Let's consider a scenario where we have $n$ observations with measurements on a set of $p$ features. PCA aims to discover a low-dimensional representation of the dataset that retains as much variation as possible. The underlying idea is that each of the $n$ observations exist in a $p$-dimensional space, but not all dimensions are equally informative. PCA seeks a small number of dimensions that capture the most interesting aspects, with interestingness measured by the amount of variability exhibited by the observations along each dimension.
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