Nonnegative matrix factorization (NMF) has become a widely used tool for the
analysis of high-dimensional data as it automatically extracts sparse and
meaningful features from a set of nonnegative data vectors. We first illustrate
this property of NMF on three applications, in image processing, text mining
and hyperspectral imaging --this is the why. Then we address the problem of
solving NMF, which is NP-hard in general. We review some standard NMF
algorithms, and also present a recent subclass of NMF problems, referred to as
near-separable NMF, that can be solved efficiently (that is, in polynomial
time), even in the presence of noise --this is the how. Finally, we briefly
describe some problems in mathematics and computer science closely related to
NMF via the nonnegative rank.

本文介绍了非负矩阵分解的稀疏特征提取功能，并探讨了如何解决通常情况下 NP 困难的 NMF 问题，介绍了一个称为近可分离 NMF 的问题子类，可以高效地解决一些在有噪声的情况下的 NMF 问题。最后简要描述了 NMF 在数学和计算机科学领域的若干相关问题。

非负矩阵分解的原因和方法

The Why and How of Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) under the separability assumption can
provably be solved efficiently, even in the presence of noise, and has been
shown to be a powerful technique in document classification and hyperspectral
unmixing. This problem is referred to as near-separable NMF and requires that
there exists a cone spanned by a small subset of the columns of the input
nonnegative matrix approximately containing all columns. In this paper, we
propose a preconditioning based on semidefinite programming making the input
matrix well-conditioned. This in turn can improve significantly the performance
of near-separable NMF algorithms which is illustrated on the popular successive
projection algorithm (SPA). The new preconditioned SPA is provably more robust
to noise, and outperforms SPA on several synthetic data sets. We also show how
an active-set method allow us to apply the preconditioning on large-scale
real-world hyperspectral images.

本研究针对非负矩阵分解以近可分离性为前提条件的问题，提出了基于半定编程的预处理方法，有效改善了连续投影算法的性能，并在多个数据集上得到实际验证，同时也探讨了主动集方法在大规模高光谱图像中的应用。