Modeling the diversity and log-normality of data

We investigate two important properties of real data: diversity and log-normality. Log-normality accounts for the fact that data follow the lognormal distribution, whereas diversity measures variations of the attributes in the data. To our knowledge, these two inherent properties have not been paid much attention from the machine learning community, especially from the topic modeling community. In this article, we fill in this gap in the framework of topic modeling. We first investigate whether or not these two properties can be captured by the most well-known Latent Dirichlet Allocation model (LDA), and find that LDA behaves inconsistently with respect to diversity. Particularly, it favors data of low diversity, but works badly on data of high diversity. Then, we argue that these two inherent properties can be captured well by endowing the topic-word distributions in LDA with the lognormal distribution. This treatment leads to a new model, named Dirichlet-lognormal topic model (DLN). Using the lognormal distribution complicates the learning and inference of DLN, compared with those of LDA. Hence, we used variational method, in which model learning and inference are reduced to solving convex optimization problems. Extensive experiments strongly suggest that (1) the predictive power of DLN is consistent with respect to diversity, and that (2) DLN works consistently better than LDA for datasets whose diversity is large, and for datasets which contain many log-normally distributed attributes. Justifications for these results require insights into the used statistical distributions and will be discussed in the article.