It was recently shown [7, 9] that "properly built" linear and polyhedral estimates nearly attain minimax accuracy bounds in the problem of recovery of unknown signal from noisy observations of linear images of the signal when the signal set is an ellitope. However, design of nearly optimal estimates relies upon solving semidefinite optimization problems with matrix variables, what puts the synthesis of such estimates beyond the rich of the standard Interior Point algorithms of semidefinite optimization even for moderate size recovery problems. Our goal is to develop First Order Optimization algorithms for the computationally efficient design of linear and polyhedral estimates. In this paper we (a) explain how to eliminate matrix variables, thus reducing dramatically the design dimension when passing from Interior Point to First Order optimization algorithms and (2) develop and analyse a dedicated algorithm of the latter type -- Composite Truncated Level method.

翻译：最新研究[7, 9]表明，当信号集为椭球体时，适当构造的线性与多面体估计在从含噪观测中恢复未知信号的问题中几乎达到极小极大精度界。然而，近优估计的设计依赖于求解含矩阵变量的半定优化问题，这使得即便针对中等规模恢复问题，此类估计的合成也已超出标准半定优化内点算法的能力范围。本文旨在开发用于高效计算线性与多面体估计的一阶优化算法，具体贡献包括：(a) 阐明如何消除矩阵变量，从而在从内点算法过渡到一阶优化算法时显著降低设计维度；(b) 开发并分析专用的一阶算法——复合截断水平方法。