Sparse and unique nonnegative matrix factorization through data preprocessing
Sparse and unique nonnegative matrix factorization through data preprocessing
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DOI：
10.5555/2503308.2503349
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发表时间：
2012-04
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影响因子：
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通讯作者：
Nicolas Gillis
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文献类型：
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作者：
Nicolas Gillis
非负矩阵分解（NMF）已成为机器学习中非常流行的技术，因为它通过稀疏和基于部分的表示自动提取有意义的特征。但是，NMF具有高度不良的缺点，也就是说，通常存在许多不同但等效的因素化。在本文中，我们引入了一种全新的方法，以获取更稀疏的解决方案的更适合的NMF问题。我们的技术基于非负输入数据矩阵的预处理，并依赖于M矩阵理论和NMF的几何解释。事实证明，在Donoho和Stodden（2003）的可分离性假设下，这种方法可导致最佳和稀疏的溶液，并且对于排名第三个矩阵，使精确的因数有限的数量。我们说明了技术对几个图像数据集的有效性。
Nonnegative matrix factorization (NMF) has become a very popular technique in machine learning because it automatically extracts meaningful features through a sparse and part-based representation. However, NMF has the drawback of being highly ill-posed, that is, there typically exist many different but equivalent factorizations. In this paper, we introduce a completely new way to obtaining more well-posed NMF problems whose solutions are sparser. Our technique is based on the preprocessing of the nonnegative input data matrix, and relies on the theory of M-matrices and the geometric interpretation of NMF. This approach provably leads to optimal and sparse solutions under the separability assumption of Donoho and Stodden (2003), and, for rank-three matrices, makes the number of exact factorizations finite. We illustrate the effectiveness of our technique on several image data sets.