[Paper Review] Sparse Quadratic Optimisation over the Stiefel Manifold with Application to Permutation Synchronisation

Florian Bernard, Daniel Cremers+1 more|arXiv (Cornell University)|Sep 30, 2021

Face and Expression Recognition4 citations

TL;DR

This paper proposes a novel algorithm that enhances the Orthogonal Iteration method to solve sparse quadratic optimisation on the Stiefel manifold, preserving global optimality while promoting sparsity through orthogonal invariance. It achieves superior performance in permutation synchronisation, outperforming prior methods in accuracy and runtime.

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.

Motivation & Objective

• To address the lack of sparsity promotion in spectral relaxations of combinatorial problems on the Stiefel manifold.
• To maintain global optimality guarantees while achieving sparse solutions in quadratic optimisation over orthogonal matrices.
• To exploit orthogonal invariance to rotate solutions toward sparser representations without sacrificing optimality.
• To improve performance in permutation synchronisation, a key problem in multi-image and 3D shape matching.
• To provide a scalable, efficient, and differentiable method suitable for integration into learning pipelines.

Proposed method

• The method modifies the Orthogonal Iteration algorithm by introducing a secondary objective that promotes sparsity through gradient-based rotation of the solution matrix.
• At each iteration, the algorithm computes a rotation matrix based on the gradient of a sparsity-promoting objective function applied to the current iterate.
• The rotation preserves the dominant subspace, ensuring the solution remains globally optimal for the original quadratic problem.
• The algorithm maintains convergence properties identical to standard Orthogonal Iteration, with convergence guaranteed almost everywhere.
• Sparsity is achieved by rotating the solution within the invariant subspace, leveraging the fact that any orthogonal transformation of a global solution remains optimal.
• The method optionally incorporates non-negativity constraints by modifying the sparsity-promoting objective function.

Experimental results

Research questions

• RQ1 Can sparsity be effectively promoted in Stiefel manifold optimisation without sacrificing global optimality?
• RQ2 Is it possible to exploit orthogonal invariance to rotate a globally optimal solution toward a sparser representation?
• RQ3 How does the proposed method compare to existing spectral relaxation techniques in permutation synchronisation tasks?
• RQ4 To what extent does the sparsity-promoting mechanism improve discrete solution quality in combinatorial problems?
• RQ5 Can the method be extended to other spectral relaxations requiring orthogonality, sparsity, and non-negativity?

Key findings

• The proposed algorithm achieves globally optimal solutions to the quadratic optimisation problem on the Stiefel manifold while promoting sparsity through orthogonal invariance.
• The method maintains the same convergence rate and global optimality guarantees as the original Orthogonal Iteration algorithm.
• In permutation synchronisation, the method achieves higher F-scores and better cycle consistency than state-of-the-art baselines.
• The algorithm demonstrates improved performance on partial permutation synchronisation tasks, particularly in noisy or incomplete data regimes.
• The method is computationally efficient and compatible with differentiable programming due to the differentiability of QR factorisation.
• Empirical results confirm that the algorithm produces solutions with fewer large entries and more near-zero elements, indicating effective sparsity promotion.