论文标题
通过锥体投影电力迭代在高维度的非sparse PCA
Non-Sparse PCA in High Dimensions via Cone Projected Power Iteration
论文作者
论文摘要
在本文中，我们提出了一种锥体预测的电力迭代算法，以从嘈杂的正半数矩阵中恢复第一个主要特征向量。当假定真正的主特征向量属于凸锥时，提出的算法是快速且具有可拖动误差的。具体而言，该方法可实现具有快速投影（例如单调锥体）的某些凸锥的多项式时间复杂性。当嘈杂的矩阵具有小圆锥限制的操作员规范时，它会达到一个小误差。我们在尖峰协方差模型下使用误差的最小值来补充上述结果。我们在模拟和真实数据上进行的数值实验表明，与普通功率迭代相比，我们的方法达到了较短的运行时间和较小的误差，如果主要特征向量位于凸锥中，则我们的方法与普通功率迭代和一些稀疏的主成分分析算法相比。
In this paper, we propose a cone projected power iteration algorithm to recover the first principal eigenvector from a noisy positive semidefinite matrix. When the true principal eigenvector is assumed to belong to a convex cone, the proposed algorithm is fast and has a tractable error. Specifically, the method achieves polynomial time complexity for certain convex cones equipped with fast projection such as the monotone cone. It attains a small error when the noisy matrix has a small cone-restricted operator norm. We supplement the above results with a minimax lower bound of the error under the spiked covariance model. Our numerical experiments on simulated and real data, show that our method achieves shorter run time and smaller error in comparison to the ordinary power iteration and some sparse principal component analysis algorithms if the principal eigenvector is in a convex cone.
