Deep generative models learn continuous representations of complex data manifolds using a finite number of samples during training. For a pre-trained generative model, the common way to evaluate the quality of the manifold representation learned, is by computing global metrics like Fr\'echet Inception Distance using a large number of generated and real samples. However, generative model performance is not uniform across the learned manifold, e.g., for \textit{foundation models} like Stable Diffusion generation performance can vary significantly based on the conditioning or initial noise vector being denoised. In this paper we study the relationship between the \textit{local geometry of the learned manifold} and downstream generation. Based on the theory of continuous piecewise-linear (CPWL) generators, we use three geometric descriptors - scaling ($\psi$), rank ($\nu$), and complexity ($\delta$) - to characterize a pre-trained generative model manifold locally. We provide quantitative and qualitative evidence showing that for a given latent, the local descriptors are correlated with generation aesthetics, artifacts, uncertainty, and even memorization. Finally we demonstrate that training a \textit{reward model} on the local geometry can allow controlling the likelihood of a generated sample under the learned distribution.

With advanced development in robotics and autonomous systems in the past decades, the opportunities and demands for more complex physical human-robot interaction (pHRI) in our everyday unconstrained environments are rising; thus, it is critical for robots to be adaptive, compliant, reactive, safe and easy to program [1, 2, 3]. In many cases, robots will need to acquire new skills to satisfy task requirements in an ever-changing environment. It is usually difficult for non-experts to program robots for complex motion tasks and even tedious for experts to reprogram them when task requirements change. A straightforward and intuitive approach for robots to develop new skills is through Learning from Demonstration (LfD) [4, 5, 6, 7, 8]. This paradigm allows robots to acquire skills, typically encoded or defined in literature as action policies, motion policies, or imitation policies, directly from motion examples provided by humans or even other robots, mirroring a teacher-student relationship. In recent years, significant progress has been made in using LfD to learn complex and diverse motion tasks.

To learn accurate representations of molecules, it is essential to consider both chemical and geometric features. To encode geometric information, many descriptors have been proposed in constrained circumstances for specific types of molecules and do not have the properties to be ``robust": 1. Invariant to rotations and translations; 2. Injective when embedding molecular structures. In this work, we propose a universal and robust Directional Node Pair (DNP) descriptor based on the graph representations of 3D molecules. Our DNP descriptor is robust compared to previous ones and can be applied to multiple molecular types. To combine the DNP descriptor and chemical features in molecules, we construct the Robust Molecular Graph Convolutional Network (RoM-GCN) which is capable to take both node and edge features into consideration when generating molecule representations. We evaluate our model on protein and small molecule datasets. Our results validate the superiority of the DNP descriptor in incorporating 3D geometric information of molecules. RoM-GCN outperforms all compared baselines.

Humans intuitively solve tasks in versatile ways, varying their behavior in terms of trajectory-based planning and for individual steps. Thus, they can easily generalize and adapt to new and changing environments. Current Imitation Learning algorithms often only consider unimodal expert demonstrations and act in a state-action-based setting, making it difficult for them to imitate human behavior in case of versatile demonstrations. Instead, we combine a mixture of movement primitives with a distribution matching objective to learn versatile behaviors that match the expert's behavior and versatility. To facilitate generalization to novel task configurations, we do not directly match the agent's and expert's trajectory distributions but rather work with concise geometric descriptors which generalize well to unseen task configurations. We empirically validate our method on various robot tasks using versatile human demonstrations and compare to imitation learning algorithms in a state-action setting as well as a trajectory-based setting. We find that the geometric descriptors greatly help in generalizing to new task configurations and that combining them with our distribution-matching objective is crucial for representing and reproducing versatile behavior.