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Machine Learning-Assisted High-Dimensional Matrix Estimation

Wan Tian, Hui Yang, Zhouhui Lian, Lingyue Zhang, Yijie Peng

Abstract

Efficient estimation of high-dimensional matrices-including covariance and precision matrices-is a cornerstone of modern multivariate statistics. Most existing studies have focused primarily on the theoretical properties of the estimators (e.g., consistency and sparsity), while largely overlooking the computational challenges inherent in high-dimensional settings. Motivated by recent advances in learning-based optimization method-which integrate data-driven structures with classical optimization algorithms-we explore high-dimensional matrix estimation assisted by machine learning. Specifically, for the optimization problem of high-dimensional matrix estimation, we first present a solution procedure based on the Linearized Alternating Direction Method of Multipliers (LADMM). We then introduce learnable parameters and model the proximal operators in the iterative scheme with neural networks, thereby improving estimation accuracy and accelerating convergence. Theoretically, we first prove the convergence of LADMM, and then establish the convergence, convergence rate, and monotonicity of its reparameterized counterpart; importantly, we show that the reparameterized LADMM enjoys a faster convergence rate. Notably, the proposed reparameterization theory and methodology are applicable to the estimation of both high-dimensional covariance and precision matrices. We validate the effectiveness of our method by comparing it with several classical optimization algorithms across different structures and dimensions of high-dimensional matrices.

Machine Learning-Assisted High-Dimensional Matrix Estimation

Abstract

Efficient estimation of high-dimensional matrices-including covariance and precision matrices-is a cornerstone of modern multivariate statistics. Most existing studies have focused primarily on the theoretical properties of the estimators (e.g., consistency and sparsity), while largely overlooking the computational challenges inherent in high-dimensional settings. Motivated by recent advances in learning-based optimization method-which integrate data-driven structures with classical optimization algorithms-we explore high-dimensional matrix estimation assisted by machine learning. Specifically, for the optimization problem of high-dimensional matrix estimation, we first present a solution procedure based on the Linearized Alternating Direction Method of Multipliers (LADMM). We then introduce learnable parameters and model the proximal operators in the iterative scheme with neural networks, thereby improving estimation accuracy and accelerating convergence. Theoretically, we first prove the convergence of LADMM, and then establish the convergence, convergence rate, and monotonicity of its reparameterized counterpart; importantly, we show that the reparameterized LADMM enjoys a faster convergence rate. Notably, the proposed reparameterization theory and methodology are applicable to the estimation of both high-dimensional covariance and precision matrices. We validate the effectiveness of our method by comparing it with several classical optimization algorithms across different structures and dimensions of high-dimensional matrices.

Paper Structure

This paper contains 23 sections, 13 theorems, 153 equations, 13 figures, 6 tables.

Table of Contents

  1. Introduction
  2. High-dimensional covariance matrix estimation
  3. High-dimensional precision matrix estimation
  4. Theoretical Properties
  5. Convergence properties of algorithms
  6. Statistical–optimization error analysis
  7. Approximation analysis of proximal operators
  8. Simulation Studies
  9. High-dimensional covariance matrix
  10. High-dimensional precision matrix
  11. Conclusion
  12. Proofs for Results
  13. Proof of Theorem \ref{['convergence_of_LADMM']}
  14. Proof of Theorem \ref{['convergence_of_neural_LADMM']}
  15. Proof of Theorem \ref{['monotonicity_of_neural_LADMM']}
  16. ...and 8 more sections

Key Result

Theorem 4.1

If $\phi_1,\phi_2>1$, then the sequence $\{X^{(k)},Y^{(k)},V^{(k)}\}$ generated by unified_LADMM converges to a KKT point of problem unified_opt.

Figures (13)

  • Figure 1: Overview of the proposed LBO algorithm framework. The left panel illustrates the forward process with total $K$ iterations of the algorithm, while the right panel shows the target loss function which is used to update learnable parameters. The operators $\eta_k,\xi_k$ are parameterized by $(w_1)_k,(w_2)_k$, respectively.
  • Figure 2: Toeplitz covariance matrices under different correlation decay coefficients $\varrho$.
  • Figure 3: Factor model under different numbers of low-rank components $m$.
  • Figure 4: Covariance matrices under different sparsity levels $q$.
  • Figure 5: Covariance matrices under different block size.
  • ...and 8 more figures

Theorems & Definitions (26)

  • Theorem 4.1: Convergence of LADMM
  • Theorem 4.2: Convergence of re-parameterized LADMM
  • Theorem 4.3: Monotonicity of re-parameterized LADMM
  • Theorem 4.4: Convergence rate of re-parameterized LADMM
  • Theorem 4.5: Superiority of re-parameterized LADMM
  • Theorem 4.6: Entrywise concentration
  • Theorem 4.7: Total error decomposition and statistically optimal early stopping
  • Theorem 4.8: Total error decomposition and statistically optimal early stopping
  • Proposition 4.1
  • Theorem 4.9
  • ...and 16 more