T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming
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T‑positive semidefiniteness of third‑order symmetric tensors and T‑semidefinite programming Meng‑Meng Zheng1 · Zheng‑Hai Huang1 · Yong Wang1 Received: 3 February 2020 / Accepted: 15 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The T-product for third-order tensors has been used extensively in the literature. In this paper, we first introduce first-order and second-order T-derivatives for the multivariable real-valued function with the tensor T-product. Inspired by an equivalent characterization of a twice continuously T-differentiable multi-variable real-valued function being convex, we present a definition of the T-positive semidefiniteness of third-order symmetric tensors. After that, we extend many properties of positive semidefinite matrices to the case of third-order symmetric tensors. In particular, analogue to the widely used semidefinite programming (SDP for short), we introduce the semidefinite programming over the space of third-order symmetric tensors (T-semidefinite programming or TSDP for short), and provide a way to solve the TSDP problem by converting it into an SDP problem in the complex domain. Furthermore, we give several TSDP examples and especially some preliminary numerical results for two unconstrained polynomial optimization problems. Experiments show that finding the global minimums of polynomials via the TSDP relaxation outperforms the traditional SDP relaxation for the test examples. Keywords T-product · T-positive semidefiniteness · T-semidefinite cone · T-semidefinite programming · Polynomial optimization Mathematics Subject Classification 15A69 · 90C22
* Zheng‑Hai Huang [email protected] Meng‑Meng Zheng [email protected] Yong Wang [email protected] 1
School of Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China
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1 Introduction With the availability of inexpensive storage and advances in instrumentation, the data collected and stored now is more complex than ever before. Especially in practical problems such as psychometrics, signal processing, computer vision, data mining, graphical analysis, neuroscience and so on, it is usually necessary to store information in a multidimensional array, and then use the multidimensional structure to compress, sort, and/or manipulate the data. Among the many problems described by high-dimensional arrays (or tensors), third-order tensors have become increasingly prevalent in recent years with the emergence of the tensor T-product, which is a new type of multiplication between third-order tensors introduced by Kilmer, Martin, and Perrone [1]. The tensor T-product has shown to be a useful tool arising in a wide variety of application areas, including, but not limited to, image processing [2–7], computer vision [8–12], signal processing, low rank tensor recovery and robust tensor PCA [13–18], and data completion and denoising [19–31], because the tensor T-product provides an effective approach to transform the tensor
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