  3D action recognition was shown to benefit from a covariance representation
of the input data (joint 3D positions). A kernel machine feed with such feature
is an effective paradigm for 3D action recognition, yielding state-of-the-art
results. Yet, the whole framework is affected by the well-known scalability
issue. In fact, in general, the kernel function has to be evaluated for all
pairs of instances inducing a Gram matrix whose complexity is quadratic in the
number of samples. In this work we reduce such complexity to be linear by
proposing a novel and explicit feature map to approximate the kernel function.
This allows to train a linear classifier with an explicit feature encoding,
which implicitly implements a Log-Euclidean machine in a scalable fashion. Not
only we prove that the proposed approximation is unbiased, but also we work out
an explicit strong bound for its variance, attesting a theoretical superiority
of our approach with respect to existing ones. Experimentally, we verify that
our representation provides a compact encoding and outperforms other
approximation schemes on a number of publicly available benchmark datasets for
3D action recognition.
