High-probability bounds for the reconstruction error of PCA
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We identify principal component analysis (PCA) as an empirical risk minimization problem and prove error bounds that hold with high probability. More precisely, we derive upper bounds for the reconstruction error of PCA that can be expressed relative to the minimal reconstruction error. The significance of these bounds is shown for the cases of functional and kernel PCA. In such scenarios, the eigenvalues of the covariance operator often decay at a polynomial or nearly exponential rate. Our results yield that the reconstruction error of PCA achieves the same rate as the minimal reconstruction error.
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