Probabilistic modeling is a core challenge in statistical machine learning. Tensor-based probabilistic graph methods address interpretability and stability concerns encountered in neural network approaches and allow tractable inference (e.g., marginal inference and conditional inference). In this paper, we introduce tensor ring decomposition for density estimation, which reduces the number of permutation candidates compared to existing methods, while simultaneously enhancing expressive power and maintaining tractable inference. Different non-negative strategies for density function results in two variants: Born TRDE offers simpler inference and sampling but with slightly lower accuracy, while Energy TRDE, though more complex, achieves superior performance. Furthermore, a mixture model that incorporates multiple permutation candidates with adaptive weights is designed, resulting in increased expressive flexibility and comprehensiveness. Unlike existing methods that focus on finding a single optimal permutation, our approach, inspired by ensemble learning, demonstrates that combining multiple suboptimal permutations can yield superior results. Experiments demonstrate that the proposed approach excels in estimating probability density functions and sampling, capturing intricate details with competitive or superior performance compared to existing state-of-the-art (SOTA) tractable density methods.

Deep Image Prior (DIP) has shown that networks with stochastic initialization and custom architectures can effectively address inverse imaging challenges. Despite its potential, DIP requires significant computational resources, whereas the lighter Implicit Neural Positional Image Prior (PIP) often yields overly smooth solutions due to exacerbated spectral bias. Research on lightweight, high-performance solutions for inverse imaging remains limited. This paper proposes a novel framework, Enhanced Positional Image Priors through High-Order Implicit Representations (HOPE), incorporating high-order interactions between layers within a conventional cascade structure. This approach reduces the spectral bias commonly seen in PIP, enhancing the model's ability to capture both low- and high-frequency components for optimal inverse problem performance. We theoretically demonstrate that HOPE's expanded representational space, narrower convergence range, and improved Neural Tangent Kernel (NTK) diagonal properties enable more precise frequency representations than PIP. Comprehensive experiments across tasks such as signal representation (audio, image, volume) and inverse image processing (denoising, super-resolution, CT reconstruction, inpainting) confirm that HOPE establishes new benchmarks for recovery quality and training efficiency.

Cooperation between temporal convolutional networks (TCN) and graph convolutional networks (GCN) as a processing module has shown promising results in skeleton-based video anomaly detection (SVAD). However, to maintain a lightweight model with low computational and storage complexity, shallow GCN and TCN blocks are constrained by small receptive fields and a lack of cross-dimension interaction capture. To tackle this limitation, we propose a lightweight module called the Dual Attention Module (DAM) for capturing cross-dimension interaction relationships in spatio-temporal skeletal data. It employs the frame attention mechanism to identify the most significant frames and the skeleton attention mechanism to capture broader relationships across fixed partitions with minimal parameters and total Floating Point Operations (FLOPs). Furthermore, the proposed Dual Attention Normalizing Flow (DA-Flow) integrates the DAM as a post-processing unit after GCN within the normalizing flow framework. Simulations show that the proposed model is robust against noise and negative samples. Experimental results show that DA-Flow reaches competitive or better performance than the existing state-of-the-art (SOTA) methods in terms of the micro AUC metric with the fewest parameters and FLOPs. Moreover, we found that even without training, simply using random projection without dimensionality reduction on skeleton data enables substantial anomaly detection capabilities.