Uniqueness and superposition of the distribution-dependent Zakai equations

The work concerns the Zakai equations from nonlinear filtering problems of McKean-Vlasov stochastic differential equations with correlated noises. First, we establish the Kushner-Stratonovich equations, the Zakai equations and the distribution-dependent Zakai equations. And then, the pathwise uniqueness, uniqueness in joint law and uniqueness in law of weak solutions for the distribution-dependent Zakai equations are shown. Finally, we prove a superposition principle between the distribution-dependent Zakai equations and distribution-dependent Fokker-Planck equations. As a by-product, we give some conditions under which distribution-dependent Fokker-Planck equations have unique weak solutions.Comment: 19 page

Shift invariant sparse coding ensemble and its application in rolling bearing fault diagnosis

This paper proposes an automatic diagnostic scheme without manual feature extraction or signal pre-processing. It directly handles the original data from sensors and determines the condition of the rolling bearing. With proper application of the new technique of shift invariant sparse coding (SISC), it is much easier to recognize the fault. Yet, this SISC, though being a powerful machine learning algorithm to train and test the original signals, is quite demanding computationally. Therefore, this paper proposes a highly efficient SISC which has been proved by experiments to be capable of representing signals better and making converges faster. For better performance, the AdaBoost algorithm is also combined with SISC classifier. Validated by the fault diagnosis of bearings and compared with other methods, this algorithm has higher accuracy rate and is more robust to load as well as to certain variation of speed

Application of a Dense Fusion Attention Network in Fault Diagnosis of Centrifugal Fan

Although the deep learning recognition model has been widely used in the condition monitoring of rotating machinery. However, it is still a challenge to understand the correspondence between the structure and function of the model and the diagnosis process. Therefore, this paper discusses embedding distributed attention modules into dense connections instead of traditional dense cascading operations. It not only decouples the influence of space and channel on fault feature adaptive recalibration feature weights, but also forms a fusion attention function. The proposed dense fusion focuses on the visualization of the network diagnosis process, which increases the interpretability of model diagnosis. How to continuously and effectively integrate different functions to enhance the ability to extract fault features and the ability to resist noise is answered. Centrifugal fan fault data is used to verify this network. Experimental results show that the network has stronger diagnostic performance than other advanced fault diagnostic models

Residual Denoising Diffusion Models

We propose residual denoising diffusion models (RDDM), a novel dual diffusion process that decouples the traditional single denoising diffusion process into residual diffusion and noise diffusion. This dual diffusion framework expands the denoising-based diffusion models, initially uninterpretable for image restoration, into a unified and interpretable model for both image generation and restoration by introducing residuals. Specifically, our residual diffusion represents directional diffusion from the target image to the degraded input image and explicitly guides the reverse generation process for image restoration, while noise diffusion represents random perturbations in the diffusion process. The residual prioritizes certainty, while the noise emphasizes diversity, enabling RDDM to effectively unify tasks with varying certainty or diversity requirements, such as image generation and restoration. We demonstrate that our sampling process is consistent with that of DDPM and DDIM through coefficient transformation, and propose a partially path-independent generation process to better understand the reverse process. Notably, our RDDM enables a generic UNet, trained with only an $\ell _1$ loss and a batch size of 1, to compete with state-of-the-art image restoration methods. We provide code and pre-trained models to encourage further exploration, application, and development of our innovative framework (https://github.com/nachifur/RDDM)