Deep Markov models (DMM) are generative models which are scalable and expressive generalization of Markov models for representation, learning, and inference problems. However, the fundamental stochastic stability guarantees of such models have not been thoroughly investigated. In this paper, we present a novel stability analysis method and provide sufficient conditions of DMM's stochastic stability. The proposed stability analysis is based on the contraction of probabilistic maps modeled by deep neural networks. We make connections between the spectral properties of neural network's weights and different types of used activation function on the stability and overall dynamic behavior of DMMs with Gaussian distributions. Based on the theory, we propose a few practical methods for designing constrained DMMs with guaranteed stability. We empirically substantiate our theoretical results via intuitive numerical experiments using the proposed stability constraints.

In this paper, we provide sufficient conditions of DMM's stochastic stability as defined in the context of dynamical systems

Deep Markov models (DMM) are generative models which are scalable and expressive generalization of Markov models for representation, learning, and inference problems. However, the fundamental stochastic stability guarantees of such models have not been thoroughly investigated. In this paper, we present a novel stability analysis method and provide sufficient conditions of DMM's stochastic stability. The proposed stability analysis is based on the contraction of probabilistic maps modeled by deep neural networks. We make connections between the spectral properties of neural network's weights and different types of used activation function on the stability and overall dynamic behavior of DMMs with Gaussian distributions.

Return-Conditioned Transformer Decision Models (RCTDM) have demonstrated the potential to enhance transformer performance in offline reinforcement learning by replacing rewards in the input sequence with returns-to-go. However, to achieve the goal of learning an optimal policy from offline datasets composed of limited suboptimal trajectories, RCTDM required alternative methods. One prominent approach, trajectory stitching, was designed to enable the network to combine multiple trajectories to find the optimal path. To implement this using only transformers without auxiliary networks, it was necessary to shorten the input sequence length to better capture the Markov property in reinforcement learnings. This, however, introduced a trade-off, as it reduced the accuracy of action inference. Our study introduces a model named Decision MetaMamba to resolve these challenges. DMM employs an input token mixer to extract patterns from short sequences and uses a State Space Model (SSM) to selectively combine information from relatively distant sequences. Inspired by Metaformer, this structure was developed by transforming Mamba's input layer into various multi-modal layers. Fortunately, with the advent of Mamba, implemented using parallel selective scanning, we achieved a high-performance sequence model capable of replacing transformers. Based on these innovations, DMM demonstrated excellent performance across various datasets in offline RL, confirming that models using SSM can improve performance by domain-specific alterations of the input layer. Additionally, it maintained its performance even in lightweight models with fewer parameters. These results suggest that decision models based on SSM can pave the way for improved outcomes in future developments.

This study tackles the efficient estimation of Kullback-Leibler (KL) Divergence in Dirichlet Mixture Models (DMM), crucial for clustering compositional data. Despite the significance of DMMs, obtaining an analytically tractable solution for KL Divergence has proven elusive. Past approaches relied on computationally demanding Monte Carlo methods, motivating our introduction of a novel variational approach. Our method offers a closed-form solution, significantly enhancing computational efficiency for swift model comparisons and robust estimation evaluations. Validation using real and simulated data showcases its superior efficiency and accuracy over traditional Monte Carlo-based methods, opening new avenues for rapid exploration of diverse DMM models and advancing statistical analyses of compositional data.

