Multi-Relational Data Characterization by Tensors: Tensor Inversion

Tensors have been applied extensively in engineering and data analytics. Pertinent problems include dimensionality reduction, data mining, data modeling, data learning, etc. Recent research attention has been paid to solve tensor equations. Existing solutions to tensor equations are mostly based on the iterative approach due to lack of sufficient theoretical framework governing how to find the inverse of an arbitrary tensor. In this work, we aim to establish a new theoretical framework missing from the literature so that a new algorithm is devised to determine the exact inverse of an arbitrary tensor, which is beyond the capability of the current iterative algorithms. To solve tensor equations, we classify a tensor as an invertible tensor or a pseudo invertible tensor in the Moore-Penrose’s sense. We present theorems to derive the general formula of both inverse and pseudo inverse of an arbitrary tensor so that the inverse of an arbitrary tensor can be constructed from the tensor itself and its partial inverse. A new tensor inversion algorithm is introduced to carry out the exact inverse or the Moore-Penrose inverse should it not be invertible. Thus, the solution to a tensor equation can be obtained immediately if such an inverse or pseudo inverse is calculated. The main contribution of our proposed approach is that we can always solve any tensor equation while additional restrictions have to be imposed for the existing iterative algorithms to converge on the other hand. The memory- and computational-complexities of our proposed new approach and existing iterative algorithms are also analyzed and compared. In order to demonstrate the applicability of our proposed tensor-inversion algorithm, we apply the tensor-inversion algorithm to the multi-relational data query problem for studying real-world web data. Moreover, numerical experiments are performed to demonstrate the effectiveness of our proposed new approach in terms of convergence together with memory- and computational-complexities.