Spectral methods

Abstract

The preceding two chapters studied the subspace clustering problem using algebraic-geometric and statistical techniques, respectively. Under the assumption that the data are not corrupted, we saw in Chapter 5 that algebraic-geometric methods are able to solve the subspace clustering problem in full generality, allowing for an arbitrary union of different subspaces of any dimensions and in any orientations, as long as sufficiently many data points in general configuration are drawn from the union of subspaces. However, while algebraic-geometric methods are able to deal with moderate amounts of noise, they are unable to deal with outliers. Moreover, even in the noise-free setting, the computational complexity of linear-algebraic methods for fitting polynomials grows exponentially with the number of subspaces and their dimensions. As a consequence, algebraic-geometric methods are most effective for low-dimensional problems with moderate amounts of noise.