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 [Formula: see text] and [Formula: see text] 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 [Formula: see text]. 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 [Formula: see text] 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 [Formula: see text] or large [Formula: see text] constrained multidimensional, sparse, least-squares problems (underdetermined or overdetermined) for all critical values of the regularization parameter [Formula: see text] 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 [Formula: see text] and [Formula: see text] 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 [Formula: see text]-sparse solutions for multilinear leastsquares problems. However, the [Formula: see text]-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.
A Method for Finding Structured Sparse Solutions to Nonnegative Least Squares Problems with Applications