Contraction rates and projection subspace estimation with Gaussian process priors in high dimension
arxiv(2024)
摘要
This work explores the dimension reduction problem for Bayesian nonparametric
regression and density estimation. More precisely, we are interested in
estimating a functional parameter f over the unit ball in ℝ^d,
which depends only on a d_0-dimensional subspace of ℝ^d, with d_0
< d.It is well-known that rescaled Gaussian process priors over the function
space achieve smoothness adaptation and posterior contraction with near
minimax-optimal rates. Moreover, hierarchical extensions of this approach,
equipped with subspace projection, can also adapt to the intrinsic dimension
d_0 ().When the ambient dimension d does not
vary with n, the minimax rate remains of the order n^-β/(2β
+d_0).
the same regardless of the ambient dimension d. However, this is up to
multiplicative constants that can become prohibitively large when d grows.
The dependences between the contraction rate and the ambient dimension have not
been fully explored yet and this work provides a first insight: we let the
dimension d grow with n and, by combining the arguments of
and , we
derive a growth rate for d that still leads to posterior consistency with
minimax rate.The optimality of this growth rate is then discussed.Additionally,
we provide a set of assumptions under which consistent estimation of f leads
to a correct estimation of the subspace projection, assuming that d_0 is
known.
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