Abstract: Given a matrix X, nonnegative matrix factorization (NMF) aims to find the smallest nonnegative matrices U and V such that X=UV'. The nonnegative rank of X is the number of columns in U (and V). Surprisingly, nobody knows how to find this number! One of the main reasons is that NMF is ill-posed with infinitely many feasible solutions. In this talk, I will introduce what we call nonnegative canonical edges (NCEs), which reveal the optimal (maximal volume) solution to NMF. This is analogous to the Moore-Penrose pseudo-inverse, which selects the optimal (minimum-norm) solution to ill-posed least-squares problems. We conjecture that NCEs hold the key to determining the nonnegative rank, and I will discuss our ideas to determine this number exactly and finally give a definitive answer to NMF.

Bio: Daniel is an Assistant Professor in the Wisconsin Institute for Discovery and the Department of Biostatistics and Medical Informatics at UW-Madison. Before that, he was an Assistant Professor in the Computer Science Department at Georgia State University. He got his PhD in Electrical and Computer Engineering at UW-Madison. His research focuses on learning from messy data – that is, data with missing values, nonnegativity constraints, mixtures, outliers, sparsity patterns, skewed classes, small sample size, and any peculiarity that breaks typical algorithms. To this end, he uses a combination of machine learning, optimization, algebraic geometry, and signal processing tools.