Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Riemannian Geometry of Symmetric Positive Definite Matrices via Cholesky
  Decomposition   We present a new Riemannian metric, termed Log-Cholesky metric, on the
manifold of symmetric positive definite (SPD) matrices via Cholesky
decomposition. We first construct a Lie group structure and a bi-invariant
metric on Cholesky space, the collection of lower triangular matrices whose
diagonal elements are all positive. Such group structure and metric are then
pushed forward to the space of SPD matrices via the inverse of Cholesky
decomposition that is a bijective map between Cholesky space and SPD matrix
space. This new Riemannian metric and Lie group structure fully circumvent
swelling effect, in the sense that the determinant of the Fr\'echet average of
a set of SPD matrices under the presented metric, called Log-Cholesky average,
is between the minimum and the maximum of the determinants of the original SPD
matrices. Comparing to existing metrics such as the affine-invariant metric and
Log-Euclidean metric, the presented metric is simpler, more computationally
efficient and numerically stabler. In particular, parallel transport along
geodesics under Log-Cholesky metric is given in a closed and easy-to-compute
form.

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The categories of this paper are: hep-ph
23
The categories of this paper are: solv-int
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
14
The categories of this paper are: math-ph
28
The categories of this paper are: alg-geom
34
The categories of this paper are: chem-ph