Sparse Quadratic Optimisation over the Stiefel Manifold with Application to Permutation Synchronisation
Bernard F, Cremers D, Thunberg J (2021)
Publication Type: Conference contribution
Publication year: 2021
Publisher: Neural information processing systems foundation
Book Volume: 30
Pages Range: 25256-25266
Conference Proceedings Title: Advances in Neural Information Processing Systems
Event location: Virtual, Online
ISBN: 9781713845393
Abstract
We address the non-convex optimisation problem of finding a sparse matrix on the Stiefel manifold (matrices with mutually orthogonal columns of unit length) that maximises (or minimises) a quadratic objective function. Optimisation problems on the Stiefel manifold occur for example in spectral relaxations of various combinatorial problems, such as graph matching, clustering, or permutation synchronisation. Although sparsity is a desirable property in such settings, it is mostly neglected in spectral formulations since existing solvers, e.g. based on eigenvalue decomposition, are unable to account for sparsity while at the same time maintaining global optimality guarantees. We fill this gap and propose a simple yet effective sparsity-promoting modification of the Orthogonal Iteration algorithm for finding the dominant eigenspace of a matrix. By doing so, we can guarantee that our method finds a Stiefel matrix that is globally optimal with respect to the quadratic objective function, while in addition being sparse. As a motivating application we consider the task of permutation synchronisation, which can be understood as a constrained clustering problem that has particular relevance for matching multiple images or 3D shapes in computer vision, computer graphics, and beyond. We demonstrate that the proposed approach outperforms previous methods in this domain.
Involved external institutions
Rheinische Friedrich-Wilhelms-Universität Bonn
Germany (DE)
Technische Universität München (TUM)
Germany (DE)
Halmstad University / Högskolan i Halmstad
Sweden (SE)
Germany (DE)
Technische Universität München (TUM)
Germany (DE)
Halmstad University / Högskolan i Halmstad
Sweden (SE)
How to cite
APA:
Bernard, F., Cremers, D., & Thunberg, J. (2021). Sparse Quadratic Optimisation over the Stiefel Manifold with Application to Permutation Synchronisation. In Marc'Aurelio Ranzato, Alina Beygelzimer, Yann Dauphin, Percy S. Liang, Jenn Wortman Vaughan (Eds.), Advances in Neural Information Processing Systems (pp. 25256-25266). Virtual, Online: Neural information processing systems foundation.
MLA:
Bernard, Florian, Daniel Cremers, and Johan Thunberg. "Sparse Quadratic Optimisation over the Stiefel Manifold with Application to Permutation Synchronisation." Proceedings of the 35th Conference on Neural Information Processing Systems, NeurIPS 2021, Virtual, Online Ed. Marc'Aurelio Ranzato, Alina Beygelzimer, Yann Dauphin, Percy S. Liang, Jenn Wortman Vaughan, Neural information processing systems foundation, 2021. 25256-25266.
BibTeX: Download