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A Newton Interior-Point Method for $\ell_0$ Factor Analysis

Linyang Wang, Wanquan Liu, Bin Zhu

TL;DR

This work tackles estimating a low-rank plus sparse covariance decomposition in factor analysis by optimizing $f(\boldsymbol L,\boldsymbol S)+C\|\boldsymbol S\|_0$ under PSD constraints, using an interior-point method that handles inequality constraints with a logarithmic barrier. The key idea is to solve unconstrained $\ell_0$-regularized problems via Newton iterations on a stationary-point reformulation, leveraging the $\ell_0$ proximal operator and a $\tau$-minimization barrier, and to characterize optima through $\gamma$-stationary points. The authors develop a Newton-based IPM that reduces the inner system, uses a geometry-driven barrier parameter update, and employs warm starts to accelerate convergence. Numerical experiments on synthetic data show that the proposed IPM achieves faster, often superlinear, convergence than ADMM and BCD, validating the second-order approach for $\ell_0$ FA. The framework also lays groundwork for extensions to dynamic factor analysis and other low-rank plus sparse graphical models.

Abstract

Factor Analysis is an effective way of dimensionality reduction achieved by revealing the low-rank plus sparse structure of the data covariance matrix. The corresponding model identification task is often formulated as an optimization problem with suitable regularizations. In particular, we use the nonconvex discontinuous $\ell_0$ norm in order to induce the sparsity of the covariance matrix of the idiosyncratic noise. This paper shows that such a challenging optimization problem can be approached via an interior-point method with inner-loop Newton iterations. To this end, we first characterize the solutions to the unconstrained $\ell_0$ regularized optimization problem through the $\ell_0$ proximal operator, and demonstrate that local optimality is equivalent to the solution of a stationary-point equation. The latter equation can then be solved using standard Newton's method, and the procedure is integrated into an interior-point algorithm so that inequality constraints of positive semidefiniteness can be handled. Finally, numerical examples validate the effectiveness of our algorithm.

A Newton Interior-Point Method for $\ell_0$ Factor Analysis

TL;DR

This work tackles estimating a low-rank plus sparse covariance decomposition in factor analysis by optimizing under PSD constraints, using an interior-point method that handles inequality constraints with a logarithmic barrier. The key idea is to solve unconstrained -regularized problems via Newton iterations on a stationary-point reformulation, leveraging the proximal operator and a -minimization barrier, and to characterize optima through -stationary points. The authors develop a Newton-based IPM that reduces the inner system, uses a geometry-driven barrier parameter update, and employs warm starts to accelerate convergence. Numerical experiments on synthetic data show that the proposed IPM achieves faster, often superlinear, convergence than ADMM and BCD, validating the second-order approach for FA. The framework also lays groundwork for extensions to dynamic factor analysis and other low-rank plus sparse graphical models.

Abstract

Factor Analysis is an effective way of dimensionality reduction achieved by revealing the low-rank plus sparse structure of the data covariance matrix. The corresponding model identification task is often formulated as an optimization problem with suitable regularizations. In particular, we use the nonconvex discontinuous norm in order to induce the sparsity of the covariance matrix of the idiosyncratic noise. This paper shows that such a challenging optimization problem can be approached via an interior-point method with inner-loop Newton iterations. To this end, we first characterize the solutions to the unconstrained regularized optimization problem through the proximal operator, and demonstrate that local optimality is equivalent to the solution of a stationary-point equation. The latter equation can then be solved using standard Newton's method, and the procedure is integrated into an interior-point algorithm so that inequality constraints of positive semidefiniteness can be handled. Finally, numerical examples validate the effectiveness of our algorithm.

Paper Structure

This paper contains 6 sections, 5 theorems, 31 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Review for $\ell_0$ Factor Analysis
  3. Optimality Theory for $\tau$-Minimization
  4. Algorithm Design for the Newton IPM
  5. Numerical Examples
  6. Conclusion

Key Result

Lemma 1

A pair $(\boldsymbol{\ell}^*,\mathbf s^*)$ is a $\gamma$-stationary point of opt_vector if and only if

Figures (1)

  • Figure 1: Comparison of the convergence rates among the three algorithms ADMM, BCD, and IPM applied to the same data set with different values of the parameter $\theta$ for the IPM.

Theorems & Definitions (9)

  • Definition 1: $\gamma$-stationary point of \ref{['opt_vector']}
  • Remark 1
  • Lemma 1
  • Proposition 1
  • Proposition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof