Convex Non-negative Matrix Factorization in the Wild

Abstract—Non-negative matrix factorization (NMF) has recently received a lot of attention in data mining, information retrieval, and computer vision. It factorizes a non-negative input matrix V into two non-negative matrix factors V = WH such that W describes ”clusters” of the datasets. Analyzing genotypes, social networks, or images, it can be beneficial to ensure V to contain meaningful “cluster centroids”, i.e., to restrict W to be convex combinations of data points. But how can we run this convex NMF in the wild, i.e., given millions of data points? Triggered by the simple observation that each data point is a convex combination of vertices of the data convex hull, we propose to restrict W further to be vertices of the convex hull. The benefits of this convex-hull NMF approach are twofold. First, the expected size of the convex hull of, for example, n random Gaussian points in the plane is Ω( √ log n), i.e., the candidate set typically grows much slower than the dat...

• Type: Conference
• Year: 2009
• Where: ICDM
• Authors: Christian Thurau, Kristian Kersting, Christian Bauckhage
• Keywords: Convex Hull, Convex Nmf, Data Mining, Data Point, ICDM 2009