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ORTHOGONAL TRACE-SUM MAXIMIZATION: APPLICATIONS, LOCAL ALGORITHMS, AND GLOBAL OPTIMALITY.

Joong-Ho Won1, Hua Zhou2, Kenneth Lange3

  • 1Department of Statistics, Seoul National University, Seoul 08826, Korea.

SIAM Journal on Matrix Analysis and Applications : a Publication of the Society for Industrial and Applied Mathematics
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PubMed
Summary
This summary is machine-generated.

This study introduces a method to find and verify global optima for orthogonal trace-sum maximization (OTSM) problems, which are crucial in fields like generalized canonical correlation analysis (CCA). The approach ensures convergence and identifies new solutions.

Keywords:
15B1049K3049M2065K0590C26Kurdyka–Łojasiewicz inequalityMAXBETMAXDIFFMM algorithmProcrustes analysisStiefel manifoldcanonical correlation analysiscryo-EMsemidefinite programming

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Area of Science:

  • Optimization
  • Linear Algebra
  • Matrix Analysis

Background:

  • The orthogonal trace-sum maximization (OTSM) problem generalizes critical applications such as generalized canonical correlation analysis (CCA), Procrustes analysis, and cryo-electron microscopy.
  • Finding global solutions for OTSM is highly desirable, but methods for identifying stationary points and testing global optimality have been lacking.

Purpose of the Study:

  • To develop a method for certifying the global optimality of stationary points for OTSM problems.
  • To propose a fix for algorithms in generalized CCA and Procrustes analysis that guarantees convergence to a stationary point.

Main Methods:

  • Leveraging Ky Fan's classical result on the sum of largest eigenvalues and semidefinite programming (SDP) relaxation.
  • Developing a certificate for global optimality by testing matrix positive semidefiniteness.
  • Modifying existing algorithms to ensure convergence to stationary points.

Main Results:

  • A simple method to certify global optimality of stationary points for OTSM problems is presented.
  • A novel strong duality is revealed as a by-product of the analysis.
  • A modified algorithm guarantees convergence to a stationary point for generalized CCA and Procrustes analysis.

Conclusions:

  • The combination of the proposed algorithm and certificate can reveal novel global optima for various OTSM instances.
  • The findings provide a robust framework for solving and verifying solutions to important optimization problems in applied mathematics and data analysis.