IEEE Robotics & Automation Magazine - June 2020 - 42

This is exploited in robotics to handle different sources of
missing data where expectations for the remaining dimensions can be computed as a multivariate distribution. These
properties make GMR an attractive tool for robotics that
can be used in a wide range of problems and combined fluently with other techniques [39].
The authors of [35] and [40] present methods for
regression from a mixture of Gaussians on Riemannian
manifolds, but they only partially exploit the manifold
structure in Gaussian conditioning. In [35], each distribution is located on its own tangent space, with the covariances encoded separately, resulting in a block-diagonal
structure in the joint distribution. In [40], a GMM is reformulated to handle the space of rotation in R 3 using
logarithms and exponential transformations on unit

quaternions, with these operations formulated in a single
tangent space (at the origin) instead of applying the transformations locally (see Figure 5). The link to Riemannian
manifolds also is not discussed.
Here, it is proposed to extend GMR to input and/or output data on SPD manifolds; see Figure 7. As the covariance
of SPD data points is a fourth-order tensor, a method is
proposed in [5] for the parallel transport of high-order
covariances on SPD manifolds by exploiting the supersymmetry properties of these fourth-order tensors. As an example application, GMR on an SPD manifold is applied to
predict wrist movement from spatial covariances computed
from sEMG data. In this application, the GMR input data
are spatial covariances that belong to the SPD manifold.
Compared to the Euclidean GMR, the GMR on an SPD

12
Input ∈ S++

Output ∈ R4

1
Rest

Reference
Standard Regression
Regression on SPD Manifold

(c)

0.5
0

Wrist
Supination

Wrist
Flexion/Extension

Wrist Flexion

0

2

4
Time (s)

6

8

1
0.5
0
0

2

4
Time (s)

6

8

Wrist Supination

(b)

Wrist Extension

(a)

1
0.5
0
0

2

4
Time (s)

6

8

2

4
Time (s)

6

8

1
0.5
0
0

(d)

Figure 8. GMR for the control of prosthetic hands within the TACT-HAND project. The (a) sEMG measurements, (b) transformation
in spatial covariances (SPD matrices), and (c) control of the corresponding hand pose. SPD signals are used as input in the form of
spatial covariances computed from sEMG sensors on participants' forearm. Activation signals corresponding to different hand poses
are used as outputs. (d) In this experiment (see [5] for details), taking the geometry of the data into account in GMR (in blue) results
in better discrimination than treating the data as if they were in a Euclidean space (in green).

42

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IEEE ROBOTICS & AUTOMATION MAGAZINE

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JUNE 2020



IEEE Robotics & Automation Magazine - June 2020

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