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We developed a new high dimensional Expectation-Maximization (EM) algorithm for latent variable models. This approach improves parameter estimation and enables optimal statistical inference in high dimensions.

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Area of Science:

  • Statistics
  • Machine Learning
  • Computational Statistics

Background:

  • High dimensional data analysis presents challenges for traditional statistical methods.
  • Latent variable models are widely used but complex to infer in high dimensions.
  • Existing Expectation-Maximization (EM) algorithms struggle with scalability and statistical guarantees in high dimensions.

Purpose of the Study:

  • To develop a general theory for the Expectation-Maximization (EM) algorithm in high dimensional settings.
  • To propose a novel EM algorithm that incorporates sparsity for improved parameter estimation.
  • To establish computationally feasible methods for optimal estimation and asymptotic inference in high dimensional latent variable models.

Main Methods:

  • A novel high dimensional EM algorithm incorporating sparsity.
  • Theoretical analysis of convergence rates (geometric rate).
  • Development of inferential procedures for hypothesis testing and confidence intervals.

Main Results:

  • The proposed EM algorithm achieves (near-)optimal statistical rates of convergence.
  • The algorithm demonstrates geometric convergence with appropriate initialization.
  • The framework provides the first computationally feasible approach for optimal estimation and asymptotic inference in high dimensions.

Conclusions:

  • The developed theory and algorithm offer a significant advancement for high dimensional latent variable modeling.
  • The approach enables efficient and statistically sound inference in complex, high dimensional datasets.
  • Numerical results validate the theoretical findings and practical utility of the proposed methods.