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.

中文翻译：

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凸正则化的有效条件数界限

我们推导出将 Renegar 的条件数与在包括 $\ell _{1}$ -analysis 设置在内的设置中控制凸正则化统计性能的数量相关联的界限。使用圆锥积分几何的结果，我们表明可以使边界仅依赖于分析算子对低维空间的随机投影或限制，并且如果这些算子是病态的，则仍然有效。作为一个应用，我们得到了复合凸正则化器欠采样相变的新界限。分析中的关键工具是 Slepian 不等式和积分几何的运动学公式。