A Compact Kernel Approximation for 3D Action Recognition

Abstract

Abstract. 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.

Scalable and compact 3D action recognition with approximated RBF kernel machines

“…This makes the computational cost of our approach substantially in line with that of other works [5], [19], [20], [21], [22].…”

“…This approach extends two previous works [5], [6]. With respect to [5], we generalize the class of feature maps thereby proposed and show that [5] is a particular case of our approximation. Besides, we extend its experimental validation on a new dataset (the NTU RGB+D) and discuss its computational cost.…”

“…The reason for that is extremely technical and we illustrate it in the Supplementary Material, Section 4. It is straightforward to see that Definition 2 actually corresponds to the generalization to the kernel (1) of the approach in [5], which is instead explicitly devised for the log-Euclidean kernel of covariance operators. Here, in fact, X and Y can be generic d×d data structures.…”

“…Here, in fact, X and Y can be generic d×d data structures. Ultimately, we can state that the approximation devised in [5] is a particular case of φ φ φ kron−e , which, in turn, is a reformulation of φ φ φ kron−π . We can also prove what follows.…”

“…Intuitively, the randomness in n can lead to "explore" all the infinite components in the exact feature map f (X) in order to accumulate enough patterns in φ φ φ kron−e and φ φ φ kron−π to properly approximate the RBF Gaussian kernel. Imposing n = 1 can be instead thought of as a sort of linearization as to approximate f (x) in (5). In such a case, there is clearly a little room for the weights W (κ) to help recovering from the compression.…”