Fourier Transporter: Bi-Equivariant Robotic Manipulation in 3D

Many complex robotic manipulation tasks can be decomposed as a sequence of pick and place actions. Training a robotic agent to learn this sequence over many different starting conditions typically requires many iterations or demonstrations, especially in 3D environments. In this work, we propose Fourier Transporter (\ours{}) which leverages the two-fold $\SE(d)\times\SE(d)$ symmetry in the pick-place problem to achieve much higher sample efficiency. \ours{} is an open-loop behavior cloning method trained using expert demonstrations to predict pick-place actions on new environments. \ours{} is constrained to incorporate symmetries of the pick and place actions independently. Our method utilizes a fiber space Fourier transformation that allows for memory-efficient construction. We test our proposed network on the RLbench benchmark and achieve state-of-the-art results across various tasks.

Multi-Irreducible Spectral Synchronization for Robust Rotation Averaging

Rotation averaging (RA) is a fundamental problem in robotics and computer vision. In RA, the goal is to estimate a set of $N$ unknown orientations $R_{1}, ..., R_{N} \in SO(3)$, given noisy measurements $R_{ij} \sim R^{-1}_{i} R_{j}$ of a subset of their pairwise relative rotations. This problem is both nonconvex and NP-hard, and thus difficult to solve in the general case. We apply harmonic analysis on compact groups to derive a (convex) spectral relaxation constructed from truncated Fourier decompositions of the individual summands appearing in the RA objective; we then recover an estimate of the RA solution by computing a few extremal eigenpairs of this relaxation, and (approximately) solving a consensus problem. Our approach affords several notable advantages versus prior RA methods: it can be used in conjunction with \emph{any} smooth loss function (including, but not limited to, robust M-estimators), does not require any initialization, and is implemented using only simple (and highly scalable) linear-algebraic computations and parallelizable optimizations over band-limited functions of individual rotational states. Moreover, under the (physically well-motivated) assumption of multiplicative Langevin measurement noise, we derive explicit performance guarantees for our spectral estimator (in the form of probabilistic tail bounds on the estimation error) that are parameterized in terms of graph-theoretic quantities of the underlying measurement network. By concretely linking estimator performance with properties of the underlying measurement graph, our results also indicate how to devise measurement networks that are \emph{guaranteed} to achieve accurate estimation, enabling such downstream tasks as sensor placement, network compression, and active sensing.

Can Euclidean Symmetry be Leveraged in Reinforcement Learning and Planning?

In robotic tasks, changes in reference frames typically do not influence the underlying physical properties of the system, which has been known as invariance of physical laws.These changes, which preserve distance, encompass isometric transformations such as translations, rotations, and reflections, collectively known as the Euclidean group. In this work, we delve into the design of improved learning algorithms for reinforcement learning and planning tasks that possess Euclidean group symmetry. We put forth a theory on that unify prior work on discrete and continuous symmetry in reinforcement learning, planning, and optimal control. Algorithm side, we further extend the 2D path planning with value-based planning to continuous MDPs and propose a pipeline for constructing equivariant sampling-based planning algorithms. Our work is substantiated with empirical evidence and illustrated through examples that explain the benefits of equivariance to Euclidean symmetry in tackling natural control problems.

Equivariant Single View Pose Prediction Via Induced and Restricted Representations

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.

Machine Learning as Ecology

Machine learning methods have had spectacular success on numerous problems. Here we show that a prominent class of learning algorithms - including Support Vector Machines (SVMs) -- have a natural interpretation in terms of ecological dynamics. We use these ideas to design new online SVM algorithms that exploit ecological invasions, and benchmark performance using the MNIST dataset. Our work provides a new ecological lens through which we can view statistical learning and opens the possibility of designing ecosystems for machine learning. Supplemental code is found at https://github.com/owenhowell20/EcoSVM.