PCA

Trying to encode m points in R^n: X, m x n matrix
points as rows

    f      g
R^n -> R^l -> R^n

with l << n

so that g(f(X)) as close as possible to X

if g is linear, g(c) = Dc, and D has orthogonal columns,
then f(x) = c = D^Tx is the best.
(in general, could it be divided by D^TD somewhere?)

Now choose D to minimize

sum ||x - g(f(x)||_2 = ||x - DD^Tx||_2

over all x rows of X.  D will be the eigenvectors of X^TX