Compressed Least-Squares Regression

We consider the problem of learning, from K data, a regression function in a linear space of high dimensionN using projections onto a random subspace of lower dimension M . From any algorithm minimizing the (possibly penalized) empirical risk, we provide bounds on the excess risk of the estimate computed in the projected subspace (compressed domain) in terms of the excess risk of the estimate built in the high-dimensional space (initial domain). We show that solving the problem in the compressed domain instead of the initial domain reduces the estimation error at the price of an increased (but controlled) approximation error. We apply the analysis to Least-Squares (LS) regression and discuss the excess risk and numerical complexity of the resulting “Compressed Least Squares Regression” (CLSR) in terms of N , K, and M . When we choose M = O( √ K), we show that CLSR has an estimation error of order O(logK/ √ K).