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We introduce Probabilistic Regular Expressions (PRE), a probabilistic analogue of regular expressions denoting probabilistic languages in which every word is assigned a probability of being generated. We present and prove the completeness of an inference system for reasoning about probabilistic language equivalence of PRE based on Salomaa's axiomatisation of Kleene Algebra.

Since the Radon transform (RT) consists in a line integral function, some modeling assumptions are made on Computed Tomography (CT) system, making image reconstruction analytical methods, such as Filtered Backprojection (FBP), sensitive to artifacts and noise. In the other hand, recently, a new integral transform, called Scale Space Radon Transform (SSRT), is introduced where, RT is a particular case. Thanks to its interesting properties, such as good scale space behavior, the SSRT has known number of new applications. In this paper, with the aim to improve the reconstructed image quality for these methods, we propose to model the X-ray beam with the Scale Space Radon Transform (SSRT) where, the assumptions done on the physical dimensions of the CT system elements reflect better the reality. After depicting the basic properties and the inversion of SSRT, the FBP algorithm is used to reconstruct the image from the SSRT sinogram where the RT spectrum used in FBP is replaced by SSRT and the Gaussian kernel, expressed in their frequency domain. PSNR and SSIM, as quality measures, are used to compare RT and SSRT-based image reconstruction on Shepp-Logan head and anthropomorphic abdominal phantoms. The first findings show that the SSRT-based method outperforms the methods based on RT, especially, when the number of projections is reduced, making it more appropriate for applications requiring low-dose radiation, such as medical X-ray CT. While SSRT-FBP and RT-FBP have utmost the same runtime, the experiments show that SSRT-FBP is more robust to Poisson-Gaussian noise corrupting CT data.

Graph partitioning is a common solution to scale up the graph algorithms, and shortest path (SP) computation is one of them. However, the existing solutions typically have a fixed partition method with a fixed path index and fixed partition structure, so it is unclear how the partition method and path index influence the pathfinding performance. Moreover, few studies have explored the index maintenance of partitioned SP (PSP) on dynamic graphs. To provide a deeper insight into the dynamic PSP indexes, we systematically deliberate on the existing works and propose a universal scheme to analyze this problem theoretically. Specifically, we first propose two novel partitioned index strategies and one optimization to improve index construction, query answering, or index maintenance of PSP index. Then we propose a path-oriented graph partitioning classification criteria for easier partition method selection. After that, we re-couple the dimensions in our scheme (partitioned index strategy, path index, and partition structure) to propose five new partitioned SP indexes that are more efficient either in the query or update on different networks. Finally, we demonstrate the effectiveness of our new indexes by comparing them with state-of-the-art PSP indexes through comprehensive evaluations.

We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. The Poisson equation is ubiquitous in scientific computing: it governs a wide array of physical phenomena, arises as a subproblem in many numerical algorithms, and serves as a model problem for the broader class of elliptic PDEs. The most popular Poisson discretizations yield large sparse linear systems. At high resolution, and for performance-critical applications, iterative solvers can be advantageous for these -- but only when paired with powerful preconditioners. The core of our solver is a neural network trained to approximate the inverse of a discrete structured-grid Laplace operator for a domain of arbitrary shape and with mixed boundary conditions. The structure of this problem motivates a novel network architecture that we demonstrate is highly effective as a preconditioner even for boundary conditions outside the training set. We show that on challenging test cases arising from an incompressible fluid simulation, our method outperforms state-of-the-art solvers like algebraic multigrid as well as some recent neural preconditioners.

The nonparametric view of Bayesian inference has transformed statistics and many of its applications. The canonical Dirichlet process and other more general families of nonparametric priors have served as a gateway to solve frontier uncertainty quantification problems of large, or infinite, nature. This success has been greatly due to available constructions and representations of such distributions, the two most useful constructions are the one based on normalization of homogeneous completely random measures and that based on stick-breaking processes. Hence, understanding their distributional features and how different random probability measures compare among themselves is a key ingredient for their proper application. In this paper, we analyse the discrepancy among some nonparametric priors employed in the literature. Initially, we compute the mean and variance of the random Kullback-Leibler divergence between the Dirichlet process and the geometric process. Subsequently, we extend our analysis to encompass a broader class of exchangeable stick-breaking processes, which includes the Dirichlet and geometric processes as extreme cases. Our results establish quantitative conditions where all the aforementioned priors are close in total variation distance. In such instances, adhering to Occam's razor principle advocates for the preference of the simpler process.

