Tensor decompositions for learning latent variable models (1210.7559v4)

Abstract: This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models---including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation---which exploits a certain tensor structure in their low-order observable moments (typically, of second- and third-order). Specifically, parameter estimation is reduced to the problem of extracting a certain (orthogonal) decomposition of a symmetric tensor derived from the moments; this decomposition can be viewed as a natural generalization of the singular value decomposition for matrices. Although tensor decompositions are generally intractable to compute, the decomposition of these specially structured tensors can be efficiently obtained by a variety of approaches, including power iterations and maximization approaches (similar to the case of matrices). A detailed analysis of a robust tensor power method is provided, establishing an analogue of Wedin's perturbation theorem for the singular vectors of matrices. This implies a robust and computationally tractable estimation approach for several popular latent variable models.

• The paper presents a unified framework that leverages tensor decompositions to efficiently estimate parameters in latent variable models such as GMMs, HMMs, and LDA.
• It employs a robust tensor power method with rigorous perturbation analysis and improved sample complexity bounds for practical, high-dimensional data.
• Practical innovations like dimensionality reduction and efficient moment representation facilitate scalable, robust applications in diverse real-world datasets.

Overview of "Tensor Decompositions for Learning Latent Variable Models"

The paper "Tensor Decompositions for Learning Latent Variable Models," authored by Animashree Anandkumar, Rong Ge, Daniel Hsu, Sham M. Kakade, and Matus Telgarsky, focuses on developing a statistically efficient and computationally tractable method for parameter estimation in a variety of popular latent variable models. These models include Gaussian mixture models (GMMs), hidden Markov models (HMMs), and latent Dirichlet allocation (LDA). The methodology exploits the tensor structure in the low-order observable moments, specifically second- and third-order moments, to perform the parameter estimation.

The core idea is to reduce the parameter estimation problem to extracting a symmetric, orthogonal decomposition of a tensor derived from these moments. This approach generalizes the singular value decomposition (SVD) for matrices to higher-order tensors.

Contributions

The paper makes several key contributions:

1. Unified Framework: The authors unify several previously disparate methods for parameter estimation in latent variable models under a single framework. They explicitly state and utilize the tensor decomposition structure, which was often implicitly applied in previous work.
2. Tensor Power Method: The paper provides a robust analysis of a tensor power method, establishing an analogue of Wedin's perturbation theorem for singular vectors in matrices. This is crucial as it provides a robust and computationally feasible approach to estimating latent variable models.
3. Practical Considerations: The authors address practical considerations, like dimensionality reduction and efficient representation of moments, enabling scalable implementations for high-dimensional data.

Numerical Results and Theoretical Implications

The paper establishes the tensor power method's effectiveness through rigorous mathematical proofs and provides theoretical bounds to back its claims. The robustness is shown with detailed perturbation analysis, and the sample complexity bounds are improved compared to previous methods. The analysis suggests that the method achieves parameter recovery efficiently, even under moderate perturbations, which is critical for applications with noisy data.

Implications and Future Directions

Practical Implications

The tensor decomposition approach has significant practical implications:

• Scalability: By avoiding the explicit construction of high-dimensional tensors and leveraging efficient linear transformations, the methods are computationally efficient and suitable for large-scale real-world applications.
• Robustness: The tensor power method's robustness to perturbations makes it highly applicable in practical scenarios where data may be noisy or incomplete.

Theoretical Implications

The theoretical implications of this work are profound:

• Unified Analysis: By unifying disparate techniques under a tensor decomposition framework, this work simplifies understanding and provides a solid mathematical basis for further developments.
• Improved Sample Complexity: The robustness and efficiency of the tensor power method lead to improved sample complexity bounds, which are crucial for theoretical guarantees of performance.

Future Directions

The promising results open up several avenues for future research:

• Further Algorithmic Development: There is scope for developing more advanced algorithms to further improve the efficiency and robustness of the tensor decomposition methods, potentially addressing the oscillatory behaviors observed in some cases.
• Applications to Other Models: Expanding the application of tensor decompositions to other types of latent variable models or even entirely different statistical models could be highly beneficial.
• Real-world Deployments: Testing these methods in a broader array of real-world datasets and scenarios to validate their practical utility and uncover any limitations in diverse applications.

In conclusion, "Tensor Decompositions for Learning Latent Variable Models" extends the frontier of parameter estimation methods in latent variable models, providing robust, theoretically grounded, and computationally feasible solutions. The unification under tensor decomposition not only simplifies the analytical landscape but also offers improved performance in both theoretical and practical domains. The insights provided could significantly influence future developments in computational statistics and machine learning.