Fast SVD free low-rank matrix recovery: Application to dynamic MRI reconstruction

This study proposes a new algorithm for recovering low-rank matrices from their under-sampled projections. Such algorithms are traditionally based on the Schatten-p (0<;p≤1) norm minimization. The minimization problem is solved directly, requiring the computing of a singular value decomposition (SVD) at each iteration. This is time consuming and greatly limits the speed of the algorithms and its applicability to real life problems. To overcome this problem, we replace the Schatten-p norm by its equivalent Ky-Fan norm. For minimizing the said norm, we derive an algorithm that does not require computing SVD´s. Instead, it computes a Cholesky decomposition - which requires many less computations than SVD. Our method yields an order of magnitude improvement in speed over existing techniques. We apply our proposed algorithm on the dynamic MRI reconstruction problem and obtain significant improvement in computational speed over the existing technique.