New approach to Bayesian high-dimensional linear regression

摘要

Consider the problem of estimating parameters $X^n in mathbb{R}^n $,generated by a stationary process, from $m$ response variables $Y^m =AX^n+Z^m$, under the assumption that the distribution of $X^n$ is known. Thisis the most general version of the Bayesian linear regression problem. The lackof computationally feasible algorithms that can employ generic priordistributions and provide a good estimate of $X^n$ has limited the set ofdistributions researchers use to model the data. In this paper, a new schemecalled Q-MAP is proposed. The new method has the following properties: (i) Ithas similarities to the popular MAP estimation under the noiseless setting.(ii) In the noiseless setting, it achieves the "asymptotically optimalperformance" when $X^n$ has independent and identically distributed components.(iii) It scales favorably with the dimensions of the problem and therefore isapplicable to high-dimensional setups. (iv) The solution of the Q-MAPoptimization can be found via a proposed iterative algorithm which is provablyrobust to the error (noise) in the response variables.