The sole aim of this book is to give a self-contained introduction to concepts and mathematical tools in Bayesian matrix decomposition in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning Bayesian matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of variational inference for conducting the optimization. We refer the reader to literature in the field of Bayesian analysis for a more detailed introduction to the related fields. This book is primarily a summary of purpose, significance of important Bayesian matrix decomposition methods, e.g., real-valued decomposition, nonnegative matrix factorization, Bayesian interpolative decomposition, and the origin and complexity of the methods which shed light on their applications. The mathematical prerequisite is a first course in statistics and linear algebra. Other than this modest background, the development is self-contained, with rigorous proof provided throughout.

Minimal adversarial perturbations added to inputs have been shown to be effective at fooling deep neural networks. In this paper, we introduce several innovations that make white-box targeted attacks follow the intuition of the attacker's goal: to trick the model to assign a higher probability to the target class than to any other, while staying within a specified distance from the original input. First, we propose a new loss function that explicitly captures the goal of targeted attacks, in particular, by using the logits of all classes instead of just a subset, as is common. We show that Auto-PGD with this loss function finds more adversarial examples than it does with other commonly used loss functions. Second, we propose a new attack method that uses a further developed version of our loss function capturing both the misclassification objective and the $L_{\infty}$ distance limit $ε$. This new attack method is relatively 1.5--4.2% more successful on the CIFAR10 dataset and relatively 8.2--14.9% more successful on the ImageNet dataset, than the next best state-of-the-art attack. We confirm using statistical tests that our attack outperforms state-of-the-art attacks on different datasets and values of $ε$ and against different defenses.

Quantum graphical models (QGMs) extend the classical framework for reasoning about uncertainty by incorporating the quantum mechanical view of probability. Prior work on QGMs has focused on hidden quantum Markov models (HQMMs), which can be formulated using quantum analogues of the sum rule and Bayes rule used in classical graphical models. Despite the focus on developing the QGM framework, there has been little progress in learning these models from data. The existing state-of-the-art approach randomly initializes parameters and iteratively finds unitary transformations that increase the likelihood of the data. While this algorithm demonstrated theoretical strengths of HQMMs over HMMs, it is slow and can only handle a small number of hidden states. In this paper, we tackle the learning problem by solving a constrained optimization problem on the Stiefel manifold using a well-known retraction-based algorithm. We demonstrate that this approach is not only faster and yields better solutions on several datasets, but also scales to larger models that were prohibitively slow to train via the earlier method.