The problem of estimating a matrix based on a set of its observed entries is commonly referred to as the matrix completion problem. In this work, we specifically address the scenario of binary observations, often termed as 1-bit matrix completion. While numerous studies have explored Bayesian and frequentist methods for real-value matrix completion, there has been a lack of theoretical exploration regarding Bayesian approaches in 1-bit matrix completion. We tackle this gap by considering a general, non-uniform sampling scheme and providing theoretical assurances on the efficacy of the fractional posterior. Our contributions include obtaining concentration results for the fractional posterior and demonstrating its effectiveness in recovering the underlying parameter matrix. We accomplish this using two distinct types of prior distributions: low-rank factorization priors and a spectral scaled Student prior, with the latter requiring fewer assumptions. Importantly, our results exhibit an adaptive nature by not mandating prior knowledge of the rank of the parameter matrix. Our findings are comparable to those found in the frequentist literature, yet demand fewer restrictive assumptions.

Causal Bayesian Networks provide an important tool for reasoning under uncertainty with potential application to many complex causal systems. Structure learning algorithms that can tell us something about the causal structure of these systems are becoming increasingly important. In the literature, the validity of these algorithms is often tested for sensitivity over varying sample sizes, hyper-parameters, and occasionally objective functions. In this paper, we show that the order in which the variables are read from data can have much greater impact on the accuracy of the algorithm than these factors. Because the variable ordering is arbitrary, any significant effect it has on learnt graph accuracy is concerning, and this raises questions about the validity of the results produced by algorithms that are sensitive to, but have not been assessed against, different variable orderings.

Machine learning models are increasingly being adopted across various fields, such as medicine, business, autonomous vehicles, and cybersecurity, to analyze vast amounts of data, detect patterns, and make predictions or recommendations. In the field of cybersecurity, these models have made significant improvements in malware detection. However, despite their ability to understand complex patterns from unstructured data, these models are susceptible to adversarial attacks that perform slight modifications in malware samples, leading to misclassification from malignant to benign. Numerous defense approaches have been proposed to either detect such adversarial attacks or improve model robustness. These approaches have resulted in a multitude of attack and defense techniques and the emergence of a field known as `adversarial machine learning.' In this survey paper, we provide a comprehensive review of adversarial machine learning in the context of Android malware classifiers. Android is the most widely used operating system globally and is an easy target for malicious agents. The paper first presents an extensive background on Android malware classifiers, followed by an examination of the latest advancements in adversarial attacks and defenses. Finally, the paper provides guidelines for designing robust malware classifiers and outlines research directions for the future.

Neurosymbolic artificial intelligence is a growing field of research aiming to combine neural network learning capabilities with the reasoning abilities of symbolic systems. Informed multi-label classification is a sub-field of neurosymbolic AI which studies how to leverage prior knowledge to improve neural classification systems. A well known family of neurosymbolic techniques for informed classification use probabilistic reasoning to integrate this knowledge during learning, inference or both. Therefore, the asymptotic complexity of probabilistic reasoning is of cardinal importance to assess the scalability of such techniques. However, this topic is rarely tackled in the neurosymbolic literature, which can lead to a poor understanding of the limits of probabilistic neurosymbolic techniques. In this paper, we introduce a formalism for informed supervised classification tasks and techniques. We then build upon this formalism to define three abstract neurosymbolic techniques based on probabilistic reasoning. Finally, we show computational complexity results on several representation languages for prior knowledge commonly found in the neurosymbolic literature.

Comparing two samples of data, we observe a change in the distribution of an outcome variable. In the presence of multiple explanatory variables, how much of the change can be explained by each possible cause? We develop a new estimation strategy that, given a causal model, combines regression and re-weighting methods to quantify the contribution of each causal mechanism. Our proposed methodology is multiply robust, meaning that it still recovers the target parameter under partial misspecification. We prove that our estimator is consistent and asymptotically normal. Moreover, it can be incorporated into existing frameworks for causal attribution, such as Shapley values, which will inherit the consistency and large-sample distribution properties. Our method demonstrates excellent performance in Monte Carlo simulations, and we show its usefulness in an empirical application.

