IEEE Robotics & Automation Magazine - June 2020 - 38

and a nonlinear generator function that maps latent points
into the input space. The nonlinearity of the generator implies
that the latent space gives a distorted view of the input space.
The latent space provides a low-dimensional representation of
the data manifold, of which it can reveal the underlying geometrical structure [30].
In neural networks such as VAEs, by using activation
functions that are C2 differentiable, an SPD matrix M = J < J
can be employed as a smoothly changing inner product structure acting as a local Mahalanobis distance measure, where J
is the Jacobian characterizing the decoder function. The
method yields a distance measure that can, for example, be
used to replace linear interpolations in the latent space by
geodesics. In [30], Arvanitidis et al. show that in the latent
space learned by a VAE, distances and interpolants can be
improved significantly under this metric, which, in turn,
improves probability distributions, sampling algorithms, and
clustering in the latent space.
One downside of manifolds with nonconstant curvature
is that the geodesic optimization problem most often corresponds to a nonconvex problem (a system of ordinary differential equations) that can be computationally heavy to
solve. Several research directions can be explored to address
this issue. In [31], the problem is circumvented by spanning
the latent space with a discrete finite graph (a k-dimensional
tree data structure with edge weights based on Riemannian
distances) used in conjunction with a classic A ) search algorithm. Recent approaches in discrete differential geometry
also address similar problems by extending continuous Riemannian manifolds to discrete formulations with fast computations [32]. Currently, these developments principally
target computer graphics applications, but they have great
potential in robotics. They could, for example, provide an
approach to link discrete robot planning problems, such as
probabilistic roadmaps, to their continuous counterparts in
Riemannian geometry.

d
S++

Riemannian Geometry and Lie Theory
A Lie group is a smooth and differentiable manifold that possesses a group structure, therefore satisfying the group axioms. There are strong links between Riemannian geometry
and Lie theory. In particular, some Lie groups, such as SO(3),
can be endowed with a bi-invariant Riemannian metric,
which gives them the structure of a Riemannian manifold. In
robotics, Lie theory is mainly exploited for applications
involving SO(3) and SE(3) groups.
In the literature, distinctive vocabulary and notation are
often employed, which hinders some of the links between the
applications from exploiting Riemannian geometry and Lie
theory. Among these differences, the Lie algebra is the tangent
space at the origin of the manifold, acting as a global reference, while u / (hat) and u 0 (vee) are used to transform elements from the Lie algebra (which can have nontrivial
structures, such as complex numbers and skew-symmetric
matrices) to vectors in R d, which are easier to manipulate.
They are the operations corresponding to the exponential and
logarithm maps in Riemannian geometry. In Lie theory, 5
and 6 are operators used to facilitate compositions with
exponential/logarithmic mapping operations. For further
reading, an excellent introduction to Lie theory for robot
applications can be found in [33].
Gaussian Distributions on Riemannian Manifolds
Several approaches have been proposed to extend Gaussian
distributions in Euclidean space to Riemannian manifolds
[34]. Here, we focus on a simple approach that consists of
estimating the mean of the Gaussian as a centroid on the
manifold (also called the Karcher/Fréchet mean) and representing the dispersion of the data as a covariance expressed in
the tangent space of the mean [4], [10], [35]. Distortions arise
when points are too far from the mean, but this distortion is
negligible in most robotics applications. In particular, this
effect is strongly attenuated when a mixture of Gaussians is

Covariance Features, Inertia and
Gain Matrices, Manipulability
Ellipsoids, Trajectory Distributions
(SPD Matrices)
Direction, Orientation
(Unit Quaternions)

SE(d )
Rigid Body Motions
(Position and Orientation)

Figure 4. Clustering on various manifolds with GMMs.

IEEE ROBOTICS & AUTOMATION MAGAZINE

JUNE 2020

IEEE Robotics & Automation Magazine - June 2020

Table of Contents for the Digital Edition of IEEE Robotics & Automation Magazine - June 2020