Hello, I would have two questions about the lecture on PCA.
First of all, I'm not sure to understand the last part of the last proof of the lecture. Specifically, I don't understand how we go from \(A_{22}\) to \(\Sigma_{22}\) (last page of the proof), if you could explain it in another way or give more details, it would be great.
For the same lecture, at the end of the PCA and Decorrelation chapter, there are two questions:
I'm not sure to understand what should be the answer to these questions. Could you give more details and the answers about them ?
The empirical covariance of the data sample \(X\) would be (up to scaling) \(X X^\top\). If we write \(X= U S V^\top\), this means that the covariance would look like \(X X^\top = U S^2 U^\top\). In expectation, \(U\) would thus correspond to the ’principal directions‘ of the distribution, \(Q\).
If the eigenvectors of the covariance \(K\) are not unique, there are two corresponding eigenvectors \(q_i, q_j\) for which \(K q_i = \lambda q_i, \, K q_j = \lambda q_j, \) and any combination \(q' = \alpha q_i + (1-\alpha) q_j\) of them also has the property \(K q = \lambda q\). In this case, you can see that the eigenvectors are not uniquely defined.
PCA
Hello, I would have two questions about the lecture on PCA.
I'm not sure to understand what should be the answer to these questions. Could you give more details and the answers about them ?
Thanks a lot for your help,
Alessio
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Hi Alessio,
Let me first answer your second question:
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Thanks a lot for your answer to my second questions it makes sense now !
Hi, does someone have any update for the first question of the second point ?
About your first question, could you point to the precise lines of the proof that you find confusing? I'll try to help.
It is this part of the proof that I do not understand.
Thank you for your help
@alessio5, can you point to a particular argument / sentence where you stop following?
@thijs and @Alesso, I think there is a typo in the lecture notes. It should be
\(diag(I, U_{22}^\top) \hat U^\top X \hat V diag(I,V_{22}) = diag(\hat \Sigma, \Sigma_{22}).\)
we should look into this (and correct me if i was wrong.)
I think I don't understand why we can say that this is a SVD of X and thus I can't understand the reasoning from that point.
Thanks !
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