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Kernel PCA is usually done via eigenvalue decomposition of the Kernel Matrix $\mathbf{K}$ and standard PCA via SVD of the input $\mathbf{X}$. In standard PCA as far as I know we can derive $\mathbf{S}$ and $\mathbf{U}$ via two eigenvalue decompositions, of the Gram and Covariance/Correlation matrices: $$ \begin{array}{c} X=U\Sigma V^T\\ C=\dfrac{X^TX}{N-1}\\ G=\dfrac{XX^T}{N-1}\\ C=VE_CV^T\\ G=UE_GU^T\\ S=\sqrt{E_C(N-1)}\\ K=U_KE_KU^T\\ ?=VE_?V^T \end{array} $$ But how does one get $\mathbf{V}$ in the case of a kernel? All posts I've ever read only discuss $\mathbf{U}$

Kernel PCA is usually done via eigenvalue decomposition of the Kernel Matrix $\mathbf{K}$ and standard PCA via SVD of the input $\mathbf{X}$. In standard PCA as far as I know we can derive $\mathbf{S}$ and $\mathbf{U}$ via two eigenvalue decompositions, of the Gram and Covariance/Correlation matrices: $$ \begin{array}{c} X=U\Sigma V^T\\ C=\dfrac{X^TX}{N-1}\\ G=\dfrac{XX^T}{N-1}\\ C=VE_CV^T\\ G=UE_GU^T\\ S=\sqrt{E_C(N-1)}\\ K=U_KE_KU^T\\ ?=VE_?V^T \end{array} $$ But how does one get $\mathbf{V}$ in the case of a kernel? All posts I've ever read only discuss $\mathbf{U}$

Note: I've read that $\mathbf{XV}=\mathbf{U\Sigma}$, however this relationship doesn't seem to hold for numpy.linalg.svd or scipy.linalg.svd

Kernel PCA is usually done via eigenvalue decomposition of the Kernel Matrix K and standard PCA via SVD of the input X. We can derive S and U from eigenvalue decomposition

How can I get the matrix V? I need it for certain calculations like biplots.

In standard PCA as far as I know we can derive U and V via two eigenvalue decompositions, of the Gram and Covariance/Correlation matrices:

Kernel PCA is usually done via eigenvalue decomposition of the Kernel Matrix $\mathbf{K}$ and standard PCA via SVD of the input $\mathbf{X}$. In standard PCA as far as I know we can derive $\mathbf{S}$ and $\mathbf{U}$ via two eigenvalue decompositions, of the Gram and Covariance/Correlation matrices: $$ \begin{array}{c} X=U\Sigma V^T\\ C=\dfrac{X^TX}{N-1}\\ G=\dfrac{XX^T}{N-1}\\ C=VE_CV^T\\ G=UE_GU^T\\ S=\sqrt{E_C(N-1)}\\ K=U_KE_KU^T\\ ?=VE_?V^T \end{array} $$ But how does one get $\mathbf{V}$ in the case of a kernel? All posts I've ever read only discuss $\mathbf{U}$

Kernel PCA is usually done via eigenvalue decomposition of the Kernel Matrix K and standard PCA via SVD of the input X. We can derive S and U from eigenvalue decomposition

How can I get the matrix V? I need it for certain calculations like biplots

Kernel PCA is usually done via eigenvalue decomposition of the Kernel Matrix K and standard PCA via SVD of the input X. We can derive S and U from eigenvalue decomposition

How can I get the matrix V? I need it for certain calculations like biplots.

In standard PCA as far as I know we can derive U and V via two eigenvalue decompositions, of the Gram and Covariance/Correlation matrices: