Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Heat kernel and intrinsic Gaussian processes on manifolds   There is an increasing interest in the problem of nonparametric regression
like Gaussian processes with predictors locating on manifold. Some recent
researches developed intrinsic Gaussian processes by using the transition
density of the Brownian motion on submanifolds of $\mathbb R^2$ and $\mathbb
R^3$ to approximate the heat kernels. {However}, when the dimension of a
manifold is bigger than two, the existing method struggled to get good
estimation of the heat kernel. In this work, we propose an intrinsic approach
of constructing the Gaussian process on \if more \fi general manifolds \if
{\color{red} in the matrix Lie groups} \fi such as orthogonal groups, unitary
groups, Stiefel manifolds and Grassmannian manifolds. The heat kernel is
estimated by simulating Brownian motion sample paths via the exponential map,
which does not depend on the embedding of the manifold. To be more precise,
this intrinsic method has the following features: (i) it is effective for high
dimensional manifolds; (ii) it is applicable to arbitrary manifolds; (iii) it
does not require the global parametrisation or embedding which may introduce
redundant parameters; (iv) results obtained by this method do not depend on the
ambient space of the manifold. Based on this method, we propose the ball
algorithm for arbitrary manifolds and the strip algorithm for manifolds with
extra symmetries, which is both theoretically proven and numerically tested to
be much more efficient than the ball algorithm. A regression example on the
projective space of dimension eight is given in this work, which demonstrates
that our intrinsic method for Gaussian process is practically effective in
great generality.

11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
5
The categories of this paper are: gr-qc
21
The categories of this paper are: adap-org
22
The categories of this paper are: chao-dyn
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
10
The categories of this paper are: q-bio
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
26
The categories of this paper are: bayes-an
8
The categories of this paper are: hep-ex