Learning Topic Models – Going beyond SVD by Sanjeev Arora, Rong Ge, and Ankur Moitra.
Topic Modeling is an approach used for automatic comprehension and classification of data in a variety of settings, and perhaps the canonical application is in uncovering thematic structure in a corpus of documents. A number of foundational works both in machine learning and in theory have suggested a probabilistic model for documents, whereby documents arise as a convex combination of (i.e. distribution on) a small number of topic vectors, each topic vector being a distribution on words (i.e. a vector of word-frequencies). Similar models have since been used in a variety of application areas; the Latent Dirichlet Allocation or LDA model of Blei et al. is especially popular.
Theoretical studies of topic modeling focus on learning the model’s parameters assuming the data is actually generated from it. Existing approaches for the most part rely on Singular Value Decomposition(SVD), and consequently have one of two limitations: these works need to either assume that each document contains only one topic, or else can only recover the span of the topic vectors instead of the topic vectors themselves.
This paper formally justifies Nonnegative Matrix Factorization(NMF) as a main tool in this context, which is an analog of SVD where all vectors are nonnegative. Using this tool we give the first polynomial-time algorithm for learning topic models without the above two limitations. The algorithm uses a fairly mild assumption about the underlying topic matrix called separability, which is usually found to hold in real-life data. A compelling feature of our algorithm is that it generalizes to models that incorporate topic-topic correlations, such as the Correlated Topic Model and the Pachinko Allocation Model.
We hope that this paper will motivate further theoretical results that use NMF as a replacement for SVD – just as NMF has come to replace SVD in many applications.
The proposal hinges on the following assumption:
Separability requires that each topic has some near-perfect indicator word – a word that we call the anchor word for this topic— that appears with reasonable probability in that topic but with negligible probability in all other topics (e.g., “soccer” could be an anchor word for the topic “sports”). We give a formal definition in Section 1.1. This property is particularly natural in the context of topic modeling, where the number of distinct words (dictionary size) is very large compared to the number of topics. In a typical application, it is common to have a dictionary size in the thousands or tens of thousands, but the number of topics is usually somewhere in the range from 50 to 100. Note that separability does not mean that the anchor word always occurs (in fact, a typical document may be very likely to contain no anchor words). Instead, it dictates that when an anchor word does occur, it is a strong indicator that the corresponding topic is in the mixture used to generate the document.
The notion of an “anchor word” (or multiple anchor words per topics as the authors point out in the conclusion) resonates with the idea of identifying a subject. It is at least a clue that an author/editor should take into account.