Information and Inference-A Journal of the Ima最新文献

英文中文

Nearly minimax-optimal rates for noisy sparse phase retrieval via early-stopped mirror descent 基于早期停止镜像下降的噪声稀疏相位检索的近似极小极大最优速率

IF 1.6 4区数学 Q2 MATHEMATICS, APPLIED

Information and Inference-A Journal of the Ima

Pub Date : 2022-08-01 DOI: 10.1093/imaiai/iaac024

Fan Wu;Patrick Rebeschini

This paper studies early-stopped mirror descent applied to noisy sparse phase retrieval, which is the problem of recovering a $k$-sparse signal $textbf{x}^star in{mathbb{R}}^n$ from a set of quadratic Gaussian measurements corrupted by sub-exponential noise. We consider the (non-convex) unregularized empirical risk minimization problem and show that early-stopped mirror descent, when equipped with the hypentropy mirror map and proper initialization, achieves a nearly minimax-optimal rate of convergence, provided the sample size is at least of order $k^2$ (modulo logarithmic term) and the minimum (in modulus) non-zero entry of the signal is on the order of $|textbf{x}^star |_2/sqrt{k}$. Our theory leads to a simple algorithm that does not rely on explicit regularization or thresholding steps to promote sparsity. More generally, our results establish a connection between mirror descent and sparsity in the non-convex problem of noisy sparse phase retrieval, adding to the literature on early stopping that has mostly focused on non-sparse, Euclidean and convex settings via gradient descent. Our proof combines a potential-based analysis of mirror descent with a quantitative control on a variational coherence property that we establish along the path of mirror descent, up to a prescribed stopping time.

本文研究了应用于噪声稀疏相位检索的早期停止镜像下降，这是从一组被亚指数噪声破坏的二次高斯测量中恢复$k$-稀疏信号$textbf｛x｝^starin｛mathbb｛R｝｝^n$的问题。我们考虑了（非凸）非规则经验风险最小化问题，并表明当配备有高熵镜像图和适当的初始化时，早期停止镜像下降实现了几乎最小最大的最优收敛速度，假设样本大小至少为$k^2$阶（模对数项），并且信号的最小（以模为单位）非零项为$|textbf｛x｝^star|_2/sqrt｛k｝$阶。我们的理论导致了一种简单的算法，该算法不依赖于显式正则化或阈值步骤来提高稀疏性。更普遍地说，我们的结果在噪声稀疏相位检索的非凸问题中建立了镜像下降和稀疏性之间的联系，增加了早期停止的文献，该文献主要关注通过梯度下降的非稀疏、欧几里得和凸设置。我们的证明将基于势的镜像下降分析与我们沿着镜像下降路径建立的变分相干性质的定量控制相结合，直到规定的停止时间。

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引用次数: 0

Perturbation bounds for (nearly) orthogonally decomposable tensors with statistical applications （近似）正交可分解张量的扰动界及其统计应用

IF 1.6 4区数学 Q2 MATHEMATICS, APPLIED

Information and Inference-A Journal of the Ima

Pub Date : 2022-08-01 DOI: 10.1093/imaiai/iaac033

Arnab Auddy;Ming Yuan

We develop deterministic perturbation bounds for singular values and vectors of orthogonally decomposable tensors, in a spirit similar to classical results for matrices such as those due to Weyl, Davis, Kahan and Wedin. Our bounds demonstrate intriguing differences between matrices and higher order tensors. Most notably, they indicate that for higher order tensors perturbation affects each essential singular value/vector in isolation, and its effect on an essential singular vector does not depend on the multiplicity of its corresponding singular value or its distance from other singular values. Our results can be readily applied and provide a unified treatment to many different problems involving higher order orthogonally decomposable tensors. In particular, we illustrate the implications of our bounds through connected yet seemingly different high-dimensional data analysis tasks: the unsupervised learning scenario of tensor SVD and the supervised task of tensor regression, leading to new insights in both of these settings.