Uncertainty quantification (UQ) in scientific machine learning (SciML) combines the powerful predictive power of SciML with methods for quantifying the reliability of the learned models. However, two major challenges remain: limited interpretability and expensive training procedures. We provide a new interpretation for UQ problems by establishing a new theoretical connection between some Bayesian inference problems arising in SciML and viscous Hamilton-Jacobi partial differential equations (HJ PDEs). Namely, we show that the posterior mean and covariance can be recovered from the spatial gradient and Hessian of the solution to a viscous HJ PDE. As a first exploration of this connection, we specialize to Bayesian inference problems with linear models, Gaussian likelihoods, and Gaussian priors. In this case, the associated viscous HJ PDEs can be solved using Riccati ODEs, and we develop a new Riccati-based methodology that provides computational advantages when continuously updating the model predictions. Specifically, our Riccati-based approach can efficiently add or remove data points to the training set invariant to the order of the data and continuously tune hyperparameters. Moreover, neither update requires retraining on or access to previously incorporated data. We provide several examples from SciML involving noisy data and \textit{epistemic uncertainty} to illustrate the potential advantages of our approach. In particular, this approach's amenability to data streaming applications demonstrates its potential for real-time inferences, which, in turn, allows for applications in which the predicted uncertainty is used to dynamically alter the learning process.

State-of-the-art neurosymbolic learning systems use probabilistic reasoning to guide neural networks towards predictions that conform to logical constraints over symbols. Many such systems assume that the probabilities of the considered symbols are conditionally independent given the input to simplify learning and reasoning. We study and criticise this assumption, highlighting how it can hinder optimisation and prevent uncertainty quantification. We prove that loss functions bias conditionally independent neural networks to become overconfident in their predictions. As a result, they are unable to represent uncertainty over multiple valid options. Furthermore, we prove that these loss functions are difficult to optimise: they are non-convex, and their minima are usually highly disconnected. Our theoretical analysis gives the foundation for replacing the conditional independence assumption and designing more expressive neurosymbolic probabilistic models.

Diffusion models, a powerful and universal generative AI technology, have achieved tremendous success in computer vision, audio, reinforcement learning, and computational biology. In these applications, diffusion models provide flexible high-dimensional data modeling, and act as a sampler for generating new samples under active guidance towards task-desired properties. Despite the significant empirical success, theory of diffusion models is very limited, potentially slowing down principled methodological innovations for further harnessing and improving diffusion models. In this paper, we review emerging applications of diffusion models, understanding their sample generation under various controls. Next, we overview the existing theories of diffusion models, covering their statistical properties and sampling capabilities. We adopt a progressive routine, beginning with unconditional diffusion models and connecting to conditional counterparts. Further, we review a new avenue in high-dimensional structured optimization through conditional diffusion models, where searching for solutions is reformulated as a conditional sampling problem and solved by diffusion models. Lastly, we discuss future directions about diffusion models. The purpose of this paper is to provide a well-rounded theoretical exposure for stimulating forward-looking theories and methods of diffusion models.

Although the existing uncertainty-based semi-supervised medical segmentation methods have achieved excellent performance, they usually only consider a single uncertainty evaluation, which often fails to solve the problem related to credibility completely. Therefore, based on the framework of evidential deep learning, this paper integrates the evidential predictive results in the cross-region of mixed and original samples to reallocate the confidence degree and uncertainty measure of each voxel, which is realized by emphasizing uncertain information of probability assignments fusion rule of traditional evidence theory. Furthermore, we design a voxel-level asymptotic learning strategy by introducing information entropy to combine with the fused uncertainty measure to estimate voxel prediction more precisely. The model will gradually pay attention to the prediction results with high uncertainty in the learning process, to learn the features that are difficult to master. The experimental results on LA, Pancreas-CT, ACDC and TBAD datasets demonstrate the superior performance of our proposed method in comparison with the existing state of the arts.

In this paper, we investigate Bayesian model compression in federated learning (FL) to construct sparse models that can achieve both communication and computation efficiencies. We propose a decentralized Turbo variational Bayesian inference (D-Turbo-VBI) FL framework where we firstly propose a hierarchical sparse prior to promote a clustered sparse structure in the weight matrix. Then, by carefully integrating message passing and VBI with a decentralized turbo framework, we propose the D-Turbo-VBI algorithm which can (i) reduce both upstream and downstream communication overhead during federated training, and (ii) reduce the computational complexity during local inference. Additionally, we establish the convergence property for thr proposed D-Turbo-VBI algorithm. Simulation results show the significant gain of our proposed algorithm over the baselines in reducing communication overhead during federated training and computational complexity of final model.