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Principal Component Analysis (PCA) is used to reduce n-dimensional data to k-dimensional data to speed things up in machine learning. After PCA is applied, one can check how much of the variance of the original dataset remains in the resulting dataset. A common goal is keeping variance between 90% and 99%.

Is it considered a good practice to try different values of the k parameter (size of the resulting dataset's dimension) and then check the results of the resulting models against some cross-validation dataset in the same way as we do to pick good values of other hyperparameters like regularization lambdas and thresholds?

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Unless you have reason to believe on a prior number of components (e.g. a hypothesis saying the number of dimensions must me around $K$ or so) tuning it is perfectly valid. Indeed, there's is recent related research, for example: Selecting the number of principal components (Choi et. al, 2015)

This has also been discussed here. This post has a nice approach in Python, and this function in a R library has a simple implementation for a tuning routine.

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