摘要翻译：
在本文中，我们探索提高降维方法可靠性的途径，如非负矩阵分解(NMF)作为解释性探索性数据分析工具。利用完全正因子分解理论，首次证明了非平凡的NMF解总是存在的，并且证明了优化问题实际上是凸的。随后，我们利用凸优化的各种思想探索了四种寻找全局最优NMF解的新方法。然后我们发展了一种新的方法--等距NMF（isoNMF）,它在保持非负性的同时也提供了等距嵌入，同时获得了两个有助于解释的性质。虽然它导致了一个更困难的优化问题，但我们的实验表明，所得到的方法是可扩展的，甚至比标准的NMF获得了更紧凑的光谱。
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英文标题：
《Non-Negative Matrix Factorization, Convexity and Isometry》
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作者：
Nikolaos Vasiloglou, Alexander G. Gray, David V. Anderson
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最新提交年份：
2009
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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：Computer Vision and Pattern Recognition        计算机视觉与模式识别
分类描述：Covers image processing, computer vision, pattern recognition, and scene understanding. Roughly includes material in ACM Subject Classes I.2.10, I.4, and I.5.
涵盖图像处理、计算机视觉、模式识别和场景理解。大致包括ACM课程I.2.10、I.4和I.5中的材料。
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英文摘要：
  In this paper we explore avenues for improving the reliability of dimensionality reduction methods such as Non-Negative Matrix Factorization (NMF) as interpretive exploratory data analysis tools. We first explore the difficulties of the optimization problem underlying NMF, showing for the first time that non-trivial NMF solutions always exist and that the optimization problem is actually convex, by using the theory of Completely Positive Factorization. We subsequently explore four novel approaches to finding globally-optimal NMF solutions using various ideas from convex optimization. We then develop a new method, isometric NMF (isoNMF), which preserves non-negativity while also providing an isometric embedding, simultaneously achieving two properties which are helpful for interpretation. Though it results in a more difficult optimization problem, we show experimentally that the resulting method is scalable and even achieves more compact spectra than standard NMF.
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PDF链接：
https://arxiv.org/pdf/0810.2311