Data inference allows to infer data from a partially revealed data set or to detect corrupted data and recover the original data. In general, to infer missing data or to correct corrupted data is theoretically impossible if given no assumption about the data. Under assumptions about the intrinsic features of the data set, algorithms can be developed to recover missing or corrupted data.
We consider data inference problems with a low-rank structure of the data matrix. By exploiting the low-rank feature of the matrix, the data inference problems can be modelled as an L1-norm optimization problem. We propose a framework to solve this kind of problems exploiting optimization theory on Grassmann manifold. We apply this framework to Smart Grid to detect false data injection attack and to predict the QoS of a cloud marketplace. The experiments show our framework achieves a good result under both scenarios.