Relaxed Triangle Inequality for Kullback-Leibler Divergence Between Multivariate Gaussian Distributions

The Kullback-Leibler (KL) divergence is not a proper distance metric and does not satisfy the triangle inequality, posing theoretical challenges in certain practical applications. Existing work has demonstrated that KL divergence between multivariate Gaussian distributions follows a relaxed triangle inequality. Given any three multivariate Gaussian distributions $\mathcal{N}_1, \mathcal{N}_2$, and $\mathcal{N}_3$, if $KL(\mathcal{N}_1, \mathcal{N}_2)\leq ε_1$ and $KL(\mathcal{N}_2, \mathcal{N}_3)\leq ε_2$, then $KL(\mathcal{N}_1, \mathcal{N}_3)< 3ε_1+3ε_2+2\sqrt{ε_1ε_2}+o(ε_1)+o(ε_2)$. However, the supremum of $KL(\mathcal{N}_1, \mathcal{N}_3)$ is still unknown. In this paper, we investigate the relaxed triangle inequality for the KL divergence between multivariate Gaussian distributions and give the supremum of $KL(\mathcal{N}_1, \mathcal{N}_3)$ as well as the conditions when the supremum can be attained. When $ε_1$ and $ε_2$ are small, the supremum is $ε_1+ε_2+\sqrt{ε_1ε_2}+o(ε_1)+o(ε_2)$. Finally, we demonstrate several applications of our results in out-of-distribution detection with flow-based generative models and safe reinforcement learning.

CSubBT: A Self-Adjusting Execution Framework for Mobile Manipulation System

With the advancements in modern intelligent technologies, mobile robots equipped with manipulators are increasingly operating in unstructured environments. These robots can plan sequences of actions for long-horizon tasks based on perceived information. However, in practice, the planned actions often fail due to discrepancies between the perceptual information used for planning and the actual conditions. In this paper, we introduce the {\itshape Conditional Subtree} (CSubBT), a general self-adjusting execution framework for mobile manipulation tasks based on Behavior Trees (BTs). CSubBT decomposes symbolic action into sub-actions and uses BTs to control their execution, addressing any potential anomalies during the process. CSubBT treats common anomalies as constraint non-satisfaction problems and continuously guides the robot in performing tasks by sampling new action parameters in the constraint space when anomalies are detected. We demonstrate the robustness of our framework through extensive manipulation experiments on different platforms, both in simulation and real-world settings.