Existing knowledge graph (KG) embedding models have primarily focused on static KGs. However, real-world KGs do not remain static, but rather evolve and grow in tandem with the development of KG applications. Consequently, new facts and previously unseen entities and relations continually emerge, necessitating an embedding model that can quickly learn and transfer new knowledge through growth. Motivated by this, we delve into an expanding field of KG embedding in this paper, i.e., lifelong KG embedding. We consider knowledge transfer and retention of the learning on growing snapshots of a KG without having to learn embeddings from scratch. The proposed model includes a masked KG autoencoder for embedding learning and update, with an embedding transfer strategy to inject the learned knowledge into the new entity and relation embeddings, and an embedding regularization method to avoid catastrophic forgetting. To investigate the impacts of different aspects of KG growth, we construct four datasets to evaluate the performance of lifelong KG embedding. Experimental results show that the proposed model outperforms the state-of-the-art inductive and lifelong embedding baselines.

Deep Learning (DL) is vulnerable to out-of-distribution and adversarial examples resulting in incorrect outputs. To make DL more robust, several posthoc anomaly detection techniques to detect (and discard) these anomalous samples have been proposed in the recent past. This survey tries to provide a structured and comprehensive overview of the research on anomaly detection for DL based applications. We provide a taxonomy for existing techniques based on their underlying assumptions and adopted approaches. We discuss various techniques in each of the categories and provide the relative strengths and weaknesses of the approaches. Our goal in this survey is to provide an easier yet better understanding of the techniques belonging to different categories in which research has been done on this topic. Finally, we highlight the unsolved research challenges while applying anomaly detection techniques in DL systems and present some high-impact future research directions.

Label Propagation (LPA) and Graph Convolutional Neural Networks (GCN) are both message passing algorithms on graphs. Both solve the task of node classification but LPA propagates node label information across the edges of the graph, while GCN propagates and transforms node feature information. However, while conceptually similar, theoretical relation between LPA and GCN has not yet been investigated. Here we study the relationship between LPA and GCN in terms of two aspects: (1) feature/label smoothing where we analyze how the feature/label of one node is spread over its neighbors; And, (2) feature/label influence of how much the initial feature/label of one node influences the final feature/label of another node. Based on our theoretical analysis, we propose an end-to-end model that unifies GCN and LPA for node classification. In our unified model, edge weights are learnable, and the LPA serves as regularization to assist the GCN in learning proper edge weights that lead to improved classification performance. Our model can also be seen as learning attention weights based on node labels, which is more task-oriented than existing feature-based attention models. In a number of experiments on real-world graphs, our model shows superiority over state-of-the-art GCN-based methods in terms of node classification accuracy.

Manually labeling objects by tracing their boundaries is a laborious process. In Polygon-RNN++ the authors proposed Polygon-RNN that produces polygonal annotations in a recurrent manner using a CNN-RNN architecture, allowing interactive correction via humans-in-the-loop. We propose a new framework that alleviates the sequential nature of Polygon-RNN, by predicting all vertices simultaneously using a Graph Convolutional Network (GCN). Our model is trained end-to-end. It supports object annotation by either polygons or splines, facilitating labeling efficiency for both line-based and curved objects. We show that Curve-GCN outperforms all existing approaches in automatic mode, including the powerful PSP-DeepLab and is significantly more efficient in interactive mode than Polygon-RNN++. Our model runs at 29.3ms in automatic, and 2.6ms in interactive mode, making it 10x and 100x faster than Polygon-RNN++.

High spectral dimensionality and the shortage of annotations make hyperspectral image (HSI) classification a challenging problem. Recent studies suggest that convolutional neural networks can learn discriminative spatial features, which play a paramount role in HSI interpretation. However, most of these methods ignore the distinctive spectral-spatial characteristic of hyperspectral data. In addition, a large amount of unlabeled data remains an unexploited gold mine for efficient data use. Therefore, we proposed an integration of generative adversarial networks (GANs) and probabilistic graphical models for HSI classification. Specifically, we used a spectral-spatial generator and a discriminator to identify land cover categories of hyperspectral cubes. Moreover, to take advantage of a large amount of unlabeled data, we adopted a conditional random field to refine the preliminary classification results generated by GANs. Experimental results obtained using two commonly studied datasets demonstrate that the proposed framework achieved encouraging classification accuracy using a small number of data for training.