Due to their stability and robustness properties, Stable Dynamical Systems (SDS) have received considerable attention as a means of representing motions in learning from demonstration tasks. Designing vector fields that fit complex trajectories while ensuring stability still remains a key challenge; although recent deep learning-based methods have shown substantial progress in this direction, their tendency to overfit to demonstration trajectories often leads to undesirable behaviors, particularly as tasks deviate from demonstrations. At a fundamental level, the only reliable way to address this lack of generalization is to provide supervision in out-of-demonstration regions. Focusing on two types of general behaviors, mimicking and contracting, we propose a Behavior-Controllable Stable Dynamics Model (BCSDM), a one-parameter family of SDS that allows users to adjust the system's overall behavior depending on user intent. We show how to extend BCSDM to accommodate demonstrations of multiple tasks, and also propose a Deep Operator Vector Field (DeepOVec) for memory-efficient encoding of multiple dynamical systems. Extensive experiments on tasks that involve mimicking or contracting behaviors demonstrate the advantages of BCSDMs over existing state-of-the-art SDS learning methods.

While conventional data are represented as discrete vectors, Implicit Neural Representations (INRs) utilize neural networks to represent data points as continuous functions. By incorporating a shared network that maps latent vectors to individual functions, one can model the distribution of functional data, which has proven effective in many applications, such as learning 3D shapes, surface reflectance, and operators. However, the infinite-dimensional nature of these representations makes them prone to overfitting, necessitating sufficient regularization. Naïve regularization methods -- those commonly used with discrete vector representations -- may enforce smoothness to increase robustness but result in a loss of data fidelity due to improper handling of function coordinates. To overcome these challenges, we start by interpreting the mapping from latent variables to INRs as a parametrization of a Riemannian manifold. We then recognize that preserving geometric quantities -- such as distances and angles -- between the latent space and the data manifold is crucial. As a result, we obtain a manifold with minimal intrinsic curvature, leading to robust representations while maintaining high-quality data fitting. Our experiments on various data modalities demonstrate that our method effectively discovers a well-structured latent space, leading to robust data representations even for challenging datasets, such as those that are small or noisy.

When using an autoencoder to learn the low-dimensional manifold of high-dimensional data, it is crucial to find the latent representations that preserve the geometry of the data manifold. However, most existing studies assume a Euclidean nature for the high-dimensional data space, which is arbitrary and often does not precisely reflect the underlying semantic or domain-specific attributes of the data. In this paper, we propose a novel autoencoder regularization framework based on the premise that the geometry of the data manifold can often be better captured with a well-designed similarity graph associated with data points. Given such a graph, we utilize a Riemannian geometric distortion measure as a regularizer to preserve the geometry derived from the graph Laplacian and make it suitable for larger-scale autoencoder training. Through extensive experiments, we show that our method outperforms existing state-of-the-art geometry-preserving and graph-based autoencoders with respect to learning accurate latent structures that preserve the graph geometry, and is particularly effective in learning dynamics in the latent space. Code is available at https://github.com/JungbinLim/GGAE-public.

Motion manifold primitives (MMP), a manifold-based approach for encoding basic motion skills, can produce diverse trajectories, enabling the system to adapt to unseen constraints. Nonetheless, we argue that current MMP models lack crucial functionalities of movement primitives, such as temporal and via-points modulation, found in traditional approaches. This shortfall primarily stems from MMP's reliance on discrete-time trajectories. To overcome these limitations, we introduce motion manifold primitives++ (MMP++), a new model that integrates the strengths of both MMP and traditional methods by incorporating parametric curve representations into the MMP framework. Furthermore, we identify a significant challenge with MMP++: performance degradation due to geometric distortions in the latent space, meaning that similar motions are not closely positioned. To address this, isometric motion manifold primitives++ (IMMP++) is proposed to ensure the latent space accurately preserves the manifold's geometry. Our experimental results across various applications, including two-DoF planar motions, seven-DoF robot arm motions, and SE(3) trajectory planning, show that MMP++ and IMMP++ outperform existing methods in trajectory generation tasks, achieving substantial improvements in some cases. Moreover, they enable the modulation of latent coordinates and via-points, thereby allowing efficient online adaptation to dynamic environments.

