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I'm studying the PCA algorithm and the theory behind it. I think I understood how does it work and the idea of dimension reduction of the data in order to find a new feature (component) that maximizes the variance of the data and minimizes the error.

My question is : in this algorithm , are the maximum variance and the minimum error reached at the same moment ?

PCA example

In this example the magenta/black line is the solution of my PCA. So I find the 1 dimension vector that reduces my 2 dimensions dataset. I found this vector because the error (length of the red lines) is minimized and the variance (distance among the red projected points) is maximized.

So if I need to apply this algorithm , if I use either only the avg error minimization , will I reach the same result of the case in which I use the variance maximization principle ?

Thanks.

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Yes, the two formulations where you either capture the maximum amount of variance of the data while reducing the dimension or where you try to minimize the distance of the data from the selected subspace (i.e. reconstruction error) are equivalent.

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