L0-regularization for high-dimensional regression with corrupted data

Corrupted data appears widely in many contemporary applications including voting behavior, high-throughput sequencing and sensor networks. In this article, we consider the sparse modeling via L-0-regularization under the framework of high-dimensional measurement error models. By utilizing the techniques of the nearest positive semi-definite matrix projection, the resulting regularization problem can be efficiently solved through a polynomial algorithm. Under some interpretable conditions, we prove that the proposed estimator can enjoy comprehensive statistical properties including the model selection consistency and the oracle inequalities. In particular, the nonoptimality of the logarithmic factor of dimensionality will be showed in the oracle inequalities. We demonstrate the effectiveness of the proposed method by simulation studies.