This paper addresses the question of learning structured distributions from batches when a constant fraction of the batches might be corrupted. This problem has been of considerable recent interest. This paper studies the setting where the underlying distribution has additional structure (namely, piece polynomial density function), in which case more sample efficient algorithms are possible. This paper develops sample and computationally efficient algorithms for such settings. The reviewers were convinced that this paper makes important technical contributions in extending recent work on this problem to the structured setting. I recommend accepting this paper.