I see that kernel PCA with a linear kernel is the same as PCA.
On Wikipedia's introduction of the kernel to PCA they suggest that there exists a non-trivial arbitrary choice of map $\Phi$ that is applied first to the feature space. From this a Gram matrix is constructed using the pairwise inner products of this map $\Phi$ applied to each feature, and then a familiar-looking exercise of SVD is performed.
Do I understand correctly that kernel PCA with a non-linear kernel constructed with the choice of map $\Phi$ is equal to linear PCA with that choice of map $\Phi$ applied first to each of the features?
I suspect they are not entirely the same. For kernel PCA with a non-linear kernel the same map $\Phi$ is applied to each feature. But in transforming the features before PCA we can apply distinct non-linear transformations to each feature.
