内点算法的平滑分析：条件数

内点算法的光滑分析：终止

Smoothed Analysis of Interior-Point Algorithms: Termination

我们进行平滑分析的终止阶段的一个通道点方法。结合Dunagan，Spielman和Teng对Renegar的邻域点算法的光滑分析，我们证明了线性规划的邻域点算法的光滑复杂性为$O (m^{3} \log (m/σ))$ .相比之下，线性规划的最坏情况复杂度的最佳已知界限是$O (m^{3} L)$，其中$L$可以与$m$一样大。我们包括一个介绍平滑分析和证明技术，已在平滑分析有用的教程。

矩阵条件数与增长因子的平滑分析

Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices

本文通过平滑分析证明，任意矩阵的轻微随机扰动具有大条件数和大增长因子的概率很低，从而改进了无主元高斯消去法的精度需求上界。

Optimal Smoothed Analysis and Quantitative Universality for the Smallest Singular Value of Random Matrices

The smallest singular value and condition number play important roles in numerical linear algebra and the analysis of algorithms. In numerical analysis with randomness, many previous works make Gaussian assumptions, which are not general enough to reflect the arbitrariness of the input. To overcome this drawback, we prove the first quantitative universality for the smallest singular value and condition number of random matrices. Moreover, motivated by the study of smoothed analysis that random perturbation makes deterministic matrices well-conditioned, we consider an analog for random matrices…

Smoothed analysis of the condition number under low-rank perturbations

Let $M$ be an arbitrary $n$ by $n$ matrix of rank $n-k$. We study the condition number of $M$ plus a \emph{low-rank} perturbation $UV^T$ where $U, V$ are $n$ by $k$ random Gaussian matrices. Under some necessary assumptions, it is shown that $M+UV^T$ is unlikely to have a large condition number. The main advantages of this kind of perturbation over the well-studied dense Gaussian perturbation, where every entry is independently perturbed, is the $O(nk)$ cost to store $U,V$ and the $O(nk)$ increase in time complexity for performing the matrix-vector multiplication $(M+UV^T)x$. This improves the…

The smoothed complexity of Frank-Wolfe methods via conditioning of random matrices and polytopes

Frank-Wolfe methods are popular for optimization over a polytope. One of the reasons is because they do not need projection onto the polytope but only linear optimization over it. To understand its complexity, Lacoste-Julien and Jaggi introduced a condition number for polytopes and showed linear convergence for several variations of the method. The actual running time can still be exponential in the worst case (when the condition number is exponential). We study the smoothed complexity of the condition number, namely the condition number of small random perturbations of the input polytope and …

对称张量的欠完全分解的线性时间算法及条件数的平滑分析

Undercomplete Decomposition of Symmetric Tensors in Linear Time, and Smoothed Analysis of the Condition Number

本文研究对称张量的欠完全分解，提出首个线性时间随机化算法，在输入张量接近精确张量时以高概率恢复分解向量，并通过条件数的平滑分析证明算法在除罕见病态输入外均在线性时间内运行。