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.

We propose a robust and reliable evaluation metric for generative models called Topological Precision and Recall (TopP&R, pronounced "topper"), which systematically estimates supports by retaining only topologically and statistically significant features with a certain level of confidence. Existing metrics, such as Inception Score (IS), Fréchet Inception Distance (FID), and various Precision and Recall(P&R) variants, rely heavily on support estimates derived from sample features. However, the reliability of these estimates has been overlooked, even though the quality of the evaluation hinges entirely on their accuracy. In this paper, we demonstrate that current methods not only fail to accurately assess sample quality when support estimation is unreliable, but also yield inconsistent results. In contrast, TopP&R reliably evaluates the sample quality and ensures statistical consistency in its results. Our theoretical and experimental findings reveal that TopP&R provides a robust evaluation, accurately capturing the true trend of change in samples, even in the presence of outliers and non-independent and identically distributed (Non-IID) perturbations where other methods result in inaccurate support estimations. To our knowledge, TopP&Ris the first evaluation metric specifically focused on the robust estimation of supports, offering statistical consistency under noise conditions.

We will now derive continuous dynamics (2) in the main paper. Let 1m = 1 if class 1 is selected at iteration mand 1m = 0 otherwise. Likewise, we can obtain the dynamics of X2j similarly. We will next prove the separation theorem in binary classification, Theorem 2.1. Given the feature vectors X1i(t), X2j(t) for i,j [n], as t and large n, 1. if α > β, they are asymptotically separable with probability tending to one, 2. if α β, they are asymptotically separable with probability tending to zero. This also aligns with our intuition that the intra-class effect should be stronger than its inter-class counterpart. On the other hand, when α>β, ignoring a null set we may assume c1 >c2 without loss of generality.

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?