PAC-Bayesian bounds for Principal Component Analysis in Hilbert spaces
PAC-Bayesian bounds for Principal Component Analysis in Hilbert spaces
复制标题
希尔伯特空间中主成分分析的 PAC-贝叶斯界限
DOI：
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发表时间：
2015
期刊：
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通讯作者：
Ilaria Giulini
中科院分区：
文献类型：
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作者：
Ilaria Giulini
基于协方差矩阵的一些新的鲁棒估计量，我们提出了主成分分析（PCA）的稳定版本，并且我们独立于环境空间的维度对其进行限定。我们首先在协方差矩阵的最大特征向量上提供正交投影的鲁棒估计器。这种估计器的行为与协方差矩阵的频谱中的间隙的大小有关，并且特别是需要大的间隙才能获得良好的近似。为了避免假设协方差矩阵的频谱中存在较大的特征间隙，我们提出了 PCA 的稳健版本，其中包括通过 Lipschitz 函数执行频谱的平滑截止。我们根据算子范数和弗罗贝尼乌斯范数提供近似误差的界限。
Based on some new robust estimators of the covariance matrix, we propose stable versions of Principal Component Analysis (PCA) and we qualify it independently of the dimension of the ambient space. We first provide a robust estimator of the orthogonal projector on the largest eigenvectors of the covariance matrix. The behavior of such an estimator is related to the size of the gap in the spectrum of the covariance matrix and in particular a large gap is needed in order to get a good approximation. To avoid the assumption of a large eigengap in the spectrum of the covariance matrix we propose a robust version of PCA that consists in performing a smooth cut-off of the spectrum via a Lipschitz function. We provide bounds on the approximation error in terms of the operator norm and of the Frobenius norm.