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摘要翻译：
在给定样本协方差矩阵的情况下，研究了当输入变量的线性组合中的非零系数个数受到限制时，最大化输入变量的线性组合所解释的方差的问题。这被称为稀疏主成分分析，在机器学习和工程中有广泛的应用。我们对该问题提出了一个新的半定松弛，并导出了一个贪婪算法，该算法对所有目标数的非零系数计算出全集的好解，总复杂度为O(n^3)，其中n是变量数。然后，我们用同样的松弛导出解的全局最优性的充分条件，每个模式可以在O(n^3)中检验。我们讨论了该算法在子集选择和稀疏恢复中的应用，并通过人工例子和生物数据证明了该算法在许多情况下都能提供全局最优解。
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英文标题：
《Optimal Solutions for Sparse Principal Component Analysis》
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作者：
Alexandre d'Aspremont, Francis Bach, Laurent El Ghaoui
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最新提交年份：
2007
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分类信息：
一级分类：Computer Science 计算机科学
二级分类：Artificial Intelligence 人工智能
分类描述：Covers all areas of AI except Vision, Robotics, Machine Learning, Multiagent Systems, and Computation and Language (Natural Language Processing), which have separate subject areas. In particular, includes Expert Systems, Theorem Proving (although this may overlap with Logic in Computer Science), Knowledge Representation, Planning, and Uncertainty in AI. Roughly includes material in ACM Subject Classes I.2.0, I.2.1, I.2.3, I.2.4, I.2.8, and I.2.11.
涵盖了人工智能的所有领域，除了视觉、机器人、机器学习、多智能体系统以及计算和语言（自然语言处理），这些领域有独立的学科领域。特别地，包括专家系统，定理证明（尽管这可能与计算机科学中的逻辑重叠），知识表示，规划，和人工智能中的不确定性。大致包括ACM学科类I.2.0、I.2.1、I.2.3、I.2.4、I.2.8和I.2.11中的材料。
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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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英文摘要：
Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse principal component analysis and has a wide array of applications in machine learning and engineering. We formulate a new semidefinite relaxation to this problem and derive a greedy algorithm that computes a full set of good solutions for all target numbers of non zero coefficients, with total complexity O(n^3), where n is the number of variables. We then use the same relaxation to derive sufficient conditions for global optimality of a solution, which can be tested in O(n^3) per pattern. We discuss applications in subset selection and sparse recovery and show on artificial examples and biological data that our algorithm does provide globally optimal solutions in many cases.
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PDF链接：
https://arxiv.org/pdf/0707.0705
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