We provide a collection of examples involving the concept of a product disintegration, which generalizes exchangeability.
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This paper considers two well-studied problems \textsc{Minimum Fill-In} (\textsc{Min Fill-In}) and \textsc{Treewidth}. Since both problems are \textsf{NP}-hard, various reduction rules simplifying an input graph have been intensively studied to better understand the structural properties relevant to these problems. Bodlaender at el. introduced the concept of a safe edge that is included in a solution of the \textsc{Minimum Fill-In} problem and showed some initial results. In this paper, we extend their result and prove a new condition for an edge set to be safe. This in turn helps us to construct a novel reduction tool for \textsc{Min Fill-In} that we use to answer other questions related to the problem. In this paper, we also study another interesting research question: Whether there exists a triangulation that answers both problems \textsc{Min Fill-In} and \textsc{Treewidth}. To formalise our study, we introduce a new parameter reflecting a distance of triangulations optimising both problems. We present some initial results regarding this parameter and study graph classes where both problems can be solved with one triangulation.
Almost all problems in applied mathematics, including the analysis of dynamical systems, deal with spaces of real-valued functions on Euclidean domains in their formulation and solution. In this paper, we describe the the tool Ariadne, which provides a rigorous calculus for working with Euclidean functions. We first introduce the Ariadne framework, which is based on a clean separation of objects as providing exact, effective, validated and approximate information. We then discuss the function calculus as implemented in \Ariadne, including polynomial function models which are the fundamental class for concrete computations. We then consider solution of some core problems of functional analysis, namely solution of algebraic equations and differential equations, and briefly discuss their use for the analysis of hybrid systems. We will give examples of C++ and Python code for performing the various calculations. Finally, we will discuss progress on extensions, including improvements to the function calculus and extensions to more complicated classes of system.
Quantum computing has emerged as a promising field with the potential to revolutionize various domains by harnessing the principles of quantum mechanics. As quantum hardware and algorithms continue to advance, the development of high-quality quantum software has become crucial. However, testing quantum programs poses unique challenges due to the distinctive characteristics of quantum systems and the complexity of multi-subroutine programs. In this paper, we address the specific testing requirements of multi-subroutine quantum programs. We begin by investigating critical properties through a survey of existing quantum libraries, providing insights into the challenges associated with testing these programs. Building upon this understanding, we present a systematic testing process tailored to the intricacies of quantum programming. The process covers unit testing and integration testing, with a focus on aspects such as IO analysis, quantum relation checking, structural testing, behavior testing, and test case generation. We also introduce novel testing principles and criteria to guide the testing process. To evaluate our proposed approach, we conduct comprehensive testing on typical quantum subroutines, including diverse mutations and randomized inputs. The analysis of failures provides valuable insights into the effectiveness of our testing methodology. Additionally, we present case studies on representative multi-subroutine quantum programs, demonstrating the practical application and effectiveness of our proposed testing processes, principles, and criteria.
Solving symbolic reasoning problems that require compositionality and systematicity is considered one of the key ingredients of human intelligence. However, symbolic reasoning is still a great challenge for deep learning models, which often cannot generalize the reasoning pattern to out-of-distribution test cases. In this work, we propose a hybrid system capable of solving arithmetic problems that require compositional and systematic reasoning over sequences of symbols. The model acquires such a skill by learning appropriate substitution rules, which are applied iteratively to the input string until the expression is completely resolved. We show that the proposed system can accurately solve nested arithmetical expressions even when trained only on a subset including the simplest cases, significantly outperforming both a sequence-to-sequence model trained end-to-end and a state-of-the-art large language model.
Models with intractable normalizing functions have numerous applications. Because the normalizing constants are functions of the parameters of interest, standard Markov chain Monte Carlo cannot be used for Bayesian inference for these models. A number of algorithms have been developed for such models. Some have the posterior distribution as their asymptotic distribution. Other ``asymptotically inexact'' algorithms do not possess this property. There is limited guidance for evaluating approximations based on these algorithms. Hence it is very hard to tune them. We propose two new diagnostics that address these problems for intractable normalizing function models. Our first diagnostic, inspired by the second Bartlett identity, is in principle broadly applicable to Monte Carlo approximations beyond the normalizing function problem. We develop an approximate version of this diagnostic that is applicable to intractable normalizing function problems. Our second diagnostic is a Monte Carlo approximation to a kernel Stein discrepancy-based diagnostic introduced by Gorham and Mackey (2017). We provide theoretical justification for our methods and apply them to several algorithms in challenging simulated and real data examples including an Ising model, an exponential random graph model, and a Conway--Maxwell--Poisson regression model, obtaining interesting insights about the algorithms in these contexts.
