Large datasets create opportunities as well as analytic challenges. A recent development is to use random projection or sketching methods for dimension reduction in statistics and machine learning. In this work, we study the statistical performance of sketching algorithms for linear regression. Suppose we randomly project the data matrix and the outcome using a random sketching matrix reducing the sample size, and do linear regression on the resulting data. How much do we lose compared to the original linear regression? The existing theory does not give a precise enough answer, and this has been a bottleneck for using random projections in practice. In this paper, we introduce a new mathematical approach to the problem, relying on very recent results from asymptotic random matrix theory and free probability theory. This is a perfect fit, as the sketching matrices are random in practice. We allow the dimension and sample sizes to have an arbitrary ratio. We study the most popular sketch 查看全文>>