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Theoretical understanding of deep learning is one of the most important tasks facing the statistics and machine learning communities. While deep neural networks (DNNs) originated as engineering methods and models of biological networks in…
This paper investigates the issues of combination and normalization of interval-valued belief structures within the framework of Dempster-Shafer theory of evidence. Existing approaches are reviewed and thoroughly analyzed. The advantages…
Stemming from de Finetti's work on finitely additive coherent probabilities, the paradigm of coherence has been applied to many uncertainty calculi in order to remove structural restrictions on the domain of the assessment. Three possible…
This article introduces a general statistical modeling principle called "Density Sharpening" and applies it to the analysis of discrete count data. The underlying foundation is based on a new theory of nonparametric approximation and…
Despite their unprecedented success, DNNs are notoriously fragile to small shifts in data distribution, demanding effective testing techniques that can assess their dependability. Despite recent advances in DNN testing, there is a lack of…
Solving nonlinear SMT problems over real numbers has wide applications in robotics and AI. While significant progress is made in solving quantifier-free SMT formulas in the domain, quantified formulas have been much less investigated. We…
Deep neural networks (DNNs) have received tremendous attention and achieved great success in various applications, such as image and video analysis, natural language processing, recommendation systems, and drug discovery. However, inherent…
Explainability of Deep Neural Networks (DNNs) has been garnering increasing attention in recent years. Of the various explainability approaches, concept-based techniques stand out for their ability to utilize human-meaningful concepts…
Rosser theories play an important role in the study of the incompleteness phenomenon and meta-mathematics of arithmetic. In this paper, we first define the notions of $n$-Rosser theories, exact $n$-Rosser theories, effectively $n$-Rosser…
Density functional theory (DFT) is an essential building block for modern theoretical physics, chemistry, and engineering, especially those concerning electronic properties. Through decades of development, various program packages for…
Deep neural networks (DNNs) have a wide range of applications, and software employing them must be thoroughly tested, especially in safety-critical domains. However, traditional software test coverage metrics cannot be applied directly to…
Number sense is a core cognitive ability supporting various adaptive behaviors and is foundational for mathematical learning. Here, we study its emergence in unsupervised generative models through the lens of rate-distortion theory (RDT), a…
The goal of this paper is twofold. First, we present a unified way of formulating numerical integration problems from both approximation theory and discrepancy theory. Second, we show how techniques, developed in approximation theory, work…
The number needed to treat (NNT) is an efficacy and effect size measure commonly used in epidemiological studies and meta-analyses. The NNT was originally defined as the average number of patients needed to be treated to observe one less…
Deep Neural Networks (DNNs) are analyzed via the theoretical framework of the information bottleneck (IB) principle. We first show that any DNN can be quantified by the mutual information between the layers and the input and output…
The inverse problem of Kohn-Sham density functional theory (DFT) is often solved in an effort to benchmark and design approximate exchange-correlation potentials. The forward and inverse problems of DFT rely on the same equations but the…
Nuclear density functional theory (DFT) is one of the main theoretical tools used to study the properties of heavy and superheavy elements, or to describe the structure of nuclei far from stability. While on-going efforts seek to better…
In this chapter, we present and discuss a new generalized proportional conflict redistribution rule. The Dezert-Smarandache extension of the Demster-Shafer theory has relaunched the studies on the combination rules especially for the…
We propose a new interpretation of measures of information and disorder by connecting these concepts to group theory in a new way. Entropy and group theory are connected here by their common relation to sets of permutations. A combinatorial…
Model counting is a fundamental problem in many practical applications, including query evaluation in probabilistic databases and failure-probability estimation of networks. In this work, we focus on a variant of this problem where the…