Computation of outer inverses of tensors using the QR decomposition
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Computation of outer inverses of tensors using the QR decomposition Jajati Keshari Sahoo1 Vasilios N. Katsikis4
· Ratikanta Behera2
· Predrag S. Stanimirovi´c3
·
Received: 24 April 2020 / Revised: 4 June 2020 / Accepted: 10 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this paper, we introduce new representations and characterizations of the outer inverse of tensors through QR decomposition. Derived representations are usable in generating corresponding representations of main tensor generalized inverses. Some results on reshape operation of a tensor are added to the existing theory. An effective algorithm for computing outer inverses of tensors is proposed and applied. The power of the proposed method is demonstrated by its application in 3D color image deblurring. Keywords Outer inverse · Generalized inverse · QR Decomposition · Image deblurring · Einstein product Mathematics Subject Classifications 15A09 · 15A10 · 15A69
Communicated by Jinyun Yuan.
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Predrag S. Stanimirovi´c [email protected] Jajati Keshari Sahoo [email protected] Ratikanta Behera [email protected] Vasilios N. Katsikis [email protected]
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Department of Mathematics, Birla Institute of Technology & Science Pilani-K.K.Birla Goa Campus, Zuarinagar, Goa, India
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Department of Mathematics, University of Central Florida, Orlando, Florida 32816, USA
3
Department of Computer Science, Faculty of Science and Mathematics, University of Niš, Niš, Serbia
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Department of Economics, Division of Mathematics and Informatics, National and Kapodistrian University of Athens, Athens, Greece 0123456789().: V,-vol
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1 Introduction The set of all order n complex (real) tensor is denoted by C I1 ×···×In (resp. R I1 ×···×In ). The element of the tensor represents Ai1 ...in . The summation of tensors A, B ∈ C I1 ×···×In ×J1 ×···×Jn is defined as: (A + B)i1 ...in j1 ... jn = Ai1 ...in j1 ... jn + Bi1 ...in j1 ... jn . (1.1) However, there is no unique way to represent product of two tensors. Different tensor products have been investigated frequently (Sun et al. 2016). This paper will concentrate on one such class of product ’Einstein product’ (see Einstein 1916; Lai et al. 2009). The Einstein product of A ∈ C I1 ×···×In ×K 1 ×···×K n and B ∈ C K 1 ×···×K n ×J1 ×···×Jm is denoted as A∗n B and is defined by: Ai1 ...in k1 ...kn Bk1 ...kn j1 ... jm ∈ C I1 ×···×In ×J1 ×···×Jm . (1.2) (A∗n B)i1 ...in j1 ... jm = k1 ,...,kn
Inverses and generalized inverses of matrices and tensors have significantly impacted many areas of theoretical and computational mathematics (Sun et al. 2018; Behera et al. 2019; Sahoo et al. 2020). Brazell et al. (2013) introduced the tensor inverse via the Einstein product. Then, Sun et al. (2016) and subsequently Behera and Mishra (2017) extended to the generalized inverse of tensor, since tensor decomposition is necessary to develop an efficient algorithm for solving multilinear systems. Liang et al. (2019) discussed
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