Tensor Manifold with Tucker Rank Constraints

Shih Yu Chang, Ziyan Luo () and Liqun Qi ()
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Shih Yu Chang: Department of Applied Data Science, San Jose State University, San Jose, CA, USA
Ziyan Luo: Department of Mathematics, Beijing Jiaotong University, Haidian District, Beijing, P. R. China
Liqun Qi: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

Asia-Pacific Journal of Operational Research (APJOR), 2022, vol. 39, issue 02, 1-30

Abstract: Low-rank tensor approximation plays a crucial role in various tensor analysis tasks ranging from science to engineering applications. There are several important problems facing low-rank tensor approximation. First, the rank of an approximating tensor is given without checking feasibility. Second, even such approximating tensors exist, however, current proposed algorithms cannot provide global optimality guarantees. In this work, we define the low-rank tensor set (LRTS) for Tucker rank which is a union of manifolds of tensors with specific Tucker rank. We propose a procedure to describe LRTS semi-algebraically and characterize the properties of this LRTS, e.g., feasibility of tensors manifold, the equations/inequations size of LRTS, algebraic dimensions, etc. Furthermore, if the cost function for tensor approximation is polynomial type, e.g., Frobenius norm, we propose an algorithm to approximate a given tensor with Tucker rank constraints and prove the global optimality of the proposed algorithm through critical sets determined by the semi-algebraic characterization of LRTS.

Keywords: Tucker decomposition; Tucker tensor rank; low-rank tensor approximation; semialgebraic set; polynomial optimization (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1142/S0217595921500226

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