arXiv 1607.060172016-07math.OCcs.DScs.LGstat.ML

快速CCA和广义特征分解的Doubly加速方法

Doubly Accelerated Methods for Faster CCA and Generalized Eigendecomposition

Institute for Advanced Study（高等研究院）；Princeton University（普林斯顿大学）

Zeyuan Allen-Zhu, Yuanzhi Li

备注：: We have now stated more clearly why this paper has outperformed relevant previous results, and included discussions for doubly-stochastic methods. arXiv admin note: text overlap with arXiv:1607.03463

摘要：

我们研究了$k$ -GenEV，找到顶部$k$广义特征向量的问题，和$k$ -CCA，在典型相关分析中找到顶部$k$向量的问题。我们提出了算法$\mathtt{LazyEV}$和$\mathtt{LazyCCA}$来解决这两个问题的运行时间线性依赖于输入大小和$k$。此外，我们的算法是双重加速的：我们的运行时间只取决于矩阵条件数的平方根，和特征间隙的平方根。这是$k$ -GenEV或$k$ -CCA的第一个此类结果。我们还提供了第一个无间隙的结果，它提供的运行时间依赖于$1/\sqrt{\varepsilon}$，而不是本征间隙。

英文摘要：

We study $k$-GenEV, the problem of finding the top $k$ generalized eigenvectors, and $k$-CCA, the problem of finding the top $k$ vectors in canonical-correlation analysis. We propose algorithms $\mathtt{LazyEV}$ and $\mathtt{LazyCCA}$ to solve the two problems with running times linearly dependent on the input size and on $k$. Furthermore, our algorithms are DOUBLY-ACCELERATED: our running times depend only on the square root of the matrix condition number, and on the square root of the eigengap. This is the first such result for both $k$-GenEV or $k$-CCA. We also provide the first gap-free results, which provide running times that depend on $1/\sqrt{\varepsilon}$ rather than the eigengap.

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