Recent advances in Keyframe Imitation Learning (IL) have enabled learning-based agents to solve a diverse range of manipulation tasks. However, most approaches ignore the rich symmetries in the problem setting and, as a consequence, are sample-inefficient. This work identifies and utilizes the bi-equivariant symmetry within Keyframe IL to design a policy that generalizes to transformations of both the workspace and the objects grasped by the gripper. We make two main contributions: First, we analyze the bi-equivariance properties of the keyframe action scheme and propose a Keyframe Transporter derived from the Transporter Networks, which evaluates actions using cross-correlation between the features of the grasped object and the features of the scene. Second, we propose a computationally efficient coarse-to-fine SE(3) action evaluation scheme for reasoning the intertwined translation and rotation action. The resulting method outperforms strong Keyframe IL baselines by an average of >10% on a wide range of simulation tasks, and by an average of 55% in 4 physical experiments.

Learning about the three-dimensional world from two-dimensional images is a fundamental problem in computer vision. An ideal neural network architecture for such tasks would leverage the fact that objects can be rotated and translated in three dimensions to make predictions about novel images. However, imposing SO(3)-equivariance on two-dimensional inputs is difficult because the group of three-dimensional rotations does not have a natural action on the two-dimensional plane. Specifically, it is possible that an element of SO(3) will rotate an image out of plane. We show that an algorithm that learns a three-dimensional representation of the world from two dimensional images must satisfy certain geometric consistency properties which we formulate as SO(2)-equivariance constraints. We use the induced and restricted representations of SO(2) on SO(3) to construct and classify architectures which satisfy these geometric consistency constraints. We prove that any architecture which respects said consistency constraints can be realized as an instance of our construction. We show that three previously proposed neural architectures for 3D pose prediction are special cases of our construction. We propose a new algorithm that is a learnable generalization of previously considered methods. We test our architecture on three pose predictions task and achieve SOTA results on both the PASCAL3D+ and SYMSOL pose estimation tasks.

Reasoning about 3D objects based on 2D images is challenging due to variations in appearance caused by viewing the object from different orientations. Tasks such as object classification are invariant to 3D rotations and other such as pose estimation are equivariant. However, imposing equivariance as a model constraint is typically not possible with 2D image input because we do not have an a priori model of how the image changes under out-of-plane object rotations. The only $\mathrm{SO}(3)$-equivariant models that currently exist require point cloud or voxel input rather than 2D images. In this paper, we propose a novel architecture based on icosahedral group convolutions that reasons in $\mathrm{SO(3)}$ by learning a projection of the input image onto an icosahedron. The resulting model is approximately equivariant to rotation in $\mathrm{SO}(3)$. We apply this model to object pose estimation and shape classification tasks and find that it outperforms reasonable baselines. Project website: \url{https://dmklee.github.io/image2icosahedral}

Abstract-- In robotic manipulation, acquiring samples is extremely expensive because it often requires interacting with the real world. Traditional image-level data augmentation has shown the potential to improve sample efficiency in various machine learning tasks. However, image-level data augmentation is insufficient for an imitation learning agent to learn good manipulation policies in a reasonable amount of demonstrations. We propose Simulation-augmented Equivariant Imitation Learning (SEIL), a method that combines a novel data augmentation strategy of supplementing expert trajectories with simulated transitions and an equivariant model that exploits the O(2) symmetry in robotic manipulation. Experimental evaluations demonstrate that our method can learn non-trivial manipulation tasks within ten demonstrations and outperforms the baselines with a significant margin.