Spectral methods for matrices and tensors

Authors:
Ravindran Kannan
Affiliations:
Microsoft Research Labs., India, Bangalore, India
Venue:
Proceedings of the forty-second ACM symposium on Theory of computing
Year:
2010

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Abstract

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 $r2$ 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.