Sparse Equisigned PCA: Algorithms and Performance Bounds in the Noisy Rank-1 Setting
Sparse Equisigned PCA: Algorithms and Performance Bounds in the Noisy Rank-1 Setting
复制标题
DOI：
10.1214/19-ejs1657
复制
发表时间：
2019-05
期刊：
影响因子：
--
通讯作者：
Arvind Prasadan;R. Nadakuditi;D. Paul
中科院分区：
文献类型：
--
作者：
Arvind Prasadan;R. Nadakuditi;D. Paul
基于奇异值分解（SVD）的主成分分析（PCA）在高维和有限的样本大小范围内分解，低于取决于系统的维数和样本数量的特定临界特征SNR。低于这个临界特征SNR，SVD返回的估计值与潜在主成分渐近不相关。我们考虑一种设置，其中底层秩一信号矩阵的左奇异向量被假设为稀疏的，并且右奇异向量被假设为等符号的，即，具有仅非负或仅非正的条目。我们考虑六种不同的算法估计稀疏主成分基于不同的统计标准，并证明，通过利用稀疏性，我们恢复一致的估计在低特征信噪比制度的SVD失败。我们的分析揭示了条件下，坐标选择方案的基础上的\textit{和型决策统计}优于计划，利用$\ell_1$和$\ell_2$基于范数的统计。我们推导出的主要左奇异向量的可检测坐标的大小的下界，并利用这些下界推导出最坏情况下的风险的下界。最后，我们用数值模拟验证了我们的研究结果，并用视频数据示例说明了性能，其中兴趣是识别对象。
Singular value decomposition (SVD) based principal component analysis (PCA) breaks down in the high-dimensional and limited sample size regime below a certain critical eigen-SNR that depends on the dimensionality of the system and the number of samples. Below this critical eigen-SNR, the estimates returned by the SVD are asymptotically uncorrelated with the latent principal components. We consider a setting where the left singular vector of the underlying rank one signal matrix is assumed to be sparse and the right singular vector is assumed to be equisigned, that is, having either only nonnegative or only nonpositive entries. We consider six different algorithms for estimating the sparse principal component based on different statistical criteria and prove that by exploiting sparsity, we recover consistent estimates in the low eigen-SNR regime where the SVD fails. Our analysis reveals conditions under which a coordinate selection scheme based on a \textit{sum-type decision statistic} outperforms schemes that utilize the $\ell_1$ and $\ell_2$ norm-based statistics. We derive lower bounds on the size of detectable coordinates of the principal left singular vector and utilize these lower bounds to derive lower bounds on the worst-case risk. Finally, we verify our findings with numerical simulations and illustrate the performance with a video data example, where the interest is in identifying objects.