  This paper presents a unified geometric framework for the statistical
analysis of a general ill-posed linear inverse model which includes as special
cases noisy compressed sensing, sign vector recovery, trace regression,
orthogonal matrix estimation, and noisy matrix completion. We propose
computationally feasible convex programs for statistical inference including
estimation, confidence intervals and hypothesis testing. A theoretical
framework is developed to characterize the local estimation rate of convergence
and to provide statistical inference guarantees. Our results are built based on
the local conic geometry and duality. The difficulty of statistical inference
is captured by the geometric characterization of the local tangent cone through
the Gaussian width and Sudakov minoration estimate.