我们为正交可分解张量的奇异值和向量开发了确定性扰动界，其精神类似于矩阵的经典结果，如Weyl、Davis、Kahan和Wedin的结果。我们的边界证明了矩阵和高阶张量之间有趣的差异。最值得注意的是，它们表明，对于高阶张量，扰动孤立地影响每个本质奇异值/向量，并且其对本质奇异向量的影响不取决于其相应奇异值的多重性或其与其他奇异值的距离。我们的结果可以很容易地应用，并为涉及高阶正交可分解张量的许多不同问题提供了统一的处理方法。特别是，我们通过连接但看似不同的高维数据分析任务来说明我们的边界的含义：张量SVD的无监督学习场景和张量回归的有监督任务，在这两种情况下都有了新的见解。

引用次数: 0

Breaking the sample complexity barrier to regret-optimal model-free reinforcement learning 打破样本复杂性障碍后悔最优无模型强化学习

IF 1.6 4区数学 Q2 MATHEMATICS, APPLIED

Information and Inference-A Journal of the Ima

Pub Date : 2022-08-01 DOI: 10.1093/imaiai/iaac034

Gen Li;Laixi Shi;Yuxin Chen;Yuejie Chi

Achieving sample efficiency in online episodic reinforcement learning (RL) requires optimally balancing exploration and exploitation. When it comes to a finite-horizon episodic Markov decision process with $S$ states, $A$ actions and horizon length $H$, substantial progress has been achieved toward characterizing the minimax-optimal regret, which scales on the order of $sqrt{H^2SAT}$ (modulo log factors) with $T$ the total number of samples. While several competing solution paradigms have been proposed to minimize regret, they are either memory-inefficient, or fall short of optimality unless the sample size exceeds an enormous threshold (e.g. $S^6A^4 ,mathrm{poly}(H)$ for existing model-free methods).To overcome such a large sample size barrier to efficient RL, we design a novel model-free algorithm, with space complexity $O(SAH)$, that achieves near-optimal regret as soon as the sample size exceeds the order of $SA,mathrm{poly}(H)$. In terms of this sample size requirement (also referred to the initial burn-in cost), our method improves—by at least a factor of $S^5A^3$—upon any prior memory-efficient algorithm that is asymptotically regret-optimal. Leveraging the recently introduced variance reduction strategy (also called reference-advantage decomposition), the proposed algorithm employs an early-settled reference update rule, with the aid of two Q-learning sequences with upper and lower confidence bounds. The design principle of our early-settled variance reduction method might be of independent interest to other RL settings that involve intricate exploration–exploitation trade-offs.

在线情景强化学习（RL）中实现样本效率需要优化地平衡探索和开发。当涉及到具有$S$状态、$a$动作和视界长度$H$的有限视界幕式马尔可夫决策过程时，在表征最小最大最优后悔方面取得了实质性进展，该最小最大最优遗憾按$sqrt｛H^2SAT｝$（模对数因子）的阶数缩放，$T$为样本总数。虽然已经提出了几种竞争解决方案范式来最大限度地减少遗憾，但它们要么记忆效率低下，要么除非样本量超过一个巨大的阈值，否则达不到最优性（例如，对于现有的无模型方法，$S^6A^4，mathrm｛poly｝（H）$）。为了克服有效RL的如此大的样本量障碍，我们设计了一种新的无模型算法，其空间复杂度为$O（SAH）$，一旦样本量超过$SA，mathrm｛poly｝（H）$的数量级，就实现了接近最优的后悔。就这个样本量要求（也称为初始老化成本）而言，我们的方法比任何先验的渐进后悔最优的内存有效算法改进了至少一倍$S^5A^3$。利用最近引入的方差减少策略（也称为参考优势分解），该算法采用了一个早期确定的参考更新规则，并借助于两个具有上下限置信度的Q学习序列。我们早期确定的方差减少方法的设计原理可能对其他涉及复杂勘探-开发权衡的RL设置独立感兴趣。

