# Orthogonal Trace-Sum Maximization: Applications, Local Algorithms, and Global Optimality

arXiv: https://arxiv.org/abs/1811.03521 · HTML: https://mlsanity.com/paper/1811.03521/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2018.

## Built on by (most cited first, 2 of 2)

- [A Versatile Framework for Attributed Network Clustering via K-Nearest Neighbor Augmentation](https://mlsanity.com/paper/2408.05459/) (2024)
- [Star-Graph Multimodal Matching Component Analysis for Data Fusion and Transfer Learning](https://mlsanity.com/paper/2210.02590/) (2022)

## Abstract (arXiv)

This paper studies the problem of maximizing the sum of traces of matrix quadratic forms on a product of Stiefel manifolds. This orthogonal trace-sum maximization (OTSM) problem generalizes many interesting problems such as generalized canonical correlation analysis (CCA), Procrustes analysis, and cryo-electron microscopy of the Nobel prize fame. For these applications finding global solutions is highly desirable but it has been unclear how to find even a stationary point, let alone testing its global optimality. Through a close inspection of Ky Fan's classical result (1949) on the variational formulation of the sum of largest eigenvalues of a symmetric matrix, and a semidefinite programming (SDP) relaxation of the latter, we first provide a simple method to certify global optimality of a given stationary point of OTSM. This method only requires testing whether a symmetric matrix is positive semidefinite. A by-product of this analysis is an unexpected strong duality between Shapiro-Botha (1988) and Zhang-Singer (2017). After showing that a popular algorithm for generalized CCA and Procrustes analysis may generate oscillating iterates, we propose a simple fix that provably guarantees convergence to a stationary point. The combination of our algorithm and certificate reveals novel global optima of various instances of OTSM.