A Method of Moments for Mixture Models and Hidden Markov Models (1203.0683v3)

Abstract: Mixture models are a fundamental tool in applied statistics and machine learning for treating data taken from multiple subpopulations. The current practice for estimating the parameters of such models relies on local search heuristics (e.g., the EM algorithm) which are prone to failure, and existing consistent methods are unfavorable due to their high computational and sample complexity which typically scale exponentially with the number of mixture components. This work develops an efficient method of moments approach to parameter estimation for a broad class of high-dimensional mixture models with many components, including multi-view mixtures of Gaussians (such as mixtures of axis-aligned Gaussians) and hidden Markov models. The new method leads to rigorous unsupervised learning results for mixture models that were not achieved by previous works; and, because of its simplicity, it offers a viable alternative to EM for practical deployment.

• The paper introduces a method of moments that efficiently estimates parameters with low-order moments, mitigating high variance in high-dimensional settings.
• It leverages spectral techniques and multi-view learning to achieve polynomial sample complexity and faster convergence compared to traditional EM approaches.
• The approach provides practical benefits for clustering and classification tasks in diverse applications, including natural language processing and computational biology.

A Method of Moments for Mixture Models and Hidden Markov Models

This paper presents an efficient method of moments approach for parameter estimation in mixture models, including mixtures of Gaussians and Hidden Markov Models (HMMs). The primary focus is to address the computational and statistical challenges present in traditional methods like the Expectation-Maximization (EM) algorithm, and to provide a polynomial sample complexity alternative to the exponential complexity often encountered in high-dimensional settings.

Overview

Mixture models are foundational tools in statistics and machine learning, often used for clustering and classification tasks. A key challenge in these models is the estimation of parameters, specifically the parameters governing the distributions of each mixture component. Traditional approaches, most notably EM, have well-recognized limitations, including slow convergence and susceptibility to local optima. Alternatively, the method of moments offers a statistically consistent approach but suffers from issues in high dimensions due to its reliance on higher-order moments.

This work circumvents these challenges by proposing a new method of moments utilizing only lower-order moments. This method uses standard numerical linear algebra routines and leverages the concept of multiple indirect "views" of the latent variables. Essentially, the authors harness multi-view data, akin to various noisy projections, to recover the model parameters effectively.

Key Results

The authors claim several advancements in parameter estimation for mixture models and HMMs:

1. Low-Order Moments: The method only requires low-order moments, mitigating the high variance typically associated with estimating high-order moments. This provides a more computationally tractable solution.
2. Spectral Techniques: By extending spectral decomposition techniques, the method ensures polynomial sample complexity with respect to the number of components—significantly improving upon existing approaches with exponential dependencies.
3. Multi-View Learning: The framework exploits the multi-view nature of data, which is prevalent in many real-world datasets. This approach notably removes the need for the separation conditions usually required for learning Gaussian mixtures.

Theoretical and Practical Implications

Theoretically, this method contributes to the ongoing development of algorithms that can efficiently estimate complex models in high-dimensional spaces without relying heavily on likelihood-based methods. It suggests that under mild conditions, polynomial sample complexity can be achieved, marking progress towards scalable unsupervised learning solutions for mixture models.

Practically, the simplicity and efficiency of the suggested approach offer improved deployment viability in real-world applications compared to methods like EM. By alleviating computational burdens and achieving fast convergence, this method could find applications across various domains where mixture models are prevalent, including natural language processing and computational biology.

Future Directions

Future work could explore extensions of this method to other types of mixture models and latent variable models. Additionally, more empirical studies would strengthen the evidence for the efficacy of this method in diverse practical scenarios. Further theoretical investigations could refine the conditions under which polynomial complexity holds and explore the limits of moment-based methods in even larger and more complex model classes.

Conclusion

This paper presents a significant development in the field of mixture models and HMMs, offering a computationally efficient and statistically consistent method for parameter estimation. By utilizing low-order moments and spectral techniques, it lays the groundwork for further research and application in high-dimensional machine learning contexts.