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引用次数: 35

Dynamic ranking and translation synchronization 动态排序和翻译同步

IF 1.6 4区数学 Q2 MATHEMATICS, APPLIED

Information and Inference-A Journal of the Ima

Pub Date : 2022-07-04 DOI: 10.1093/imaiai/iaad029

E. Araya, Eglantine Karl'e, Hemant Tyagi

In many applications, such as sport tournaments or recommendation systems, we have at our disposal data consisting of pairwise comparisons between a set of $n$ items (or players). The objective is to use these data to infer the latent strength of each item and/or their ranking. Existing results for this problem predominantly focus on the setting consisting of a single comparison graph $G$. However, there exist scenarios (e.g. sports tournaments) where the pairwise comparison data evolve with time. Theoretical results for this dynamic setting are relatively limited, and are the focus of this paper. We study an extension of the translation synchronization problem, to the dynamic setting. In this set-up, we are given a sequence of comparison graphs $(G_t)_{tin{{mathscr{T}}}}$, where $ {{mathscr{T}}} subset [0,1]$ is a grid representing the time domain, and for each item $i$ and time $tin{{mathscr{T}}}$ there is an associated unknown strength parameter $z^*_{t,i}in{{mathbb{R}}}$. We aim to recover, for $tin{{mathscr{T}}}$, the strength vector $z^*_t=(z^*_{t,1},dots ,z^*_{t,n})$ from noisy measurements of $z^*_{t,i}-z^*_{t,j}$, where $left {{i,j}right }$ is an edge in $G_t$. Assuming that $z^*_t$ evolves smoothly in $t$, we propose two estimators—one based on a smoothness-penalized least squares approach and the other based on projection onto the low-frequency eigenspace of a suitable smoothness operator. For both estimators, we provide finite sample bounds for the $ell _2$ estimation error under the assumption that $G_t$ is connected for all $tin{{mathscr{T}}}$, thus proving the consistency of the proposed methods in terms of the grid size $|mathscr{T}|$. We complement our theoretical findings with experiments on synthetic and real data.

在许多应用程序中，例如体育比赛或推荐系统，我们可以处理由一组$n$项目(或玩家)之间的两两比较组成的数据。目的是使用这些数据来推断每个项目的潜在强度和/或它们的排名。该问题的现有结果主要集中在由单个比较图$G$组成的设置上。然而，在某些情况下(例如体育比赛)，两两比较数据会随着时间的推移而变化。这一动态设置的理论结果相对有限，是本文的重点。本文将翻译同步问题推广到动态环境。在这个设置中，我们得到了一系列比较图$(G_t)_{tin{{mathscr{T}}}}$，其中$ {{mathscr{T}}} subset [0,1]$是表示时域的网格，对于每个项目$i$和时间$tin{{mathscr{T}}}$，都有一个相关的未知强度参数$z^*_{t,i}in{{mathbb{R}}}$。对于$tin{{mathscr{T}}}$，我们的目标是从$z^*_{t,i}-z^*_{t,j}$的噪声测量中恢复强度向量$z^*_t=(z^*_{t,1},dots ,z^*_{t,n})$，其中$left {{i,j}right }$是$G_t$的一条边。假设$z^*_t$在$t$中平滑演化，我们提出了两个估计器——一个基于平滑惩罚最小二乘方法，另一个基于投影到合适的平滑算子的低频特征空间。对于这两个估计器，我们在假设$G_t$对所有$tin{{mathscr{T}}}$都是连通的情况下为$ell _2$估计误差提供了有限的样本边界，从而证明了所提出的方法在网格大小$|mathscr{T}|$方面的一致性。我们用合成和真实数据的实验来补充我们的理论发现。

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引用次数: 1

Super-resolution multi-reference alignment. 超分辨率多参考对齐。

IF 1.4 4区数学 Q2 MATHEMATICS, APPLIED

Information and Inference-A Journal of the Ima

Pub Date : 2022-06-01 Epub Date: 2021-02-18 DOI: 10.1093/imaiai/iaab003

Tamir Bendory, Ariel Jaffe, William Leeb, Nir Sharon, Amit Singer

We study super-resolution multi-reference alignment, the problem of estimating a signal from many circularly shifted, down-sampled and noisy observations. We focus on the low SNR regime, and show that a signal in $ℝ^{M}$ is uniquely determined when the number L of samples per observation is of the order of the square root of the signal's length ( $L = O (\sqrt{M})$ ). Phrased more informally, one can square the resolution. This result holds if the number of observations is proportional to 1/SNR³. In contrast, with fewer observations recovery is impossible even when the observations are not down-sampled (L = M). The analysis combines tools from statistical signal processing and invariant theory. We design an expectation-maximization algorithm and demonstrate that it can super-resolve the signal in challenging SNR regimes.

