We derive bounds relating Renegar's condition number to quantities that govern the statistical performance of convex regularization in settings that include the $\ell_1$-analysis setting. Using results from conic integral geometry, we show that the bounds can be made to depend only on a random projection, or restriction, of the analysis operator to a lower dimensional space, and can still be effective if these operators are ill-conditioned. As an application, we get new bounds for the undersampling phase transition of composite convex regularizers. Key tools in the analysis are Slepian's inequality and the kinematic formula from integral geometry.
Heuristic parameter-choice rules for convex variational regularization based on error estimates
Zentrum für Technomathematik, Universität Bremen（不来梅大学技术数学中心）；Institute for Analysis and Algebra, Pockelsstr. 14, D-38118 Braunschweig, Germany（不伦瑞克工业大学分析与代数研究所）
Ground States and Singular Vectors of Convex Variational Regularization Methods
Magnetic Resonance Research Centre, Department of Chemical Engineering and Biotechnology（磁共振研究中心，化学工程与生物系）；Westfälische Wilhelms-Universität Münster, Institut für Numerische und Angewandte Mathematik（明斯特西法利希威廉大学，数值与应用数学研究所）
