
A probabilistic framework for approximating functions in active subspaces
This paper develops a comprehensive probabilistic setup to compute appro...
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Privately Learning Subspaces
Private data analysis suffers a costly curse of dimensionality. However,...
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Generalized bounds for active subspaces
The active subspace method, as a dimension reduction technique, can subs...
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Reducing Subspace Models for LargeScale Covariance Regression
We develop an envelope model for joint mean and covariance regression in...
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Subspace Learning with Partial Information
The goal of subspace learning is to find a kdimensional subspace of R^d...
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Active Manifolds: A nonlinear analogue to Active Subspaces
We present an approach to analyze C^1(R^m) functions that addresses limi...
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Learning multivariate functions with lowdimensional structures using polynomial bases
In this paper we study the multivariate ANOVA decomposition for function...
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Learning functions varying along an active subspace
Many functions of interest are in a highdimensional space but exhibit lowdimensional structures. This paper studies regression of a sHölder function f in R^D which varies along an active subspace of dimension d while d≪ D. A direct approximation of f in R^D with an ε accuracy requires the number of samples n in the order of ε^(2s+D)/s. this paper, we modify the Generalized Contour Regression (GCR) algorithm to estimate the active subspace and use piecewise polynomials for function approximation. GCR is among the best estimators for the active subspace, but its sample complexity is an open question. Our modified GCR improves the efficiency over the original GCR and leads to an mean squared estimation error of O(n^1) for the active subspace, when n is sufficiently large. The mean squared regression error of f is proved to be in the order of (n/log n)^2s/2s+d where the exponent depends on the dimension of the active subspace d instead of the ambient space D. This result demonstrates that GCR is effective in learning lowdimensional active subspaces. The convergence rate is validated through several numerical experiments.
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