Tensor Manifold with Tucker Rank Constraints

Publication Date

4-1-2022

Document Type

Article

Publication Title

Asia-Pacific Journal of Operational Research

Volume

39

Issue

2

DOI

10.1142/S0217595921500226

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.

Funding Number

11771038

Funding Sponsor

National Natural Science Foundation of China

Keywords

low-rank tensor approximation, polynomial optimization, semialgebraic set, Tucker decomposition, Tucker tensor rank

Department

Applied Data Science

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