In this paper, we introduce several geometric characterizations for strong minima of optimization problems. Applying these results to nuclear norm minimization problems allows us to obtain new necessary and sufficient quantitative conditions for this important property. Our characterizations for strong minima are weaker than the Restricted Injectivity and Nondegenerate Source Condition, which are usually used to identify solution uniqueness of nuclear norm minimization problems. Consequently, we obtain the minimum (tight) bound on the number of measurements for (strong) exact recovery of low-rank matrices.

翻译：本文针对优化问题中的强最小值引入了几种几何刻画。将这些结果应用于核范数极小化问题，使我们能够为该重要性质建立新的充要定量条件。我们提出的强最小值几何刻画条件比通常用于识别核范数极小化问题解唯一性的限制性单射性和非退化源条件更弱。基于此，我们获得了低秩矩阵（强）精确恢复所需的最少（紧）测量数下界。