On the Probabilistic Latent Semantic Analysis Generalization as the Singular Value Decomposition Probabilistic Image

Abstract

The Probabilistic Latent Semantic Analysis has been related with the Singular Value Decomposition. Several problems occur when this comparative is done. Data class restrictions and the existence of several local optima mask the relation, being a formal analogy without any real significance. Moreover, the computational difficulty in terms of time and memory limits the technique applicability. In this work, we use the Nonnegative Matrix Factorization with the Kullback–Leibler divergence to prove, when the number of model components is enough and a limit condition is reached, that the Singular Value Decomposition and the Probabilistic Latent Semantic Analysis empirical distributions are arbitrary close. Under such conditions, the Nonnegative Matrix Factorization and the Probabilistic Latent Semantic Analysis equality is obtained. With this result, the Singular Value Decomposition of every nonnegative entries matrix converges to the general case Probabilistic Latent Semantic Analysis results and constitutes the unique probabilistic image. Moreover, a faster algorithm for the Probabilistic Latent Semantic Analysis is provided.

Keywords

• Singular value decomposition
• Probabilistic latent semantic analysis
• Nonnegative matrix factorization
• Kullback–Leibler divergence