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Maximizing sum of coupled traces with applications

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Abstract

This paper concerns maximizing the sum of coupled traces of quadratic and linear matrix forms. The coupling comes from requiring the matrix variables in the quadratic and linear matrix forms to be packed together to have orthonormal columns. At a maximum, the KKT condition becomes a nonlinear polar decomposition (NPD) of a matrix-valued function with dependency on the orthogonal polar factor. A self-consistent-field iteration, along with a locally optimal conjugate gradient (LOCG) acceleration, are proposed to compute the NPD. It is proved that both methods are convergent and it is demonstrated numerically that the LOCG acceleration is very effective. As applications, we demonstrate our methods on the MAXBET subproblem and the multi-view partially shared subspace learning (MvPS) subproblem, both of which sit at the computational kernels of two multi-view subspace learning models. In particular, we also demonstrate MvPS on several real world data sets.

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Notes

  1. Computing the SVD of an \(n\times k\) matrix takes \(6nk^2+20k^3\) flops [18, p.493].

  2. A MATLAB toolbox for OPTimization on MANifolds available online at https://www.manopt.org/.

  3. A toolbox for STiefel manifold OPtimization available online at https://stmopt.gitee.io/.

  4. All numerical demonstrations in this paper were done in MATLAB on a laptop with Intel(R) Core(TM) i7-1165G7 CPU 2.80 GHz and 32GB memory, except those in Sect. 5.2.2 which were performed on an EXXACT workstation (www.exxactcorp.com).

  5. Up to this point, \(X_j\) as in (1.1) has been used for an orthonormal projection matrix. For the rest of this section, we will use them as data matrices as is done conventionally. Hopefully, no confusion will arise.

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Acknowledgements

The authors wish to thank the two anonymous referees for their constructive suggestions that greatly improved the presentation of this paper. They are indebted to Prof. M. Overton of New York University for his numerous minor but important corrections across the manuscript. Wang was supported in part by NSF DMS-2009689; Zhang was supported in part by the National Natural Science Foundation of China NSFC-12071332; Li was supported in part by NSF DMS-1719620 and DMS-2009689.

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Authors and Affiliations

  1. Department of Mathematics, University of Texas at Arlington, P.O. Box 19408, Arlington, TX, 76019-0408, USA

    Li Wang & Ren-Cang Li

  2. School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu, 215006, China

    Lei-Hong Zhang

  3. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Kowloon, Hong Kong

    Ren-Cang Li

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  1. Li Wang
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Wang, L., Zhang, LH. & Li, RC. Maximizing sum of coupled traces with applications. Numer. Math. 152, 587–629 (2022). https://doi.org/10.1007/s00211-022-01322-y

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