Message passing neural networks (MPNNs) have emerged as the most popular framework of graph neural networks (GNNs) in recent years. However, their expressive power is limited by the 1-dimensional Weisfeiler-Lehman (1-WL) test. Some works are inspired by k-WL/FWL (Folklore WL) and design the corresponding neural versions. Despite the high expressive power, there are serious limitations in this line of research. In particular, (1) k-WL/FWL requires at least O(nk)space complexity, which is impractical for large graphs even when k = 3; (2) The design space of k-WL/FWL is rigid, with the only adjustable hyper-parameter being k. To tackle the first limitation, we propose an extension, (k,t)-FWL. We theoretically prove that even if we fix the space complexity to O(nk) (for any k 2) in (k,t)-FWL, we can construct an expressiveness hierarchy up to solving the graph isomorphism problem. To tackle the second problem, we propose k-FWL+, which considers any equivariant set as neighbors instead of all nodes, thereby greatly expanding the design space of k-FWL.

Current black-box variational inference (BBVI) methods require the user to make numerous design choices--such as the selection of variational objective and approximating family--yet there is little principled guidance on how to do so. We develop a conceptual framework and set of experimental tools to understand the effects of these choices, which we leverage to propose best practices for maximizing posterior approximation accuracy. Our approach is based on studying the pre-asymptotic tail behavior of the density ratios between the joint distribution and the variational approximation, then exploiting insights and tools from the importance sampling literature. Our framework and supporting experiments help to distinguish between the behavior of BBVI methods for approximating low-dimensional versus moderate-to-high-dimensional posteriors. In the latter case, we show that mass-covering variational objectives are difficult to optimize and do not improve accuracy, but flexible variational families can improve accuracy and the effectiveness of importance sampling--at the cost of additional optimization challenges. Therefore, for moderate-to-high-dimensional posteriors we recommend using the (mode-seeking) exclusive KL divergence since it is the easiest to optimize, and improving the variational family or using model parameter transformations to make the posterior and optimal variational approximation more similar. On the other hand, in low-dimensional settings, we show that heavy-tailed variational families and mass-covering divergences are effective and can increase the chances that the approximation can be improved by importance sampling.

Every image is encoded using an ImageNet pre-trained ResNet18 to a latent feature map z R4 4 256. A flattened version of z is processed with one linear layer with output channels equal to N 3to get the predictions for points u and visibility v. We apply the sigmoid function to the visibility predictions v to enforce a numerical range [0,1]. Our models are trained using Adam optimizer with learning rate equal to 1e-4. In detail, scale is sampled from the range [0.7, 1.2], vertical translation is up to 38 pixels and we also apply 2D rotation up to 40 degrees. For camera equivariance the image is simply flipped horizontally and given as input to the network to estimate the pose.

There are no explicit equations for the cerebellum traditionally also has access to a desired state ds (in particular, one can consider this a and forward DNI, respectively; L denotes the loss function. In addition, the inverse model of the of a motor area and sensory area, respectively; CB,CF denotes the computation of backward DNI Notation is largely consistent with section 2 of the main text: hm,hs denotes the hidden activity properties of the inverse model of the cerebellum can be set against those of forward DNI (red). Controller Neocortex Main model Cerebellum Synthesiser Forward Model Backward DNIInverse Model Forward DNI be summarised in table S1. In general, the likeness in formulation between DNI and the cerebellar internal model hypothesis can backward DNI where the main model is an motor-associated RNN. In fact, it was recently suggested that the cerebellum out that though the temporal case of forward DNI was not originally considered in [5], there remain learns to mimic the forward computations which then take place in the neocortex.

Overall, our work offers a novel perspective on the cerebellum as a brainneuronal observations while making several testable predictions across multiple mental observations. Moreover, our model also explains recent behavioural and learning while reducing ataxia-like behaviours, consistent with classical experishown to be cerebellar-dependent. In all tasks, we observe that ccRNNs facilitates and cognitive tasks (pattern recognition and caption generation) that have been network (ccRNN) model on a number of sensorimotor (line and digit drawing) tions from a cerebellar module. We test this cortico-cerebellar recurrent neural in which a recurrent cortical network receives online temporal feedback predicdemonstrate the potential of this framework we introduce a systems-level model lum, helps the cerebral cortex solve similar locking problems akin to DNIs.