An Augmented Regression Model for Tensors With Missing Values

Heterogeneous but complementary sources of data provide an unprecedented opportunity for developing accurate statistical models of systems. Although the existing methods have shown promising results, they are mostly applicable to situations where the system output is measured in its complete form. In reality, however, it may not be feasible to obtain the complete output measurement of a system, which results in observations that contain missing values. This article introduces a general framework that integrates tensor regression with tensor completion and proposes an efficient optimization framework that alternates between two steps for parameter estimation. Through multiple simulations and a case study, we evaluate the performance of the proposed method. The results indicate the superiority of the proposed method in comparison to a benchmark. Note to Practitioners—The proposed method aims to obtain an accurate estimation of the regression model when certain entries of the response are inaccessible. By considering both the information from multiple inputs and the structure of the response, our proposed method can achieve more accurate estimation of the output tensor. In order to apply the proposed method in practice, two assumptions should hold. First, the response tensor should be low-rank, meaning that fewer variation patterns should exist in the response than its dimensions. Second, the relationship between the input tensors and the response should be linear or approximately linear. The presented method in this article uses tensor decomposition techniques to exploit the correlation structures of the high-dimensional data and prevent overfitting. Another benefit of our integrated framework is that the rank of the response tensor converges automatically, which can be used directly in the parameter estimation.