我们研究的是超分辨率多参考对齐，即从许多圆周位移、下采样和噪声观测中估计信号的问题。我们将重点放在低信噪比机制上，并证明当每个观测点的采样数目为信号长度的平方根数量级（L = O ( M )）时，ℝ M 中的信号是唯一确定的。换个非正式的说法，我们可以将分辨率平方化。如果观测数据的数量与 1/SNR3 成正比，则这一结果成立。相反，如果观测值较少，即使观测值没有降低采样（L = M），也不可能恢复。分析结合了统计信号处理和不变理论的工具。我们设计了一种期望最大化算法，并证明它能在具有挑战性的信噪比情况下超级解译信号。

引用次数: 0

Minimax optimal clustering of bipartite graphs with a generalized power method 二部图的极小极大最优聚类的广义幂方法

IF 1.6 4区数学 Q2 MATHEMATICS, APPLIED

Information and Inference-A Journal of the Ima

Pub Date : 2022-05-24 DOI: 10.1093/imaiai/iaad006

Guillaume Braun, Hemant Tyagi

Clustering bipartite graphs is a fundamental task in network analysis. In the high-dimensional regime where the number of rows $n_{1}$ and the number of columns $n_{2}$ of the associated adjacency matrix are of different order, the existing methods derived from the ones used for symmetric graphs can come with sub-optimal guarantees. Due to increasing number of applications for bipartite graphs in the high-dimensional regime, it is of fundamental importance to design optimal algorithms for this setting. The recent work of Ndaoud et al. (2022, IEEE Trans. Inf. Theory, 68, 1960–1975) improves the existing upper-bound for the misclustering rate in the special case where the columns (resp. rows) can be partitioned into $L = 2$ (resp. $K = 2$) communities. Unfortunately, their algorithm cannot be extended to the more general setting where $K neq L geq 2$. We overcome this limitation by introducing a new algorithm based on the power method. We derive conditions for exact recovery in the general setting where $K neq L geq 2$, and show that it recovers the result in Ndaoud et al. (2022, IEEE Trans. Inf. Theory, 68, 1960–1975). We also derive a minimax lower bound on the misclustering error when $K=L$ under a symmetric version of our model, which matches the corresponding upper bound up to a factor depending on $K$.

二部图聚类是网络分析中的一项基本任务。在高维状态下，相关邻接矩阵的行数$n_{1}$和列数$n_{2}$的顺序不同，从对称图中派生出来的现有方法可能会带来次优保证。由于高维区域中二部图的应用越来越多，因此设计最优算法具有重要的基础意义。Ndaoud et al. (2022, IEEE Trans.)Inf. Theory, 68, 1960-1975)在列(对应的列)的特殊情况下，改进了现有的错误聚类率上限。行)可以分区到$L = 2$(参见。$K = 2$)社区。不幸的是，他们的算法不能扩展到更一般的设置$K neq L geq 2$。我们通过引入一种基于幂方法的新算法来克服这一限制。我们推导了在$K neq L geq 2$的一般设置下精确恢复的条件，并表明它恢复了Ndaoud等人(2022,IEEE Trans.)的结果。参考理论，68,1960-1975)。在我们模型的对称版本下，我们还导出了在$K=L$时错误聚类误差的最小最大下界，它与依赖于$K$的因子的相应上界相匹配。

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引用次数: 5

An analysis of classical multidimensional scaling with applications to clustering. 经典多维尺度分析与聚类应用。

IF 1.6 4区数学 Q2 MATHEMATICS, APPLIED

Information and Inference-A Journal of the Ima

Pub Date : 2022-04-23 eCollection Date: 2023-03-01 DOI: 10.1093/imaiai/iaac004

Anna Little, Yuying Xie, Qiang Sun

Classical multidimensional scaling is a widely used dimension reduction technique. Yet few theoretical results characterizing its statistical performance exist. This paper provides a theoretical framework for analyzing the quality of embedded samples produced by classical multidimensional scaling. This lays a foundation for various downstream statistical analyses, and we focus on clustering noisy data. Our results provide scaling conditions on the signal-to-noise ratio under which classical multidimensional scaling followed by a distance-based clustering algorithm can recover the cluster labels of all samples. Simulation studies confirm these scaling conditions are sharp. Applications to the cancer gene-expression data, the single-cell RNA sequencing data and the natural language data lend strong support to the methodology and theory.

