Continuous diffusion models are commonly acknowledged to display a deterministic probability flow, whereas discrete diffusion models do not. In this paper, we aim to establish the fundamental theory for the probability flow of discrete diffusion models. Specifically, we first prove that the continuous probability flow is the Monge optimal transport map under certain conditions, and also present an equivalent evidence for discrete cases.

Continuous diffusion models are commonly acknowledged to display a deterministic probability flow, whereas discrete diffusion models do not. In this paper, we aim to establish the fundamental theory for the probability flow of discrete diffusion models. Specifically, we first prove that the continuous probability flow is the Monge optimal transport map under certain conditions, and also present an equivalent evidence for discrete cases.

Decision trees offer this inter-pretability, especially when they are small.

Soft robots are distinguished by their flexible and adaptable, allowing them to perform tasks that are nearly impossible for rigid robots. However, controlling their configuration is challenging due to their nonlinear material response and infinite deflection degrees of freedom. A potential solution is to discretize the infinite-dimensional configuration space of soft robots into a finite but sufficiently large number of functional shapes. This study explores a co-design strategy for pneumatically actuated soft grippers with multiple encoded stable states, enabling desired functional shape and stiffness reconfiguration. An energy based analytical model for soft multistable grippers is presented, mapping the robots' infinite-dimensional configuration space into discrete stable states, allowing for prediction of the systems final state and dynamic behavior. Our approach introduces a general method to capture the soft robots' response with the lattice lumped parameters using automatic relevance determination regression, facilitating inverse co-design. The resulting computationally efficient model enables us to explore the configuration space in a tractable manner, allowing the inverse co-design of our robots by setting desired targeted positions with optimized stiffness of the set targets. This strategy offers a framework for controlling soft robots by exploiting the nonlinear mechanics of multistable structures, thus embodying mechanical intelligence into soft structures.

Differentiable particle filters combine the flexibility of neural networks with the probabilistic nature of sequential Monte Carlo methods. However, traditional approaches rely on the availability of labelled data, i.e., the ground truth latent state information, which is often difficult to obtain in real-world applications. This paper compares the effectiveness of two semi-supervised training objectives for differentiable particle filters. We present results in two simulated environments where labelled data are scarce.

Continuous diffusion models are commonly acknowledged to display a deterministic probability flow, whereas discrete diffusion models do not. In this paper, we aim to establish the fundamental theory for the probability flow of discrete diffusion models. Specifically, we first prove that the continuous probability flow is the Monge optimal transport map under certain conditions, and also present an equivalent evidence for discrete cases. In view of these findings, we are then able to define the discrete probability flow in line with the principles of optimal transport. Finally, drawing upon our newly established definitions, we propose a novel sampling method that surpasses previous discrete diffusion models in its ability to generate more certain outcomes. Extensive experiments on the synthetic toy dataset and the CIFAR-10 dataset have validated the effectiveness of our proposed discrete probability flow.