New approach to Bayesian high-dimensional linear regression

Consider the problem of estimating parameters X-n is an element of R-n, from m response variables Y-m = AX(n) + Z(m), under the assumption that the distribution of X-n is known. Lack of computationally feasible algorithms that employ generic prior distributions and provide a good estimate of X-n has limited the set of distributions researchers use to model the data. To address this challenge, in this article, a new estimation scheme named quantized maximum a posteriori (Q-MAP) is proposed. The new method has the following properties: (i) In the noiseless setting, it has similarities to maximum a posteriori (MAP) estimation. (ii) In the noiseless setting, when X-1, ... , X-n are independent and identically distributed, asymptotically, as n grows to infinity, its required sampling rate (m/n) for an almost zero-distortion recovery approaches the fundamental limits. (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 that is provably robust to error (noise) in response variables.