  While Spectral Methods have long been used for Principal Component Analysis,
this survey focusses on work over the last 15 years with three salient
features: (i) Spectral methods are useful not only for numerical problems, but
also discrete optimization problems (Constraint Optimization Problems - CSP's)
like the max. cut problem and similar mathematical considerations underlie both
areas. (ii) Spectral methods can be extended to tensors. The theory and
algorithms for tensors are not as simple/clean as for matrices, but the survey
describes methods for low-rank approximation which extend to tensors. These
tensor approximations help us solve Max-$r$-CSP's for $r>2$ as well as
numerical tensor problems. (iii) Sampling on the fly plays a prominent role in
these methods. A primary result is that for any matrix, a random submatrix of
rows/columns picked with probabilities proportional to the squared lengths (of
rows/columns), yields estimates of the singular values as well as an
approximation to the whole matrix.