Grasping previously unseen objects for the first time, in which only partially occluded views of the object are available, remains a difficult challenge. Despite their recent successes, deep learning-based end-to-end methods remain impractical when training data and resources are limited and multiple grippers are used. Two-step methods that first identify the object shape and structure using deformable shape templates, then plan and execute the grasp, are free from those limitations, but also have difficulty with partially occluded objects. In this paper, we propose a two-step method that merges a richer set of shape primitives, the deformable superquadrics, with a deep learning network,DSQNet, that is trained to identify complete object shapes from partial point cloud data. Grasps are then generated that take into account the kinematic and structural properties of the gripper while exploiting the closed-form equations available for deformable superquadrics. A seven-dof robotic arm equipped with a parallel jaw gripper is used to conduct experiments involving a collection of household objects, achieving average grasp success rates of 93% (compared to 86% for existing methods), with object recognition times that are ten times faster. Code is available athttps://github.com/seungyeon-k/DSQNet-publicNote to Practitioners—This paper provides a comprehensive two-step method for grasping previously unseen objects, in which only partially occluded views of the object may be available. End-to-end deep learning-based methods typically require large amounts of training data, in the form of images of the objects taken from different angles and with different levels of occlusion, and grasping experiments that record the success and failure of each attempt; if a new gripper is used, more often than not the training data must be recollected and a new set of experiments performed. Two-step methods that first identify the object structure and shape using deformable shape templates, then plan the grasp based on knowledge of the object shape, are currently a more practical solution, but also have difficulty when only occluded views of the object are available. Our newly proposed two-step method takes advantage of a more flexible set of shape primitives, and also uses a supervised deep learning network to identify the object from occluded views. Our experimental results indicate improved grasp success rates against the state-of-the-art, with recognition rates that are up to ten times faster. Our method shows high recognition and grasping performance so is well applicable on most of the general household objects, but it cannot be directly applied to more diverse public 3D datasets since it requires some human-annotated segmentation labels. In future research, we will develop our deep learning network to automatically learn segmentation without human-annotated labels, allowing it to recognize more complex and diverse object shapes.

Many problems in machine learning involve data sets in which each data point is a point cloud in $\mathbb{R}^D$. A growing number of applications require a means of measuring not only distances between point clouds, but also angles, volumes, derivatives, and other more advanced concepts. To formulate and quantify these concepts in a coordinate-invariant way, we develop a Riemannian geometric framework for point cloud data. By interpreting each point in a point cloud as a sample drawn from some given underlying probability density, the space of point cloud data can be given the structure of a statistical manifold – each point on this manifold represents a point cloud – with the Fisher information metric acting as a natural Riemannian metric. Two autoencoder applications of our framework are presented: (i) smoothly deforming one 3D object into another via interpolation between the two corresponding point clouds; (ii) learning an optimal set of latent space coordinates for point cloud data that best preserves angles and distances, and thus produces a more discriminative representation space. Experiments with large-scale standard benchmark point cloud data show greatly improved classification accuracy vis-á-vis existing methods. Code is available at https://github.com/seungyeon-k/SMF-public.

Vanilla autoencoders often produce manifolds that overfit to noisy training data, or have the wrong local connectivity and geometry. Autoencoder regularization techniques, e.g., the denoising autoencoder, have had some success in reducing overfitting, whereas recent graph-based methods that exploit local connectivity information provided by neighborhood graphs have had some success in mitigating local connectivity errors. Neither of these two approaches satisfactorily reduce both overfitting and connectivity errors; moreover, graph-based methods typically involve considerable preprocessing and tuning. To simultaneously address the two issues of overfitting and local connectivity, we propose a new graph-based autoencoder, the Neighborhood Reconstructing Autoencoder (NRAE). Unlike existing graph-based methods that attempt to encode the training data to some prescribed latent space distribution -- one consequence being that only the encoder is the object of the regularization -- NRAE merges local connectivity information contained in the neighborhood graphs with local quadratic approximations of the decoder function to formulate a new neighborhood reconstruction loss. Compared to existing graph-based methods, our new loss function is simple and easy to implement, and the resulting algorithm is scalable and computationally efficient; the only required preprocessing step is the construction of the neighborhood graph. Extensive experiments with standard datasets demonstrate that, compared to existing methods, NRAE improves both overfitting and local connectivity in the learned manifold, in some cases by significant margins. Code for NRAE is available at https://github.com/Gabe-YHLee/NRAE-public.

Abstract We present the iterative medial axis transform (IMAT), an iterative descent method that constructs a medial axis transform (MAT) for a sparse, noisy, oriented point cloud sampled from an object's boundary. We first establish the equivalence between the traditional definition of the MAT of an object, i.e., the set of centres and corresponding radii of all balls maximally inscribed inside the object, with an alternative characterization matching the boundary enclosing the union of the balls with the object boundary. Based on this boundary equivalence characterization, a new MAT algorithm is proposed, in which an error function that reflects the difference between the two boundaries is minimized while restricting the number of balls to within some a priori specified upper limit. An iterative descent method with guaranteed local convergence is developed for the minimization that is also amenable to parallelization. Both quantitative and qualitative analyses of diverse 2D and 3D objects demonstrate the noise robustness, shape fidelity, and representation efficiency of the resulting MAT.