A.1 Rotations in 2 dimensions In 2-dimensional settings, there exists a single scalar angular position, the yaw angle θ. In order to perform the transformation, we have to express the angular positions in a format suitable for linear transformations; we do so by transforming them to rotation matrices, perform a matrix multiplication, and then transform the angular positions back to angle format. In 2 dimensions, we use eq. After the rotation, we can convert them back to angle format using the 2-argument arc-tangent function: θ = atan2(sinθ,cosθ) (14) Simplified rotations In 2 dimensions, the computations can be simplified since rotations commute. First, we show that chained rotations result in angle addition/subtraction, that is: Q(θi) Q(θj) = cosθi sinθi sinθicosθi cosθj sinθj sinθjcosθj (15) = cosθicosθj sinθisinθj cosθisinθj sinθicosθj sinθicosθj +cosθisinθj sinθisinθj +cosθicosθj (16) = cos(θi +θj) sin(θi +θj) sin(θi +θj) cos(θi +θj) (17) = Q(θi +θj) (18) Following the same approach, we compute the inverse rotation: Q (θi) Q(θj) = Q( θi) Q(θj) = Q(θj θi) (19) Thus, instead of rotating the angular positions (expressed in rotation matrix form) using the rotation matrix Q, in practice we perform the transformation directly to the angles via addition/subtraction, and replace the matrix Qwith the identity matrix I1 1.

Modelling interactions is critical in learning complex dynamical systems, namely systems of interacting objects with highly non-linear and time-dependent behaviour. A large class of such systems can be formalized as geometric graphs, i.e., graphs with nodes positioned in the Euclidean space given an arbitrarily chosen global coordinate system, for instance vehicles in a traffic scene. Notwithstanding the arbitrary global coordinate system, the governing dynamics of the respective dynamical systems are invariant to rotations and translations, also known as Galilean invariance. As ignoring these invariances leads to worse generalization, in this work we propose local coordinate frames per node-object to induce roto-translation invariance to the geometric graph of the interacting dynamical system. Further, the local coordinate frames allow for a natural definition of anisotropic filtering in graph neural networks. Experiments in traffic scenes, 3D motion capture, and colliding particles demonstrate that the proposed approach comfortably outperforms the recent state-of-the-art.

Understanding the training dynamics of deep learning models is perhaps a necessary step toward demystifying the effectiveness of these models. In particular, how do data from different classes gradually become separable in their feature spaces when training neural networks using stochastic gradient descent?

With a principled representation of uncertainty and closed form posterior updates, Gaussian processes (GPs) are a natural choice for online decision making. However, Gaussian processes typically require at least O(n2) computations for n training points, limiting their general applicability. Stochastic variational Gaussian processes (SVGPs) can provide scalable inference for a dataset of fixed size, but are difficult to efficiently condition on new data. We propose online variational conditioning (OVC), a procedure for efficiently conditioning SVGPs in an online setting that does not require re-training through the evidence lower bound with the addition of new data. OVC enables the pairing of SVGPs with advanced lookahead acquisition functions for black-box optimization, even with non-Gaussian likelihoods. We show OVC provides compelling performance in a range of applications including active learning of malaria incidence, and reinforcement learning on MuJoCo simulated robotic control tasks.