Geometric Inference for General High-Dimensional Linear Inverse Problems
This provides a foundational approach for researchers in statistics and machine learning dealing with ill-posed inverse problems, though it is incremental in extending geometric methods to a broader class of models.
The paper tackles the problem of statistical inference for general high-dimensional linear inverse problems, including compressed sensing and matrix completion, by developing a unified geometric framework and convex programs, achieving theoretical guarantees for estimation rates and inference.
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.