Smoothed analysis is a powerful paradigm in overcoming worst-case intractability in high-dimensional data analysis and unsupervised learning. While polynomial time smoothed analysis guarantees have been obtained for worst-case intractable problems like tensor decomposition and learning mixtures of Gaussians, such guarantees have been hard to obtain for several other important problems in data analysis. A core technical challenge in analyzing algorithms is obtaining lower bounds on the least singular value for random matrix ensembles with dependent entries, that are given by low-degree polynomials of a few base underlying random variables. In this work, we address this challenge by obtaining high-confidence lower bounds on the least singular value of new classes of structured random matrix ensembles of the above kind. We then use these bounds to design algorithms with polynomial time smoothed analysis guarantees for the following three important problems in high-dimensional data analysis:

中文翻译：

---

无监督学习中张量方法的平滑分析

平滑分析是克服高维数据分析和无监督学习中最坏情况难以处理的强大范例。虽然对于最坏情况下的棘手问题（如张量分解和高斯学习混合）已经获得多项式时间平滑分析保证，但对于数据分析中的其他几个重要问题，很难获得这种保证。分析算法的一个核心技术挑战是获得具有相关项的随机矩阵系综的最小奇异值的下限，这些项由几个基础随机变量的低次多项式给出。在这项工作中，我们通过获得上述类型的结构化随机矩阵集合的新类别的最小奇异值的高置信度下界来解决这一挑战。