经典的多维缩放是一种广泛使用的降维技术。然而，描述其统计性能的理论结果却寥寥无几。本文为分析经典多维缩放产生的嵌入样本的质量提供了一个理论框架。这为各种下游统计分析奠定了基础，我们重点关注噪声数据的聚类。我们的研究结果提供了信噪比的缩放条件，在这些条件下，经典多维缩放和基于距离的聚类算法可以恢复所有样本的聚类标签。仿真研究证实这些缩放条件非常精确。癌症基因表达数据、单细胞 RNA 测序数据和自然语言数据的应用为该方法和理论提供了有力支持。

引用次数: 0

Linear convergence of the subspace constrained mean shift algorithm: from Euclidean to directional data. 子空间约束均值移动算法的线性收敛：从欧几里得数据到方向数据。

IF 1.4 4区数学 Q2 MATHEMATICS, APPLIED

Information and Inference-A Journal of the Ima

Pub Date : 2022-04-09 eCollection Date: 2023-03-01 DOI: 10.1093/imaiai/iaac005

Yikun Zhang, Yen-Chi Chen

This paper studies the linear convergence of the subspace constrained mean shift (SCMS) algorithm, a well-known algorithm for identifying a density ridge defined by a kernel density estimator. By arguing that the SCMS algorithm is a special variant of a subspace constrained gradient ascent (SCGA) algorithm with an adaptive step size, we derive the linear convergence of such SCGA algorithm. While the existing research focuses mainly on density ridges in the Euclidean space, we generalize density ridges and the SCMS algorithm to directional data. In particular, we establish the stability theorem of density ridges with directional data and prove the linear convergence of our proposed directional SCMS algorithm.

本文研究了子空间约束均值移动（SCMS）算法的线性收敛性，这是一种著名的算法，用于识别由核密度估计器定义的密度脊。通过论证 SCMS 算法是具有自适应步长的子空间约束梯度上升（SCGA）算法的特殊变体，我们推导出了这种 SCGA 算法的线性收敛性。现有研究主要关注欧几里得空间中的密度脊，而我们将密度脊和 SCMS 算法推广到了方向性数据。特别是，我们建立了具有方向性数据的密度脊稳定性定理，并证明了我们提出的方向性 SCMS 算法的线性收敛性。

引用次数: 0

Zero-truncated Poisson regression for sparse multiway count data corrupted by false zeros 被假零损坏的稀疏多路计数数据的零截断泊松回归

IF 1.6 4区数学 Q2 MATHEMATICS, APPLIED

Information and Inference-A Journal of the Ima

Pub Date : 2022-01-25 DOI: 10.1093/imaiai/iaad016

Oscar L'opez, Daniel M. Dunlavy, R. Lehoucq

We propose a novel statistical inference methodology for multiway count data that is corrupted by false zeros that are indistinguishable from true zero counts. Our approach consists of zero-truncating the Poisson distribution to neglect all zero values. This simple truncated approach dispenses with the need to distinguish between true and false zero counts and reduces the amount of data to be processed. Inference is accomplished via tensor completion that imposes low-rank tensor structure on the Poisson parameter space. Our main result shows that an $N$-way rank-$R$ parametric tensor $boldsymbol{mathscr{M}}in (0,infty )^{Itimes cdots times I}$ generating Poisson observations can be accurately estimated by zero-truncated Poisson regression from approximately $IR^2log _2^2(I)$ non-zero counts under the nonnegative canonical polyadic decomposition. Our result also quantifies the error made by zero-truncating the Poisson distribution when the parameter is uniformly bounded from below. Therefore, under a low-rank multiparameter model, we propose an implementable approach guaranteed to achieve accurate regression in under-determined scenarios with substantial corruption by false zeros. Several numerical experiments are presented to explore the theoretical results.

我们提出了一种新的统计推断方法，用于多路计数数据，这些数据被假零损坏，与真零计数无法区分。我们的方法包括对泊松分布进行零截断以忽略所有零值。这种简单的截断方法不需要区分真零计数和假零计数，并减少了要处理的数据量。推理是通过张量补全来完成的，它在泊松参数空间上施加了低秩张量结构。我们的主要结果表明 $N$-路阶-$R$ 参数张量 $boldsymbol{mathscr{M}}in (0,infty )^{Itimes cdots times I}$ 通过零截断泊松回归可以精确地估计泊松观测值的产生 $IR^2log _2^2(I)$ 非负正则多进分解下的非零计数。我们的结果还量化了当参数从下面均匀有界时，对泊松分布进行零截断所产生的误差。因此，在低秩多参数模型下，我们提出了一种可实现的方法，保证在假零严重破坏的欠确定场景下实现准确的回归。通过几个数值实验来验证理论结果。

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引用次数: 1

OUP accepted manuscript OUP接受稿件

IF 1.6 4区数学 Q2 MATHEMATICS, APPLIED

Information and Inference-A Journal of the Ima

Pub Date : 2022-01-01 DOI: 10.1093/imaiai/iaac007

引用次数: 0

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