A Spectral Algorithm for Latent Dirichlet Allocation

TL;DR

This paper introduces a spectral algorithm called Excess Correlation Analysis (ECA) for efficiently recovering latent topics in LDA models using only third-order moments, improving scalability and accuracy in topic modeling.

Contribution

The paper presents a novel spectral method for learning LDA parameters using low-order moments, which is simple, scalable, and guarantees recovery of topic distributions.

Findings

Successfully recovers topic vectors and priors using trigram statistics.

Uses spectral decomposition on low-order moments for parameter estimation.

Algorithm is scalable with respect to the number of topics.

Abstract

The problem of topic modeling can be seen as a generalization of the clustering problem, in that it posits that observations are generated due to multiple latent factors (e.g., the words in each document are generated as a mixture of several active topics, as opposed to just one). This increased representational power comes at the cost of a more challenging unsupervised learning problem of estimating the topic probability vectors (the distributions over words for each topic), when only the words are observed and the corresponding topics are hidden. We provide a simple and efficient learning procedure that is guaranteed to recover the parameters for a wide class of mixture models, including the popular latent Dirichlet allocation (LDA) model. For LDA, the procedure correctly recovers both the topic probability vectors and the prior over the topics, using only trigram statistics (i.e., third order moments, which may be estimated with documents containing just three words). The method, termed Excess Correlation Analysis (ECA), is based on a spectral decomposition of low order moments (third and fourth order) via two singular value decompositions (SVDs). Moreover, the algorithm is scalable since the SVD operations are carried out on $k\times k$ matrices, where $k$ is the number of latent factors (e.g. the number of topics), rather than in the $d$-dimensional observed space (typically $d \gg k$).