Algorithms for Nonnegative Matrix Factorization with the β-Divergence
Algorithms for Nonnegative Matrix Factorization with the β-Divergence
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DOI：
10.1162/neco_a_00168
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发表时间：
2011-09-01
影响因子：
2.9
通讯作者：
Idier, Jerome
中科院分区：
文献类型：
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作者：
Fevotte, Cedric;Idier, Jerome
这封信描述了具有β散度的非负矩阵分解(NMF)的算法(Beta-NMF)。β-散度是由单个形状参数β参数参数化的代价函数族，它将欧几里德距离、Kullback-Leibler散度和Itakura-Saito散度作为特例(分别为β=2、1、0)。所提出的算法基于代理辅助函数(准则函数的局部优化)。我们首先描述了一种导致乘性更新的优化最小化算法，该算法不同于标准启发式乘性更新，其不同于标准启发式乘性更新的幂指数依赖。然而，使用所提出的辅助函数可以证明启发式算法的单调性，因为β是(0，1)的元素。然后介绍了优化均衡(ME)算法的概念，该算法产生的更新沿辅助函数的恒定水平集移动，并导致比MM更大的步长。对合成和真实数据的模拟表明，ME方法的收敛速度更快。这封信还描述了所提出的算法如何适用于NMF的两种常见变体：惩罚NMF(当将因子的惩罚函数添加到准则函数中时)和凸NMF(当假设字典属于已知子空间时)。
This letter describes algorithms for nonnegative matrix factorization (NMF) with the beta-divergence (beta-NMF). The beta-divergence is a family of cost functions parameterized by a single shape parameter beta that takes the Euclidean distance, the Kullback-Leibler divergence, and the Itakura-Saito divergence as special cases (beta = 2, 1, 0 respectively). The proposed algorithms are based on a surrogate auxiliary function (a local majorization of the criterion function). We first describe a majorization-minimization algorithm that leads to multiplicative updates, which differ from standard heuristic multiplicative updates by a beta-dependent power exponent. The monotonicity of the heuristic algorithm can, however, be proven for beta is an element of (0, 1) using the proposed auxiliary function. Then we introduce the concept of the majorization-equalization (ME) algorithm, which produces updates that move along constant level sets of the auxiliary function and lead to larger steps than MM. Simulations on synthetic and real data illustrate the faster convergence of the ME approach. The letter also describes how the proposed algorithms can be adapted to two common variants of NMF: penalized NMF (when a penalty function of the factors is added to the criterion function) and convex NMF (when the dictionary is assumed to belong to a known subspace).