A sparse system identification by using adaptively-weighted total variation via a primal-dual splitting approach

Observing that sparse systems are almost smooth, we propose to utilize the newly-introduced adaptively-weighted total variation (AWTV) for sparse system identification. In our formulation, a sparse system identification problem is posed as a sequential suppression of a time-varying cost function: the sum of AWTV and a data-fidelity term. In order to handle such a non-differentiable cost function efficiently, we propose a time-varying extension of a primal-dual splitting type algorithm, named the adaptive primal-dual splitting method (APDS). APDS is free from operator inversion or other highly complex operations, resulting in computationally efficient implementation in online manner. Moreover, APDS realizes that the sequence defined in a certain product space monotonically approaches the solution set of the current cost function, i.e., the sequence generated by APDS pursues desired replicas of the unknown system in each time-step. Our scheme is applied to a network echo cancellation problem where it shows excellent performance compared with conventional methods.