Existing equivariant neural networks require prior knowledge of the symmetry group and discretization for continuous groups. We propose to work with Lie algebras (infinitesimal generators) instead of Lie groups. Our model, the Lie algebra convolutional network (L-conv) can automatically discover symmetries and does not require discretization of the group. We show that L-conv can serve as a building block to construct any group equivariant feedforward architecture. Both CNNs and Graph Convolutional Networks can be expressed as L-conv with appropriate groups. We discover direct connections between L-conv and physics: (1) group invariant loss generalizes field theory (2) Euler-Lagrange equation measures the robustness, and (3) equivariance leads to conservation laws and Noether current.

Most existing point cloud completion methods assume the input partial point cloud is clean, which is not the case in practice, and are generally based on supervised learning. In this paper, we present an unsupervised generative adversarial autoencoding network, named UGAAN, which completes the partial point cloud contaminated by surroundings from real scenes and cutouts the object simultaneously, only using artificial CAD models as assistance. The generator of UGAAN learns to predict the complete point clouds on real data from both the discriminator and the autoencoding process of artificial data. The latent codes from generator are also fed to discriminator which makes encoder only extract object features rather than noises. We also devise a refiner for generating better complete cloud with a segmentation module to separate the object from background. We train our UGAAN with one real scene dataset and evaluate it with the other two. Extensive experiments and visualization demonstrate our superiority, generalization and robustness. Comparisons against the previous method show that our method achieves the state-of-the-art performance on unsupervised point cloud completion and segmentation on real data.

A burgeoning paradigm in algorithm design is the field of algorithms with predictions, in which algorithms can take advantage of a possibly-imperfect prediction of some aspect of the problem. While much work has focused on using predictions to improve competitive ratios, running times, or other performance measures, less effort has been devoted to the question of how to obtain the predictions themselves, especially in the critical online setting. We introduce a general design approach for algorithms that learn predictors: (1) identify a functional dependence of the performance measure on the prediction quality and (2) apply techniques from online learning to learn predictors, tune robustness-consistency trade-offs, and bound the sample complexity. We demonstrate the effectiveness of our approach by applying it to bipartite matching, ski-rental, page migration, and job scheduling. In several settings we improve upon multiple existing results while utilizing a much simpler analysis, while in the others we provide the first learning-theoretic guarantees.

We derive the first finite-time logarithmic Bayes regret upper bounds for Bayesian bandits. In a multi-armed bandit, we obtain O(c logn)and O(ch log2 n)upper bounds for an upper confidence bound algorithm, where ch and c are constants depending on the prior distribution and the gaps of bandit instances sampled from it, respectively. The latter bound asymptotically matches the lower bound of Lai (1987). Our proofs are a major technical departure from prior works, while being simple and general. To show the generality of our techniques, we apply them to linear bandits. Our results provide insights on the value of prior in the Bayesian setting, both in the objective and as a side information given to the learner. They significantly improve upon existing O( n)bounds, which have become standard in the literature despite the logarithmic lower bound of Lai (1987).