Sparse Quadratic Optimisation over the Stiefel Manifold with Application to Permutation Synchronisation

Florian Bernard, Daniel Cremers, Anders Johan Thunberg

Keywords: Stiefel manifold, quadratic optimisation, permutation synchronisation, sparsity, multi-matching, correspondence problems, manifold optimisation, QR decomposition, orthogonal iteration algorithm

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.

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TL;DR: A method for finding a globally optimal solution of a quadratic objective function over the Stiefel manifold that is sparse.

Code: https://github.com/fbernardpi/SparseStiefelOpt