Simplex Decompositions using SVD and PLSA

Probabilistic Latent Semantic Analysis (PLSA) is a popular technique to analyze non-negative data where multinomial distributions underlying every data vector are expressed as linear combinations of a set of basis distributions. These learned basis distributions that characterize the dataset lie on the standard simplex and themselves represent corners of a simplex within which all data approximations lie. In this paper, we describe a novel method to extend the PLSA decomposition where the bases are not constrained to lie on the standard simplex and thus are better able to characterize the data. The locations of PLSA basis distributions on the standard simplex depend on how the dataset is aligned with respect to the standard simplex. If the directions of maximum variance of the dataset are orthogonal to the standard simplex, then the PLSA bases will give a poor representation of the dataset. Our approach overcomes this drawback by utilizing Singular Values Decomposition (SVD) to identify the directions of maximum variance, and transforming the dataset to align these directions parallel to the standard simplex before performing PLSA. The learned PLSA features are then transformed back into the data space. The effectiveness of the proposed approach is demonstrated with experiments on synthetic data.