A Compact Kernel Approximation for 3D Action Recognition

3D action recognition was shown to benefit from a covariance representation of the input data (3D positions of the joints). A kernel machine fed 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.