Despite the recent success of graph neural networks (GNN), common architectures often exhibit significant limitations, including sensitivity to oversmoothing, long-range dependencies, and spurious edges, e.g., as can occur as a result of graph heterophily or adversarial attacks. To at least partially address these issues within a simple transparent framework, we consider a new family of GNN layers designed to mimic and integrate the update rules of two classical iterative algorithms, namely, proximal gradient descent and iterative reweighted least squares (IRLS). The former defines an extensible base GNN architecture that is immune to oversmoothing while nonetheless capturing long-range dependencies by allowing arbitrary propagation steps. In contrast, the latter produces a novel attention mechanism that is explicitly anchored to an underlying end-to-end energy function, contributing stability with respect to edge uncertainty. When combined we obtain an extremely simple yet robust model that we evaluate across disparate scenarios including standardized benchmarks, adversarially-perturbated graphs, graphs with heterophily, and graphs involving long-range dependencies. In doing so, we compare against SOTA GNN approaches that have been explicitly designed for the respective task, achieving competitive or superior node classification accuracy. Our code is available at https://github.com/FFTYYY/TWIRLS. And for an extended version of this work, please see https://arxiv.org/abs/2103.06064.

Steerable CNN imposes the prior knowledge of transformation invariance or equivariance in the network architecture to enhance the the network robustness on geometry transformation of data and reduce overfitting. It has been an intuitive and widely used technique to construct a steerable filter by augmenting a filter with its transformed copies in the past decades, which is named as filter transform in this paper. Recently, the problem of steerable CNN has been studied from aspect of group representation theory, which reveals the function space structure of a steerable kernel function. However, it is not yet clear on how this theory is related to the filter transform technique. In this paper, we show that kernel constructed by filter transform can also be interpreted in the group representation theory. This interpretation help complete the puzzle of steerable CNN theory and provides a novel and simple approach to implement steerable convolution operators. Experiments are executed on multiple datasets to verify the feasibility of the proposed approach.

Link prediction is an important learning task for graph-structured data. In this paper, we propose a novel topological approach to characterize interactions between two nodes. Our topological feature, based on the extended persistent homology, encodes rich structural information regarding the multi-hop paths connecting nodes. Based on this feature, we propose a graph neural network method that outperforms state-of-the-arts on different benchmarks. As another contribution, we propose a novel algorithm to more efficiently compute the extended persistence diagrams for graphs. This algorithm can be generally applied to accelerate many other topological methods for graph learning tasks.

In this paper, we propose conjugate energy-based models (CEBMs), a new class of energy-based models that define a joint density over data and latent variables. The joint density of a CEBM decomposes into an intractable distribution over data and a tractable posterior over latent variables. CEBMs have similar use cases as variational autoencoders, in the sense that they learn an unsupervised mapping from data to latent variables. However, these models omit a generator network, which allows them to learn more flexible notions of similarity between data points. Our experiments demonstrate that conjugate EBMs achieve competitive results in terms of image modelling, predictive power of latent space, and out-of-domain detection on a variety of datasets.

In the deep learning era, the vast majority of methods to predict pose from a single image are trained to classify or regress to a single given ground truth pose per image. Such methods have two main shortcomings, i) they cannot represent uncertainty about the predictions, and ii) they cannot handle symmetric objects, where multiple (potentially infinite) poses may be correct. Only recently these shortcomings have been addressed, but current approaches as limited in that they cannot express the full rich space of distributions on the rotation manifold. To this end, we introduce a method to estimate arbitrary, non-parametric distributions on SO(3). Our key idea is to represent the distributions implicitly, with a neural network that estimates the probability density, given the input image and a candidate pose. At inference time, grid sampling or gradient ascent can be used to find the most likely pose, but it is also possible to evaluate the density at any pose, enabling reasoning about symmetries and uncertainty. This is the most general way of representing distributions on manifolds, and to demonstrate its expressive power we introduce a new dataset containing symmetric and nearly-symmetric objects. Our method also shows advantages on the popular object pose estimation benchmarks ModelNet10-SO(3) and T-LESS. Code, data, and visualizations may be found at implicit-pdf.github.io.

Pixelizations of Platonic solids such as the cube and icosahedron have been widely used to represent spherical data, from climate records to Cosmic Microwave Background maps. Platonic solids have well-known global symmetries. Once we pixelize each face of the solid, each face also possesses its own local symmetries in the form of Euclidean isometries. One way to combine these symmetries is through a hierarchy. However, this approach does not adequately model the interplay between the two levels of symmetry transformations. We show how to model this interplay using ideas from group theory, identify the equivariant linear maps, and introduce equivariant padding that respects these symmetries. Deep networks that use these maps as their building blocks generalize gauge equivariant CNNs on pixelized spheres. These deep networks achieve state-of-the-art results on semantic segmentation for climate data and omnidirectional image processing. Code is available at https://git.io/JGiZA.

For machine learning models to make reliable predictions in deployment, one needs to ensure the previously unknown test samples need to be sufficiently similar to the training data. The commonly used shift-invariant kernels do not have the compositionality and fail to capture invariances in high-dimensional data in computer vision. We propose a shift-invariant convolutional neural tangent kernel (SCNTK) based outlier detector and two-sample tests with maximum mean discrepancy (MMD) that is O(n) in the number of samples due to using the random feature approximation. On MNIST and CIFAR10 with various types of dataset shifts, we empirically show that statistical tests with such compositional kernels, inherited from infinitely wide neural networks, achieve higher detection accuracy than existing non-parametric methods. Our method also provides a competitive alternative to adapted kernel methods that require a training phase.