I am reading this deep learning book by Ian goodfellow. In the PCA formulation in the first chapter i.e Linear Algebra, he mentions the following:
we need to choose the encoding matrix D. To do so, we revisit the idea of minimizing the L2 distance between inputs and reconstructions. Since we will use the same matrix D to decode all the points, we can no longer consider the points in isolation. Instead, we must minimize the Frobenius norm of the matrix of errors computed over all dimensions and all points.
I can't understand the explanation for why can we no longer see the points in isolation. Would really appreciate if someone could help me out here.