矩阵平衡的随机Osborne算法的近线性收敛
Near-linear convergence of the Random Osborne algorithm for Matrix Balancing
摘要：
我们重新审视矩阵平衡，一个普遍用于计算特征值和矩阵指数的预处理任务。自1960年以来，奥斯本算法一直是从业者的首选算法，现在在大多数数值软件包中实现。然而，它的理论性质还没有得到很好的理解。在这里，我们表明，一个简单的随机变量奥斯本的算法收敛在近线性时间的输入稀疏。具体而言，它在$O(mε^{-2}\logκ)$算术运算之后平衡$K\in\mathbb{R}_{\geq 0}^{n\times n}$，其中$m$是$K$中非零的数目，$ε$是$\ell_1$精度，而$κ=\sum_{ij}K_{ij}/(\min_{ij:K_{ij}\neq 0}K_{ij})$测量$K$的条件。以前的工作已经建立了接近线性的运行时间，要么只为$\ell_2$的准确性（一个较弱的标准，是不太相关的应用程序），或通过一个完全不同的算法（目前）不切实际的拉普拉斯求解器的基础上。我们进一步证明了，如果具有邻接矩阵$K$的图是适度连通的，例如，如果$K$至少有一个正行/列对--那么Osborne的算法最初会以指数速度收敛，从而提高运行时间$O(mε^{-1}\logκ)$。我们还解决了数字精度，显示这些运行时的界限仍然持有时，使用$O(\log(nκ/ε))$位数。我们的研究结果是通过一个直观的潜在的参数，利用凸优化的角度奥斯本的算法，并涉及到每迭代的进展，以目前的不平衡，在海林格距离测量。与以前的分析不同，我们严格利用对数凸的潜力。我们的分析扩展到其他变种奥斯本的算法：一路上，我们建立显着改善循环，贪婪和并行化的变种运行时界限。
英文摘要：
We revisit Matrix Balancing, a pre-conditioning task used ubiquitously for computing eigenvalues and matrix exponentials. Since 1960, Osborne's algorithm has been the practitioners' algorithm of choice and is now implemented in most numerical software packages. However, its theoretical properties are not well understood. Here, we show that a simple random variant of Osborne's algorithm converges in near-linear time in the input sparsity. Specifically, it balances $K\in\mathbb{R}_{\geq 0}^{n\times n}$ after $O(mε^{-2}\logκ)$ arithmetic operations, where $m$ is the number of nonzeros in $K$, $ε$ is the $\ell_1$ accuracy, and $κ=\sum_{ij}K_{ij}/(\min_{ij:K_{ij}\neq 0}K_{ij})$ measures the conditioning of $K$. Previous work had established near-linear runtimes either only for $\ell_2$ accuracy (a weaker criterion which is less relevant for applications), or through an entirely different algorithm based on (currently) impractical Laplacian solvers. We further show that if the graph with adjacency matrix $K$ is moderately connected--e.g., if $K$ has at least one positive row/column pair--then Osborne's algorithm initially converges exponentially fast, yielding an improved runtime $O(mε^{-1}\logκ)$. We also address numerical precision by showing that these runtime bounds still hold when using $O(\log(nκ/ε))$-bit numbers. Our results are established through an intuitive potential argument that leverages a convex optimization perspective of Osborne's algorithm, and relates the per-iteration progress to the current imbalance as measured in Hellinger distance. Unlike previous analyses, we critically exploit log-convexity of the potential. Our analysis extends to other variants of Osborne's algorithm: along the way, we establish significantly improved runtime bounds for cyclic, greedy, and parallelized variants.
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