Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On The Convergence of Gradient Descent for Finding the Riemannian Center
  of Mass   We study the problem of finding the global Riemannian center of mass of a set
of data points on a Riemannian manifold. Specifically, we investigate the
convergence of constant step-size gradient descent algorithms for solving this
problem. The challenge is that often the underlying cost function is neither
globally differentiable nor convex, and despite this one would like to have
guaranteed convergence to the global minimizer. After some necessary
preparations we state a conjecture which we argue is the best (in a sense
described) convergence condition one can hope for. The conjecture specifies
conditions on the spread of the data points, step-size range, and the location
of the initial condition (i.e., the region of convergence) of the algorithm.
These conditions depend on the topology and the curvature of the manifold and
can be conveniently described in terms of the injectivity radius and the
sectional curvatures of the manifold. For manifolds of constant nonnegative
curvature (e.g., the sphere and the rotation group in $\mathbb{R}^{3}$) we show
that the conjecture holds true (we do this by proving and using a comparison
theorem which seems to be of a different nature from the standard comparison
theorems in Riemannian geometry). For manifolds of arbitrary curvature we prove
convergence results which are weaker than the conjectured one (but still
superior over the available results). We also briefly study the effect of the
configuration of the data points on the speed of convergence.

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The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
20
The categories of this paper are: acc-phys
0
The categories of this paper are: hep-ph
28
The categories of this paper are: alg-geom
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
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The categories of this paper are: patt-sol
15
The categories of this paper are: nucl-ex
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
9
The categories of this paper are: nlin
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The categories of this paper are: eess
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The categories of this paper are: adap-org