Learning Topic Models - Going beyond SVD (1204.1956v2)

Published 9 Apr 2012 in cs.LG, cs.DS, and cs.IR

Abstract: 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.

Authors (3)

Sanjeev Arora (93 papers)
Rong Ge (92 papers)
Ankur Moitra (88 papers)

Citations (427)

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Summary

Overview of "Learning Topic Models — Going beyond SVD"

The paper "Learning Topic Models — Going beyond SVD" by Arora, Ge, and Moitra presents an innovative approach to topic modeling that leverages Nonnegative Matrix Factorization (NMF) as a core computational method, instead of the traditional Singular Value Decomposition (SVD). The authors address the limitations of existing SVD-based methodologies, which restrict documents to a single topic or only recover the span of the topic vectors. Their work introduces a polynomial-time algorithm capable of learning topic models without these restrictions, under the assumption of a separability condition.

Key Contributions

The authors' primary contribution is a novel algorithm that formalizes the use of NMF in the context of topic modeling. This approach employs a condition known as separability within the topic matrix, where each topic has an "anchor word" unique to it that appears predominantly within that topic. This condition allows the algorithm to effectively factorize the document-topic matrix, overcoming challenges faced by SVD-based methods in real-world data scenarios.

Another significant advancement is the capability of the proposed algorithm to handle topic models with correlated topics, such as the Correlated Topic Model (CTM) and the Pachinko Allocation Model (PAM), enhancing the practical utility and robustness of the model in capturing real-world thematic structures.

Theoretical Implications

The theoretical implications center around the authors' ability to learn both the topic matrix and the parameters of the generative distribution for the documents efficiently, provided the dataset exhibits the aforementioned separable structure. The algorithm's assumptions and mathematical rigor demonstrate a potential shift towards NMF as a favored tool in situations where SVD is traditionally applied but lacks theoretical guarantees for parameter recovery under noisy conditions.

Numerical Results and Claims

The paper presents strong numerical results underscoring the algorithm's capacity to recover the topic matrix to an additive error epsilon with a polynomial number of documents under realistic assumptions about separability. The condition for separability introduces a relaxed demand compared to prior models that necessitated each document to belong to only one topic.

Practical Implications

Practically, this research pushes topic modeling closer to robust, scalable applications across various domains, from document classification to genetic data interpretation, by reducing the reliance on heuristics and local search methods which do not offer provable guarantees. Moreover, the adaptability to correlated topic models expands its utility in capturing nuanced inter-topic relationships.

Future Directions

The authors express aspirations for their work to inspire further theoretical explorations and practical applications of NMF within machine learning research, emphasizing its capacity to supplant SVD in scenarios where data naturalness aligns with separability. The separation condition, found naturally in empirical topic data, provides an intriguing area for ongoing research to better understand data structures that meet or exceed the requirements for successful factorization.

Conclusion

In conclusion, the paper addresses significant gaps in the field of topic modeling by replacing SVD with NMF under plausible separability conditions, opening avenues for more accurate and applicable learning of document-topic structures. This provides a foundation for further innovation and application, particularly in domains requiring consideration of topic correlation and noise-resilient learning methods.

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