Randomized algorithms for tensor response regression

In this paper, we consider the estimation algorithm of tensor response on vector covariate regression model. Based on projection theory of tensor and the idea of randomized algorithm for tensor decomposition, three new algorithms named SHOLRR, RHOLRR and RSHOLRR are proposed under the low‐rank Tucker decomposition and some theoretical analyses for two randomized algorithms are also provided. To explore the nonlinear relationship between tensor response and vector covariate, we develop the KRSHOLRR algorithm based on kernel trick and RSHOLRR algorithm. Our proposed algorithms can not only guarantee high estimation accuracy but also have the advantage of fast computing speed, especially for higher‐order tensor response. Through extensive synthesized data analyses and applications to two real datasets, we demonstrate the outperformance of our proposed algorithms over the stat‐of‐art.