In this paper, we revisit the LASSO sparse representation problem, which has been studied and used in a variety of different areas, ranging from signal processing and information theory to computer vision and machine learning. In the vision community, it found its way into many important applications, including face recognition, tracking, super resolution, image denoising, to name a few. Despite advances in efficient sparse algorithms, solving large-scale LASSO problems remains a challenge. To circumvent this difficulty, people tend to downsample and subsample the problem (e.g. via dimensionality reduction) to maintain a manageable sized LASSO, which usually comes at the cost of losing solution accuracy. This paper proposes a novel circulant reformulation of the LASSO that lifts the problem to a higher dimension, where ADMM can be efficiently applied to its dual form. Because of this lifting, all optimization variables are updated using only basic element-wise operations, the most computationally expensive of which is a 1D FFT. In this way, there is no need for a linear system solver nor matrix-vector multiplication. Since all operations in our FFTLasso method are element-wise, the subproblems are completely independent and can be trivially parallelized (e.g. on a GPU). The attractive computational properties of FFTLasso are verified by extensive experiments on synthetic and real data and on the face recognition task. They demonstrate that FFTLasso scales much more effectively than a state-of-the-art solver.
Supplementary material can be found here.