Product Analysis: Learning to Model Observations as Products of Hidden Variables

Factor analysis and principal components analysis can be used to model linear relationships between observed variables and linearly map high-dimensional data to a lower-dimensional hidden space. In factor analysis, the observations are modeled as a linear combination of normally distributed hidden variables. We describe a nonlinear generalization of factor analysis, called "product analysis", that models the observed variables as a linear combination of products of normally distributed hidden variables. Just as factor analysis can be viewed as unsupervised linear regression on unobserved, normally distributed hidden variables, product analysis can be viewed as unsupervised linear regression on products of unobserved, normally distributed hidden variables. The mapping between the data and the hidden space is nonlinear, so we use an approximate variational technique for inference and learning. Since product analysis is a generalization of factor analysis, product analysis always finds a higher data likelihood than factor analysis. We give results on pattern recognition and illumination-invariant image clustering.