Tensor Least Angle Regression for Sparse Representations of Multidimensional Signals

Sparse signal representations have gained much interest recently in both signal processing and statistical communities. Compared to orthogonal matching pursuit (OMP) and basis pursuit, which solve the $$ and $$ constrained sparse least-squares problems, respectively, least angle regression (LARS) is a computationally efficient method to solve both problems for all critical values of the regularization parameter $$. However, all of these methods are not suitable for solving large multidimensional sparse least-squares problems, as they would require extensive computational power and memory. An earlier generalization of OMP, known as Kronecker-OMP, was developed to solve the $$ problem for large multidimensional sparse least-squares problems. However, its memory usage and computation time increase quickly with the number of problem dimensions and iterations. In this letter, we develop a generalization of LARS, tensor least angle regression (T-LARS) that could efficiently solve either large $$ or large $$ constrained multidimensional, sparse, least-squares problems (underdetermined or overdetermined) for all critical values of the regularization parameter $$ and with lower computational complexity and memory usage than Kronecker-OMP. To demonstrate the validity and performance of our T-LARS algorithm, we used it to successfully obtain different sparse representations of two relatively large 3D brain images, using fixed and learned separable overcomplete dictionaries, by solving both $$ and $$ constrained sparse least-squares problems. Our numerical experiments demonstrate that our T-LARS algorithm is significantly faster (46 to 70 times) than Kronecker-OMP in obtaining $$-sparse solutions for multilinear leastsquares problems. However, the $$-sparse solutions obtained using Kronecker-OMP always have a slightly lower residual error (1.55% to 2.25%) than ones obtained by T-LARS. Therefore, T-LARS could be an important tool for numerous multidimensional biomedical signal processing applications.