Page 66 in this slideset discusses and presents an example for the following idea: given the idea that a (Laplacian, hence square) matrix can be exactly decomposed as
$$ A = U^\top U $$
(which could be obtained by Gaussian elimination), define the approximated decomposition problem
$$ A \approx V^\top V$$
where $V$ is lower rank (notionally even a column vector). The slides give a numerical example rather than the algorithm itself, but there are steps I can't follow (it starts with "Find the rank-1 matrix that agrees on" (in the sense of producing the approximated $V_1^\top V_1$ matrix reproducing) "the first row and column" -- how?).
I guess my question is more like "does this have a name? Is it implemented in linear algebra packages?" than "how exactly is this computed" (because my naive implementation of a detailed explanation would probably suck). Barring that, some light on how "find the low-rank matrix that agrees work" would be immensely appreciated.