# Trace Ratio Optimization with an Application to Multi-view Learning

arXiv: https://arxiv.org/abs/2101.04292 · HTML: https://mlsanity.com/paper/2101.04292/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 66 references that are in the map.
- First posted on arXiv 12 Jan 2021.

## Builds on (most cited first, 5 of 5)

- [Multimodal Machine Learning: A Survey and Taxonomy](https://mlsanity.com/paper/1705.09406/) (2017)
- [CM-GANs: Cross-modal Generative Adversarial Networks for Common Representation Learning](https://mlsanity.com/paper/1710.05106/) (2017)
- [A generalized power iteration method for solving quadratic problem on the Stiefel manifold](https://mlsanity.com/paper/1701.00381/) (2017)
- [Generalized Multi-view Embedding for Visual Recognition and Cross-modal Retrieval](https://mlsanity.com/paper/1605.09696/) (2016)
- [A Self-consistent-field Iteration for Orthogonal Canonical Correlation Analysis](https://mlsanity.com/paper/1909.11527/) (2019)

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

- [Nested subspace learning with flags](https://mlsanity.com/paper/2502.06022/) (2025)
- [On the Spectral Structure and Objective Equivalence of Orthogonal Multilabel Fisher Discriminants](https://mlsanity.com/paper/2605.03283/) (2026)

## Abstract (arXiv)

A trace ratio optimization problem over the Stiefel manifold is investigated from the perspectives of both theory and numerical computations. At least three special cases of the problem have arisen from Fisher linear discriminant analysis, canonical correlation analysis, and unbalanced Procrustes problem, respectively. Necessary conditions in the form of nonlinear eigenvalue problem with eigenvector dependency are established and a numerical method based on the self-consistent field (SCF) iteration is designed and proved to be always convergent. As an application to multi-view subspace learning, a new framework and its instantiated concrete models are proposed and demonstrated on real world data sets. Numerical results show that the efficiency of the proposed numerical methods and effectiveness of the new multi-view subspace learning models.