Practical approximation algorithms for ℓ_1 -regularized sparse rank-1 approximation to higher-order tensors

Two approximation algorithms are proposed for ℓ_1 -regularized sparse rank-1 approximation to higher-order tensors. The algorithms are based on multilinear relaxation and sparsification, which are easily implemented and well scalable. In particular, the second one scales linearly with the size of the input tensor. Based on a careful estimation of the ℓ_1 -regularized sparsification, theoretical approximation lower bounds are derived. Our theoretical results also suggest an explicit way of choosing the regularization parameters. Numerical examples are provided to verify the proposed algorithms.