Abstract
Many functions of interest are in a high-dimensional space but exhibit low-dimensional structures. This paper studies regression of an π -HΓΆlder function π in βπ· which varies along a central subspace of dimension π while πβͺπ·. A direct approximation of π in βπ· with an π accuracy requires the number of samples π in the order of πβ(2β’π +π·)/π . In this paper, we analyze the generalized contour regression (GCR) algorithm for the estimation of the central subspace and use piecewise polynomials for function approximation. GCR is among the best estimators for the central subspace, but its sample complexity is an open question. In this paper, we partially answer this questions by proving that if a variance quantity is exactly known, GCR leads to a mean squared estimation error of πβ‘(πβ1) for the central subspace. The estimation error of this variance quantity is also given in this paper. The mean squared regression error of π is proved to be in the order of (π/logβ‘π)β2β’π /2β’π +π, where the exponent depends on the dimension of the central subspace π instead of the ambient space π·. This result demonstrates that GCR is effective in learning the low-dimensional central subspace. We also propose a modified GCR with improved efficiency. The convergence rate is validated through several numerical experiments.
Original language | English |
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Pages (from-to) | 343-371 |
Number of pages | 29 |
Journal | SIAM Journal on Mathematics of Data Science |
Volume | 6 |
Issue number | 2 |
Early online date | 22 Apr 2024 |
DOIs | |
Publication status | Published - Jun 2024 |
User-Defined Keywords
- Central subspace
- Dimension reduction
- Nonparametric regression
- Sample complexity