Kolmogorov widths, Grassmann manifolds and unfoldings of time series

V. M. Buchstaber

Sbornik. Mathematics, Tome 216 (2025) no. 3, pp. 314-332

Résumé

Problems in Kolmogorov's theory of widths and the theory of unfoldings of time series are considered. These theories are related by means of the theory of extremal problems on the Grassmann manifolds $G(n,q)$ of $q$-dimensional linear subspaces of $\mathbb R^n$. The necessary information on the manifolds $G(n,q)$ is provided. Using an unfolding of a time series, the concept of the $q$-width of this series is introduced, and the $q$-width of a time series is calculated in the case of the functional of component analysis of the nodes of the unfolding. Using the Schubert basis of a $q$-dimensional linear subspace of $\mathbb R^n$ the concept of time series regression is introduced and its properties are described. An algorithm for the projection of a piecewise linear curve in $\mathbb R^n$ onto the space of unfoldings of time series is described and, on this basis, the concept of an $L$-approximation of a time series is introduced, where $L$ is an arbitrary $q$-dimensional subspace of $\mathbb R^n$. The results of calculations for discretizations of model functions and for time series obtained at a station monitoring the concentration of atmospheric $\mathrm{CO}_2$ are presented. Bibliography: 32 titles.

Détail

Keywords: manifold of linear subspaces, algorithm for optimizing functions on a Grassmann manifold, $q$-width of a time series, Schubert basis, $L$-approximation of a time series.