Nonlinear PCA: Estimation of Algebraic Varieties

Music City Center 

An algebraic variety is defined as the set of solutions of a system of polynomial equations over the reals. In this paper we consider the goal of recovering an unknown algebraic variety from noisy measurements of latent variables lying on the algebraic variety. Note that, estimation of an algebraic variety --- which generalizes the notion of solutions to a linear system of equations --- generalizes the concept of principal component analysis (PCA) to nonlinear structures (i.e., solutions of a system of polynomial equations). Our estimation strategy proceeds via three steps: (i) construction of the {\it moment matrix} from the vandermonde matrix associated with the data set and the degree of the fitted polynomial, (ii) debiasing the moment matrix, and (iii) eigenvalue decomposition of the moment matrix to recover the underling algebraic variety. We present theoretical results regarding the recovery guarantees of the underlying algebraic variety. We illustrate the power and usefulness of our methodology via simulation and real data examples.

debiasing, moment matrix, singular value decomposition