Estimation of Algebraic Sets: Extending PCA Beyond Linearity

TL;DR

This paper extends PCA to estimate unknown algebraic sets from noisy data by constructing and debiasing a moment matrix, enabling the recovery of the set's defining polynomials with provable accuracy.

Contribution

It introduces a novel method for estimating algebraic sets from noisy observations, generalizing PCA beyond linear subspaces with theoretical guarantees.

Findings

Consistent estimators for polynomial coefficients of the algebraic set.

Three strategies for reconstructing the algebraic set from estimated polynomials.

Nearly parametric error bounds for the set recovery process.

Abstract

An algebraic set is defined as the zero locus of a system of real polynomial equations. In this paper we address the problem of recovering an unknown algebraic set $\mathcal{A}$ from noisy observations of latent points lying on $\mathcal{A}$ -- a task that extends principal component analysis, which corresponds to the purely linear case. Our procedure consists of three steps: (i) constructing the {\it moment matrix} from the Vandermonde matrix associated with the data set and the degree of the fitted polynomials, (ii) debiasing this moment matrix to remove the noise-induced bias, (iii) extracting its kernel via an eigenvalue decomposition of the debiased moment matrix. These steps yield $n^{-1/2}$-consistent estimators of the coefficients of a set of generators for the ideal of polynomials vanishing on $\mathcal{A}$. To reconstruct $\mathcal{A}$ itself, we propose three complementary strategies: (a) compute the zero set of the fitted polynomials; (b) build a semi-algebraic approximation that encloses $\mathcal{A}$; (c) when structural prior information is available, project the estimated coefficients onto the corresponding constrained space. We prove (nearly) parametric asymptotic error bounds and show that each approach recovers $\mathcal{A}$ under mild regularity conditions.