{"ID":2886628,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.01976","arxiv_id":"2508.01976","title":"Estimation of Algebraic Sets: Extending PCA Beyond Linearity","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.","short_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...","url_abs":"https://arxiv.org/abs/2508.01976","url_pdf":"https://arxiv.org/pdf/2508.01976v1","authors":"[\"Alberto González-Sanz\",\"Gilles Mordant\",\"Álvaro Samperio\",\"Bodhisattva Sen\"]","published":"2025-08-04T01:26:36Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}