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I (think I) understand PCA, but it still blows my mind that you can get such good reconstruction of the faces with only like 20 eigenface components. How is that possible?? Why should it be the case that faces lie more or less on a 20-dimensional hyperplane?

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Because all the faces derive from the same genetic code, or "program", which evolution has established should look a certain way; a forward-looking pair of eyes, a centrally-placed nose with a pair of nostrils, etc. Maybe you should try running eigenfaces with all our primate cousins and see how many components you need?

If you're interested in pursuing this train of thought from a more analytical perspective look up the "manifold hypothesis".

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    $\begingroup$ +1. Or: run eigenfaces with flies. Or with octopi. Or with skyscrapers, or beer bottles. That something is identifiable as a face (rather than as a beer bottle) already says that it shares a lot of features with other things we recognize as faces. $\endgroup$ Dec 1, 2014 at 20:55
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    $\begingroup$ It's funny you mention beer bottles because I watched deep learning demo the other day that did confuse them for humans. If you squint, they do look rather feminine from a distance :) $\endgroup$
    – Emre
    Dec 1, 2014 at 21:07
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    $\begingroup$ I will not tell my wife she looks like a beer bottle. $\endgroup$ Dec 1, 2014 at 21:13

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