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As I understand it, PCA can pick out structure if the data is linear. Given that it works on pictures of human faces to produce so called eigenfaces, does this mean these pictures of human faces are linear in the high dimensional space? If so, why?

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Essentially, yes.

Pictures of human faces are closely approximated by 9-dimensional linear subspaces under varying lighting conditions. The paper referenced below shows this under the assumptions that light reflects off of faces according to the Lambertian model and the fact that human faces can be approximated by a convex surface. The approximation is quite good, as the paper shows that the 9-dimensional model captures about 98% of the energy of the actual images. The "why" is the subject of the paper. It's been a while since I've read it, but if you have questions about the paper I can do my best to answer them.

Reference: R. Basri and D. Jacobs, “Lambertian reflection and linear subspaces” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 25, no. 3,pp. 218–233, 2003.

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