Geometrizing Local Rates of Convergence for Linear Inverse Problems

Ill-posed inverse problems ($p \gg n$, parameter dimension much larger than sample size) with additional low complexity structures pose many challenges for engineers, applied mathematicians and statisticians. Over the past decade, techniques based on Dantzig selector, Lasso and nuclear norm minimization have been developed to attack each problem independently. We consider the general linear inverse problem in the noisy setting, and are interested in statistical estimation rate. Our contributions are two-fold. First, we propose a general computationally feasible convex program that provides near optimal rate of convergence simultaneously in a collection of problems. Second, a simple and clean theoretical framework is built based on local geometry and duality. We also study minimax lower bounds for linear inverse problems restricted to a general convex cone. The local rate of convergence is captured by the complexity of the local tangent cone through geometric quantities such as Gaussian width, volume ratio and Sudakov minoration.