Randomized Robust Subspace Recovery and Outlier Detection for High Dimensional Data Matrices

This paper explores and analyzes two randomized designs for robust principal component analysis employing lowdimensional data sketching. In one design, a data sketch is constructed using random column sampling followed by lowdimensional embedding, while in the other, sketching is based on random column and rowsampling. Both designs are shown to bring about substantial savings in complexity andmemory requirements for robust subspace learning over conventional approaches that use the full scale data. A characterization of the sample and computational complexity of both designs is derived in the context of two distinct outliermodels, namely, sparse and independent outlier models. The proposed randomized approach can provably recover the correct subspace with computational and sample complexity which depend only weakly on the size of the data (only through the coherence parameters). The results of the mathematical analysis are confirmed through numerical simulations using both synthetic and real data.