We introduce a non-parametric density estimator deemed Radial Voronoi Density Estimator (RVDE). RVDE is grounded in the geometry of Voronoi tessellations and as such benefits from local geometric adaptiveness and broad convergence properties. Due to its radial definition RVDE is continuous and computable in linear time with respect to the dataset size. This amends for the main shortcomings of previously studied VDEs, which are highly discontinuous and computationally expensive. We provide a theoretical study of the modes of RVDE as well as an empirical investigation of its performance on high-dimensional data. Results show that RVDE outperforms other non-parametric density estimators, including recently introduced VDEs.

We introduce a general method for learning representations that are equivariant to symmetries of data. Our central idea is to decompose the latent space into an invariant factor and the symmetry group itself. The components semantically correspond to intrinsic data classes and poses respectively. The learner is trained on a loss encouraging equivariance based on supervision from relative symmetry information. The approach is motivated by theoretical results from group theory and guarantees representations that are lossless, interpretable and disentangled. We provide an empirical investigation via experiments involving datasets with a variety of symmetries. Results show that our representations capture the geometry of data and outperform other equivariant representation learning frameworks.

We introduce an algorithm for active function approximation based on nearest neighbor regression. Our Active Nearest Neighbor Regressor (ANNR) relies on the Voronoi-Delaunay framework from computational geometry to subdivide the space into cells with constant estimated function value and select novel query points in a way that takes the geometry of the function graph into account. We consider the recent state-of-the-art active function approximator called DEFER, which is based on incremental rectangular partitioning of the space, as the main baseline. The ANNR addresses a number of limitations that arise from the space subdivision strategy used in DEFER. We provide a computationally efficient implementation of our method, as well as theoretical halting guarantees. Empirical results show that ANNR outperforms the baseline for both closed-form functions and real-world examples, such as gravitational wave parameter inference and exploration of the latent space of a generative model.

The Voronoi Density Estimator (VDE) is an established density estimation technique that adapts to the local geometry of data. However, its applicability has been so far limited to problems in two and three dimensions. This is because Voronoi cells rapidly increase in complexity as dimensions grow, making the necessary explicit computations infeasible. We define a variant of the VDE deemed Compactified Voronoi Density Estimator (CVDE), suitable for higher dimensions. We propose computationally efficient algorithms for numerical approximation of the CVDE and formally prove convergence of the estimated density to the original one. We implement and empirically validate the CVDE through a comparison with the Kernel Density Estimator (KDE). Our results indicate that the CVDE outperforms the KDE on sound and image data.

Evaluating the quality of learned representations without relying on a downstream task remains one of the challenges in representation learning. In this work, we present Geometric Component Analysis (GeomCA) algorithm that evaluates representation spaces based on their geometric and topological properties. GeomCA can be applied to representations of any dimension, independently of the model that generated them. We demonstrate its applicability by analyzing representations obtained from a variety of scenarios, such as contrastive learning models, generative models and supervised learning models.

Capturing scene dynamics and predicting the future scene state is challenging but essential for robotic manipulation tasks, especially when the scene contains both rigid and deformable objects. In this work, we contribute a simulation environment and generate a novel dataset for task-specific manipulation, involving interactions between rigid objects and a deformable bag. The dataset incorporates a rich variety of scenarios including different object sizes, object numbers and manipulation actions. We approach dynamics learning by proposing an object-centric graph representation and two modules which are Active Prediction Module (APM) and Position Prediction Module (PPM) based on graph neural networks with an encode-process-decode architecture. At the inference stage, we build a two-stage model based on the learned modules for single time step prediction. We combine modules with different prediction horizons into a mixed-horizon model which addresses long-term prediction. In an ablation study, we show the benefits of the two-stage model for single time step prediction and the effectiveness of the mixed-horizon model for long-term prediction tasks. Supplementary material is available at https://github.com/wengzehang/deformable_rigid_interaction_prediction

We present a framework for visual action planning of complex manipulation tasks with high-dimensional state spaces such as manipulation of deformable objects. Planning is performed in a low-dimensional latent state space that embeds images. We define and implement a Latent Space Roadmap (LSR) which is a graph-based structure that globally captures the latent system dynamics. Our framework consists of two main components: a Visual Foresight Module (VFM) that generates a visual plan as a sequence of images, and an Action Proposal Network (APN) that predicts the actions between them. We show the effectiveness of the method on a simulated box stacking task as well as a T-shirt folding task performed with a real robot.

Caging grasps limit the mobility of an object to a bounded component of configuration space. We introduce a notion of partial cage quality based on maximal clearance of an escaping path. As this is a computationally demanding task even in a two-dimensional scenario, we propose a deep learning approach. We design two convolutional neural networks and construct a pipeline for real-time partial cage quality estimation directly from 2D images of object models and planar caging tools. One neural network, CageMaskNN, is used to identify caging tool locations that can support partial cages, while a second network that we call CageClearanceNN is trained to predict the quality of those configurations. A dataset of 3811 images of objects and more than 19 million caging tool configurations is used to train and evaluate these networks on previously unseen objects and caging tool configurations. Furthermore, the networks are trained jointly on configurations for both 3 and 4 caging tool configurations whose shape varies along a 1-parameter family of increasing elongation. In experiments, we study how the networks' performance depends on the size of the training dataset, as well as how to efficiently deal with unevenly distributed training data. In further analysis, we show that the evaluation pipeline can approximately identify connected regions of successful caging tool placements and we evaluate the continuity of the cage quality score evaluation along caging tool trajectories. Experiments show that evaluation of a given configuration on a GeForce GTX 1080 GPU takes less than 6 ms.

Caging provides an alternative to point-contact-based rigid grasping, relying on reasoning about the global free configuration space of an object under consideration. While substantial progress has been made toward the analysis, verification, and synthesis of cages of polygonal objects in the plane, the use of caging as a tool for manipulating general complex objects in 3-D remains challenging. In this work, we introduce the problem of caging rigid and partially deformable 3-D objects, which exhibit geometric features we call double forks and necks. Our approach is based on the linking number-a classical topological invariant, allowing us to determine sufficient conditions for caging objects with these features even in the case when the object under consideration is partially deformable under a set of neck or double fork preserving deformations. We present synthesis and verification algorithms and demonstrations of applying these algorithms to cage 3-D meshes.