Subsequence clustering of time series is an essential task in data mining, and interpreting the resulting clusters is also crucial since we generally do not have prior knowledge of the data. Thus, given a large collection of tensor time series consisting of multiple modes, including timestamps, how can we achieve subsequence clustering for tensor time series and provide interpretable insights? In this paper, we propose a new method, Dynamic Multi-network Mining (DMM), that converts a tensor time series into a set of segment groups of various lengths (i.e., clusters) characterized by a dependency network constrained with l1-norm. Our method has the following properties. (a) Interpretable: it characterizes the cluster with multiple networks, each of which is a sparse dependency network of a corresponding non-temporal mode, and thus provides visible and interpretable insights into the key relationships. (b) Accurate: it discovers the clusters with distinct networks from tensor time series according to the minimum description length (MDL). (c) Scalable: it scales linearly in terms of the input data size when solving a non-convex problem to optimize the number of segments and clusters, and thus it is applicable to long-range and high-dimensional tensors. Extensive experiments with synthetic datasets confirm that our method outperforms the state-of-the-art methods in terms of clustering accuracy. We then use real datasets to demonstrate that DMM is useful for providing interpretable insights from tensor time series.

Recently, neural networks have been extensively employed to solve partial differential equations (PDEs) in physical system modeling. While major studies focus on learning system evolution on predefined static mesh discretizations, some methods utilize reinforcement learning or supervised learning techniques to create adaptive and dynamic meshes, due to the dynamic nature of these systems. However, these approaches face two primary challenges: (1) the need for expensive optimal mesh data, and (2) the change of the solution space's degree of freedom and topology during mesh refinement. To address these challenges, this paper proposes a neural PDE solver with a neural mesh adapter. To begin with, we introduce a novel data-free neural mesh adaptor, called Data-free Mesh Mover (DMM), with two main innovations. Firstly, it is an operator that maps the solution to adaptive meshes and is trained using the Monge-Ampère equation without optimal mesh data. Secondly, it dynamically changes the mesh by moving existing nodes rather than adding or deleting nodes and edges. Theoretical analysis shows that meshes generated by DMM have the lowest interpolation error bound. Based on DMM, to efficiently and accurately model dynamic systems, we develop a moving mesh based neural PDE solver (MM-PDE) that embeds the moving mesh with a twobranch architecture and a learnable interpolation framework to preserve information within the data. Empirical experiments demonstrate that our method generates suitable meshes and considerably enhances accuracy when modeling widely considered PDE systems. The simulation of physical phenomena is a popular research topic in many disciplines, ranging from weather forecasting (Schalkwijk et al., 2015), structural mechanics (Panthi et al., 2007) to turbulence modeling (Garnier et al., 2021). Meanwhile, due to the rapid development of deep learning techniques, there are emerging neural network based approaches designed for simulating physical systems (Li et al., 2020; Raissi et al., 2019; Brandstetter et al., 2022).

Humanoid robots have the potential to help human workers by realizing physically demanding manipulation tasks such as moving large boxes within warehouses. We define such tasks as Dynamic Mobile Manipulation (DMM). This paper presents a framework for DMM via whole-body teleoperation, built upon three key contributions: Firstly, a teleoperation framework employing a Human Machine Interface (HMI) and a bi-wheeled humanoid, SATYRR, is proposed. Secondly, the study introduces a dynamic locomotion mapping, utilizing human-robot reduced order models, and a kinematic retargeting strategy for manipulation tasks. Additionally, the paper discusses the role of whole-body haptic feedback for wheeled humanoid control. Finally, the system's effectiveness and mappings for DMM are validated through locomanipulation experiments and heavy box pushing tasks. Here we show two forms of DMM: grasping a target moving at an average speed of 0.4 m/s, and pushing boxes weighing up to 105\% of the robot's weight. By simultaneously adjusting their pitch and using their arms, the pilot adjusts the robot pose to apply larger contact forces and move a heavy box at a constant velocity of 0.2 m/s.

Denoising diffusions are state-of-the-art generative models exhibiting remarkable empirical performance. They work by diffusing the data distribution into a Gaussian distribution and then learning to reverse this noising process to obtain synthetic datapoints. The denoising diffusion relies on approximations of the logarithmic derivatives of the noised data densities using score matching. Such models can also be used to perform approximate posterior simulation when one can only sample from the prior and likelihood. We propose a unifying framework generalising this approach to a wide class of spaces and leading to an original extension of score matching. We illustrate the resulting models on various applications.