摘要翻译：
低秩矩阵恢复在机器学习、信号处理、协同过滤、系统辨识、欧几里德嵌入等科学和工程领域有着广泛的应用。但低秩矩阵恢复问题是一个NP难题，具有挑战性。一种常用的启发式方法是核范数最小化。文[12,14,15]给出了核范数极小化恢复所有可能的秩至多为r的低秩矩阵的充要条件（强零空间条件）。此外，在[12]中，Oymak等。利用核范数极小化建立了一个成功恢复给定低秩矩阵的零空间条件（弱零空间条件），并导出了核范数极小化的相变。本文证明了文[12]中的弱零空间条件仅仅是核范数极小化矩阵恢复成功的充分条件，而不是文[12]所说的必要条件。本文进一步给出了低秩矩阵恢复的弱零空间条件，这是核范数极小化成功的必要条件和充分条件。在我们的推导的核心是一个刻划块矩阵核范数的不等式，以及在该不等式中相等成立的条件。
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英文标题：
《Necessary and Sufficient Null Space Condition for Nuclear Norm
  Minimization in Low-Rank Matrix Recovery》
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作者：
Jirong Yi and Weiyu Xu
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最新提交年份：
2018
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分类信息：

一级分类：Mathematics        数学
二级分类：Optimization and Control        优化与控制
分类描述：Operations research, linear programming, control theory, systems theory, optimal control, game theory
运筹学，线性规划，控制论，系统论，最优控制，博弈论
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一级分类：Computer Science        计算机科学
二级分类：Information Theory        信息论
分类描述：Covers theoretical and experimental aspects of information theory and coding. Includes material in ACM Subject Class E.4 and intersects with H.1.1.
涵盖信息论和编码的理论和实验方面。包括ACM学科类E.4中的材料，并与H.1.1有交集。
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一级分类：Computer Science        计算机科学
二级分类：Machine Learning        机器学习
分类描述：Papers on all aspects of machine learning research (supervised, unsupervised, reinforcement learning, bandit problems, and so on) including also robustness, explanation, fairness, and methodology. cs.LG is also an appropriate primary category for applications of machine learning methods.
关于机器学习研究的所有方面的论文（有监督的，无监督的，强化学习，强盗问题，等等），包括健壮性，解释性，公平性和方法论。对于机器学习方法的应用，CS.LG也是一个合适的主要类别。
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一级分类：Electrical Engineering and Systems Science        电气工程与系统科学
二级分类：Signal Processing        信号处理
分类描述：Theory, algorithms, performance analysis and applications of signal and data analysis, including physical modeling, processing, detection and parameter estimation, learning, mining, retrieval, and information extraction. The term "signal" includes speech, audio, sonar, radar, geophysical, physiological, (bio-) medical, image, video, and multimodal natural and man-made signals, including communication signals and data. Topics of interest include: statistical signal processing, spectral estimation and system identification; filter design, adaptive filtering / stochastic learning; (compressive) sampling, sensing, and transform-domain methods including fast algorithms; signal processing for machine learning and machine learning for signal processing applications; in-network and graph signal processing; convex and nonconvex optimization methods for signal processing applications; radar, sonar, and sensor array beamforming and direction finding; communications signal processing; low power, multi-core and system-on-chip signal processing; sensing, communication, analysis and optimization for cyber-physical systems such as power grids and the Internet of Things.
信号和数据分析的理论、算法、性能分析和应用，包括物理建模、处理、检测和参数估计、学习、挖掘、检索和信息提取。“信号”一词包括语音、音频、声纳、雷达、地球物理、生理、（生物）医学、图像、视频和多模态自然和人为信号，包括通信信号和数据。感兴趣的主题包括：统计信号处理、谱估计和系统辨识；滤波器设计；自适应滤波/随机学习；（压缩）采样、传感和变换域方法，包括快速算法；用于机器学习的信号处理和用于信号处理应用的机器学习；网络与图形信号处理；信号处理中的凸和非凸优化方法；雷达、声纳和传感器阵列波束形成和测向；通信信号处理；低功耗、多核、片上系统信号处理；信息物理系统的传感、通信、分析和优化，如电网和物联网。
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一级分类：Mathematics        数学
二级分类：Information Theory        信息论
分类描述：math.IT is an alias for cs.IT. Covers theoretical and experimental aspects of information theory and coding.
它是cs.it的别名。涵盖信息论和编码的理论和实验方面。
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一级分类：Statistics        统计学
二级分类：Machine Learning        机器学习
分类描述：Covers machine learning papers (supervised, unsupervised, semi-supervised learning, graphical models, reinforcement learning, bandits, high dimensional inference, etc.) with a statistical or theoretical grounding
覆盖机器学习论文（监督，无监督，半监督学习，图形模型，强化学习，强盗，高维推理等）与统计或理论基础
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英文摘要：
  Low-rank matrix recovery has found many applications in science and engineering such as machine learning, signal processing, collaborative filtering, system identification, and Euclidean embedding. But the low-rank matrix recovery problem is an NP hard problem and thus challenging. A commonly used heuristic approach is the nuclear norm minimization. In [12,14,15], the authors established the necessary and sufficient null space conditions for nuclear norm minimization to recover every possible low-rank matrix with rank at most r (the strong null space condition). In addition, in [12], Oymak et al. established a null space condition for successful recovery of a given low-rank matrix (the weak null space condition) using nuclear norm minimization, and derived the phase transition for the nuclear norm minimization. In this paper, we show that the weak null space condition in [12] is only a sufficient condition for successful matrix recovery using nuclear norm minimization, and is not a necessary condition as claimed in [12]. In this paper, we further give a weak null space condition for low-rank matrix recovery, which is both necessary and sufficient for the success of nuclear norm minimization. At the core of our derivation are an inequality for characterizing the nuclear norms of block matrices, and the conditions for equality to hold in that inequality.
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PDF链接：
https://arxiv.org/pdf/1802.05234