Local search is an effective method for solving large-scale combinatorial optimization problems, and it has made remarkable progress in recent years through several subtle mechanisms. In this paper, we found two ways to improve the local search algorithms in solving Pseudo-Boolean Optimization (PBO): Firstly, some of those mechanisms such as unit propagation are merely used in solving MaxSAT before, which can be generalized to solve PBO as well; Secondly, the existing local search algorithms utilize the heuristic on variables, so-called score, to mainly guide the search. We attempt to gain more insights into the clause, as it plays the role of a middleman who builds a bridge between variables and the given formula. Hence, we first extended the combination of unit propagation-based decimation algorithm to PBO problem, giving a further generalized definition of unit clause for PBO problem, and apply it to the existing solver LS-PBO for constructing an initial assignment; then, we introduced a new heuristic on clauses, dubbed care, to set a higher priority for the clauses that are less satisfied in current iterations. Experiments on benchmarks from the most recent PB Competition, as well as three real-world application benchmarks including minimum-width confidence band, wireless sensor network optimization, and seating arrangement problems show that our algorithm DeciLS-PBO has a promising performance compared to the state-of-the-art algorithms.
We propose a causal framework for decomposing a group disparity in an outcome in terms of an intermediate treatment variable. Our framework captures the contributions of group differences in baseline potential outcome, treatment prevalence, average treatment effect, and selection into treatment. This framework is counterfactually formulated and readily informs policy interventions. The decomposition component for differential selection into treatment is particularly novel, revealing a new mechanism for explaining and ameliorating disparities. This framework reformulates the classic Kitagawa-Blinder-Oaxaca decomposition in causal terms, supplements causal mediation analysis by explaining group disparities instead of group effects, and resolves conceptual difficulties of recent random equalization decompositions. We also provide a conditional decomposition that allows researchers to incorporate covariates in defining the estimands and corresponding interventions. We develop nonparametric estimators based on efficient influence functions of the decompositions. We show that, under mild conditions, these estimators are $\sqrt{n}$-consistent, asymptotically normal, semiparametrically efficient, and doubly robust. We apply our framework to study the causal role of education in intergenerational income persistence. We find that both differential prevalence of and differential selection into college graduation significantly contribute to the disparity in income attainment between income origin groups.
This paper presents a novel approach for safe control synthesis using the dual formulation of the navigation problem. The main contribution of this paper is in the analytical construction of density functions for almost everywhere navigation with safety constraints. In contrast to the existing approaches, where density functions are used for the analysis of navigation problems, we use density functions for the synthesis of safe controllers. We provide convergence proof using the proposed density functions for navigation with safety. Further, we use these density functions to design feedback controllers capable of navigating in cluttered environments and high-dimensional configuration spaces. The proposed analytical construction of density functions overcomes the problem associated with navigation functions, which are known to exist but challenging to construct, and potential functions, which suffer from local minima. Application of the developed framework is demonstrated on simple integrator dynamics and fully actuated robotic systems.
Foundation models pretrained on diverse data at scale have demonstrated extraordinary capabilities in a wide range of vision and language tasks. When such models are deployed in real world environments, they inevitably interface with other entities and agents. For example, language models are often used to interact with human beings through dialogue, and visual perception models are used to autonomously navigate neighborhood streets. In response to these developments, new paradigms are emerging for training foundation models to interact with other agents and perform long-term reasoning. These paradigms leverage the existence of ever-larger datasets curated for multimodal, multitask, and generalist interaction. Research at the intersection of foundation models and decision making holds tremendous promise for creating powerful new systems that can interact effectively across a diverse range of applications such as dialogue, autonomous driving, healthcare, education, and robotics. In this manuscript, we examine the scope of foundation models for decision making, and provide conceptual tools and technical background for understanding the problem space and exploring new research directions. We review recent approaches that ground foundation models in practical decision making applications through a variety of methods such as prompting, conditional generative modeling, planning, optimal control, and reinforcement learning, and discuss common challenges and open problems in the field.
In this article, we will look at autoencoders. This article covers the mathematics and the fundamental concepts of autoencoders. We will discuss what they are, what the limitations are, the typical use cases, and we will look at some examples. We will start with a general introduction to autoencoders, and we will discuss the role of the activation function in the output layer and the loss function. We will then discuss what the reconstruction error is. Finally, we will look at typical applications as dimensionality reduction, classification, denoising, and anomaly detection. This paper contains the notes of a PhD-level lecture on autoencoders given in 2021.
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