New approach to Bayesian high-dimensional linear regression
Consider the problem of estimating parameters Xn E Rn, generated by a stationary process, from m response variables Y m = AXn + Zm, under the assumption that the distribution of Xn is known. This is the most general version of the Bayesian linear regression problem. The lack of computationally feasible algorithms that can employ generic prior distributions and provide a good estimate of Xn has limited the set of distributions researchers use to model the data. In this paper, a new scheme called Q-MAP is proposed. The new method has the following properties: (i) It has similarities to the popular MAP estimation under the noiseless setting. (ii) In the noiseless setting, it achieves the asymptotically optimal performance when Xn has independent and identically distributed components. (iii) It scales favorably with the dimensions of the problem and therefore is applicable to high-dimensional setups. (iv) The solution of the Q-MAP optimization can be found via a proposed iterative algorithm which is provably robust to the error (noise) in the response variables.