IEEE Robotics & Automation Magazine - June 2020 - 39

considered, as each Gaussian is employed to model a limited
region of the manifold. In the general case of a manifold M,
such a model is a distribution maximizing the entropy in the
tangent space. It is defined as
N M ^x

n , R h = ^ ( 2r )

d

R h 2 e - 2 Log
1

1

n

(x) R -1 Log n (x)

,

where x ! M is a point of the manifold, n ! M is the mean
of the distribution (origin of the tangent space), and
R ! Tn M is the covariance defined in this tangent space.
For a set of N data points, this geometric mean corresponds to the minimization,
N

min
/ Log n (x n) < Log n (x n),
n
n=1

which can be solved by a simple and fast Gauss-Newton iterative algorithm. The algorithm starts from an initial estimate
on the manifold and an associated tangent space. The data
points {x n} Nn = 1 are projected in this tangent space to compute
a direction vector, which provides an updated estimate of the
mean. This process is repeated by iterating
u= 1
N

N

/ Log

n=1

n

(x n),

n

! Exp n (u),

until convergence. In practice, such an algorithm converges
very fast in only a couple of iterations (typically fewer than 10
for the accuracy required by the applications presented here).
After convergence, a covariance is computed in the tangent
space as R = 1/N R Nn = 1 Log n (x n) Log 


IEEE Robotics & Automation Magazine - June 2020
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