L1 minimization using recursive reduction of dimensionality

In this paper, L1 minimization refers to finding the minimum L1-norm solution to an overdetermined linear system y = X·p. The underdetermined variant of the same problem has recently received much attention, mainly due to the new compressive sensing theory that shows, under wide conditions, the minimum L1-norm solution is also the sparsest solution to the system of linear equations. Overdetermined case is mostly related to system identification, linear regression or robust adaptation. In the robust wavelet adaptation, it has been shown that it also leads to sparse solutions. Although the underlying problem is a linear program, conventional algorithms suffer from poor scalability for big data problems. In this paper, we provide an L1 minimization method that recursively reduces and increases dimensionality of the observed subspace and uses weighted median to efficiently find the global minimum. It overperforms state-of-the art competitive methods when the number of equations is very high, and the number of unknown parameters is relatively small. It is often the case in parametric modelling of multidimensional data sets. In particular, we give examples of sliding-window robust system identification and L1 regression using the proposed method. MATLAB implementations of the algorithms described in this paper have been made publicly available.