行为(在尖刺的Wigner模型中),以广泛的信噪比。
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套索是一种高维回归的方法,当时,当协变量$ p $的订单数量或大于观测值$ n $时,通常使用它。由于两个基本原因,经典的渐近态性理论不适用于该模型:$(1)$正规风险是非平滑的; $(2)$估算器$ \ wideHat {\ boldsymbol {\ theta}} $与true参数vector $ \ boldsymbol {\ theta}^*$无法忽略。结果,标准的扰动论点是渐近正态性的传统基础。另一方面,套索估计器可以精确地以$ n $和$ p $大,$ n/p $的订单为一。这种表征首先是在使用I.I.D的高斯设计的情况下获得的。协变量:在这里,我们将其推广到具有非偏差协方差结构的高斯相关设计。这是根据更简单的``固定设计''模型表示的。我们在两个模型中各种数量的分布之间的距离上建立了非反应界限,它们在合适的稀疏类别中均匀地固定在信号上$ \ boldsymbol {\ theta}^*$。作为应用程序,我们研究了借助拉索的分布,并表明需要校正程度对于计算有效的置信区间是必要的。
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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。
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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 .
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在深度学习中的优化分析是连续的,专注于(变体)梯度流动,或离散,直接处理(变体)梯度下降。梯度流程可符合理论分析,但是风格化并忽略计算效率。它代表梯度下降的程度是深度学习理论的一个开放问题。目前的论文研究了这个问题。将梯度下降视为梯度流量初始值问题的近似数值问题,发现近似程度取决于梯度流动轨迹周围的曲率。然后,我们表明,在具有均匀激活的深度神经网络中,梯度流动轨迹享有有利的曲率,表明它们通过梯度下降近似地近似。该发现允许我们将深度线性神经网络的梯度流分析转换为保证梯度下降,其几乎肯定会在随机初始化下有效地收敛到全局最小值。实验表明,在简单的深度神经网络中,具有传统步长的梯度下降确实接近梯度流。我们假设梯度流动理论将解开深入学习背后的奥秘。
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随机块模型(SBM)是一个随机图模型,其连接不同的顶点组不同。它被广泛用作研究聚类和社区检测的规范模型,并提供了肥沃的基础来研究组合统计和更普遍的数据科学中出现的信息理论和计算权衡。该专着调查了最近在SBM中建立社区检测的基本限制的最新发展,无论是在信息理论和计算方案方面,以及各种恢复要求,例如精确,部分和弱恢复。讨论的主要结果是在Chernoff-Hellinger阈值中进行精确恢复的相转换,Kesten-Stigum阈值弱恢复的相变,最佳的SNR - 单位信息折衷的部分恢复以及信息理论和信息理论之间的差距计算阈值。该专着给出了在寻求限制时开发的主要算法的原则推导,特别是通过绘制绘制,半定义编程,(线性化)信念传播,经典/非背带频谱和图形供电。还讨论了其他块模型的扩展,例如几何模型和一些开放问题。
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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。
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The Forster transform is a method of regularizing a dataset by placing it in {\em radial isotropic position} while maintaining some of its essential properties. Forster transforms have played a key role in a diverse range of settings spanning computer science and functional analysis. Prior work had given {\em weakly} polynomial time algorithms for computing Forster transforms, when they exist. Our main result is the first {\em strongly polynomial time} algorithm to compute an approximate Forster transform of a given dataset or certify that no such transformation exists. By leveraging our strongly polynomial Forster algorithm, we obtain the first strongly polynomial time algorithm for {\em distribution-free} PAC learning of halfspaces. This learning result is surprising because {\em proper} PAC learning of halfspaces is {\em equivalent} to linear programming. Our learning approach extends to give a strongly polynomial halfspace learner in the presence of random classification noise and, more generally, Massart noise.
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The stochastic block model (SBM) is a random graph model with planted clusters. It is widely employed as a canonical model to study clustering and community detection, and provides generally a fertile ground to study the statistical and computational tradeoffs that arise in network and data sciences.This note surveys the recent developments that establish the fundamental limits for community detection in the SBM, both with respect to information-theoretic and computational thresholds, and for various recovery requirements such as exact, partial and weak recovery (a.k.a., detection). The main results discussed are the phase transitions for exact recovery at the Chernoff-Hellinger threshold, the phase transition for weak recovery at the Kesten-Stigum threshold, the optimal distortion-SNR tradeoff for partial recovery, the learning of the SBM parameters and the gap between information-theoretic and computational thresholds.The note also covers some of the algorithms developed in the quest of achieving the limits, in particular two-round algorithms via graph-splitting, semi-definite programming, linearized belief propagation, classical and nonbacktracking spectral methods. A few open problems are also discussed.
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张量模型在许多领域中起着越来越重要的作用,特别是在机器学习中。在几种应用中,例如社区检测,主题建模和高斯混合物学习,必须估算噪声张量的低级别信号。因此,了解该信号的估计器的基本限制不可避免地要求研究随机张量。最近,在大维限制中,该主题取得了实质性进展。然而,其中一些最重要的结果(尤其是对突然的相变(相对于信噪比)的精确表征),该表现控制着对称等级的最大可能性(ML)估计器的性能 - 具有高斯噪声的模型 - 基于平均场自旋玻璃理论得出,非专家不容易访问。在这项工作中,我们依靠标准但强大的工具开发出一种截然不同,更基本的方法,这是由随机矩阵理论的多年进步带来的。关键思想是研究由给定随机张量的收缩引起的随机矩阵的光谱。我们展示了如何访问随机张量本身的光谱属性。对于上述排名衡量模型,我们的技术产生了迄今未知的固定点方程,其解决方案与第三阶情况下的相变阈值高于相变阈值的ML估计器的渐近性能。数值验证提供了证据,表明订单4和5相同,导致我们猜想,对于任何顺序,我们的定点方程等于已知的ML估计性能的表征,这些表现通过依靠旋转玻璃而获得。此外,我们的方法阐明了ML问题景观的某些特性,可以扩展到其他模型,例如不对称和非高斯。
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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.
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Outier-bubust估计是一个基本问题,已由统计学家和从业人员进行了广泛的研究。在过去的几年中,整个研究领域的融合都倾向于“算法稳定统计”,该统计数据的重点是开发可拖动的异常体 - 固定技术来解决高维估计问题。尽管存在这种融合,但跨领域的研究工作主要彼此断开。本文桥接了有关可认证的异常抗衡器估计的最新工作,该估计是机器人技术和计算机视觉中的几何感知,并在健壮的统计数据中并行工作。特别是,我们适应并扩展了最新结果对可靠的线性回归(适用于<< 50%异常值的低外壳案例)和列表可解码的回归(适用于>> 50%异常值的高淘汰案例)在机器人和视觉中通常发现的设置,其中(i)变量(例如旋转,姿势)属于非convex域,(ii)测量值是矢量值,并且(iii)未知的异常值是先验的。这里的重点是绩效保证:我们没有提出新算法,而是为投入测量提供条件,在该输入测量值下,保证现代估计算法可以在存在异常值的情况下恢复接近地面真相的估计值。这些条件是我们所谓的“估计合同”。除了现有结果的拟议扩展外,我们认为本文的主要贡献是(i)通过指出共同点和差异来统一平行的研究行,(ii)在介绍先进材料(例如,证明总和证明)中的统一行为。对从业者的可访问和独立的演讲,(iii)指出一些即时的机会和开放问题,以发出异常的几何感知。
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