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PCA solutions are not unique, and are not continuous, i.e. we don't have that if we decompose $X^TX=U\Sigma V$ and then change $X^TX$ by a small amount, the updated $U,\Sigma,V$ will be close to the originals. Is there some way to add penalty terms in order to make the solutions unique and continuous?

kjetil b halvorsen
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mlstudent
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    Your initial assertion is true only when there is a pair of nearby eigenvalues. The usual regularization methods won't change that. For an example of what can be done to make the SVD continuously track small changes in $X,$ see https://stats.stackexchange.com/questions/34396. – whuber Aug 05 '20 at 17:02

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