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Article Dans Une Revue SIAM Journal on Matrix Analysis and Applications Année : 2023

Convergence of gradient-based block coordinate descent algorithms for non-orthogonal joint approximate diagonalization of matrices

Résumé

In this paper, we propose a gradient based block coordinate descent (BCD-G) framework to solve the joint approximate diagonalization of matrices defined on the product of the complex Stiefel manifold and the special linear group. Instead of the cyclic fashion, we choose the block for optimization in a way based on the Riemannian gradient. To update the first block variable in the complex Stiefel manifold, we use the well-known line search descent method. To update the second block variable in the special linear group, based on four different kinds of elementary transformations, we construct three classes: GLU, GQU and GU, and then get three BCD-G algorithms: BCD-GLU, BCD-GQU and BCD-GU. We establish the global convergence and weak convergence of these three algorithms using the Lojasiewicz gradient inequality under the assumption that the iterates are bounded. We also propose a gradient based Jacobi-type framework to solve the joint approximate diagonalization of matrices defined on the special linear group. Similar as in the BCD-G case, using the GLU and GQU classes of elementary transformations, we focus on the Jacobi-GLU and Jacobi-GQU algorithms, and establish their global convergence and weak convergence as well. All the algorithms and convergence results in this paper also apply to the real case.
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Dates et versions

hal-03408912 , version 1 (04-11-2021)
hal-03408912 , version 2 (05-05-2023)

Identifiants

Citer

Jianze Li, Konstantin Usevich, Pierre Comon. Convergence of gradient-based block coordinate descent algorithms for non-orthogonal joint approximate diagonalization of matrices. SIAM Journal on Matrix Analysis and Applications, 2023, 44 (2), pp.592-621. ⟨10.1137/21M1456972⟩. ⟨hal-03408912v2⟩
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