Sketch‐and‐project methods for tensor linear systems

Abstract

For tensor linear systems with respect to the popular t-product, we first present the sketch-and-project method and its adaptive variants. Their Fourier domain versions are also investigated. Then, considering that the existing sketching tensor or way for sampling has some limitations, we propose two improved strategies. Convergence analyses for the methods mentioned above are provided. We compare our methods with the existing ones using synthetic and real data.Numerical results show that they have quite decent performance in terms of the number of iterations and running time.

On sketch-and-project methods for solving tensor equations

“…Similar to the previous works [24][25][26], we take the point which is closest to the current iteration X t and solve a sketched version of the oringinal tensor equation (1) as the next iteration X t+1 , that is…”

“…Since the same distributions D S and D V are used in each iteration, this may lead to poor selection of S and V in some iterations, resulting in slow convergence. With this in mind, similar to [25,26], we will give three adaptive sampling strategies, which utilize the information of the current iteration. It is worth mentioning that we will derive these adaptive sampling strategies on two finite sets of sketching tubal matrices preselected from two distributions.…”

“…2 ⌉ in line 8 of Algorithm 5 will be the same. As pointed out in [26,29], it would be better to use different sketching matrices for these ⌈ l+1 2 ⌉ independent matrix equations.…”

“…Similar to [25,26], we implement the nonadaptive and adaptive TESP methods including the NTESP, ATESP-MD, ATESP-PR and ATESP-CS methods in their corresponding fast versions, for example, the fast version of the ATESP-PR method is given in Algorithm 6 in the appendix.…”

“…Furthermore, Du et al [23] proposed the randomized block coordinate descent methods for solving the matrix least-squares problem min X∈R r×s C−AXB F , where • F denotes the Frobenius norm of a matrix. As we know, the randomized Kaczmarz-type and coordinate descent methods can be unified into the sketch-and-project method and its adaptive variants [24][25][26]. So, more specifically, the stochastic iterative methods we consider for solving (1) in this paper are the sketch-and-project methods.…”