1,657 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems

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    It is often desirable to summarise a probability measure on a space XX in terms of a mode, or MAP estimator, i.e.\ a point of maximum probability. Such points can be rigorously defined using masses of metric balls in the small-radius limit. However, the theory is not entirely straightforward: the literature contains multiple notions of mode and various examples of pathological measures that have no mode in any sense. Since the masses of balls induce natural orderings on the points of XX, this article aims to shed light on some of the problems in non-parametric MAP estimation by taking an order-theoretic perspective, which appears to be a new one in the inverse problems community. This point of view opens up attractive proof strategies based upon the Cantor and Kuratowski intersection theorems; it also reveals that many of the pathologies arise from the distinction between greatest and maximal elements of an order, and from the existence of incomparable elements of XX, which we show can be dense in XX, even for an absolutely continuous measure on X=RX = \mathbb{R}.Comment: 38 page

    Best approximation results and essential boundary conditions for novel types of weak adversarial network discretizations for PDEs

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    In this paper, we provide a theoretical analysis of the recently introduced weakly adversarial networks (WAN) method, used to approximate partial differential equations in high dimensions. We address the existence and stability of the solution, as well as approximation bounds. More precisely, we prove the existence of discrete solutions, intended in a suitable weak sense, for which we prove a quasi-best approximation estimate similar to Cea's lemma, a result commonly found in finite element methods. We also propose two new stabilized WAN-based formulas that avoid the need for direct normalization. Furthermore, we analyze the method's effectiveness for the Dirichlet boundary problem that employs the implicit representation of the geometry. The key requirement for achieving the best approximation outcome is to ensure that the space for the test network satisfies a specific condition, known as the inf-sup condition, essentially requiring that the test network set is sufficiently large when compared to the trial space. The method's accuracy, however, is only determined by the space of the trial network. We also devise a pseudo-time XNODE neural network class for static PDE problems, yielding significantly faster convergence results than the classical DNN network.Comment: 30 pages, 7 figure

    Reinforcement learning in large state action spaces

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    Reinforcement learning (RL) is a promising framework for training intelligent agents which learn to optimize long term utility by directly interacting with the environment. Creating RL methods which scale to large state-action spaces is a critical problem towards ensuring real world deployment of RL systems. However, several challenges limit the applicability of RL to large scale settings. These include difficulties with exploration, low sample efficiency, computational intractability, task constraints like decentralization and lack of guarantees about important properties like performance, generalization and robustness in potentially unseen scenarios. This thesis is motivated towards bridging the aforementioned gap. We propose several principled algorithms and frameworks for studying and addressing the above challenges RL. The proposed methods cover a wide range of RL settings (single and multi-agent systems (MAS) with all the variations in the latter, prediction and control, model-based and model-free methods, value-based and policy-based methods). In this work we propose the first results on several different problems: e.g. tensorization of the Bellman equation which allows exponential sample efficiency gains (Chapter 4), provable suboptimality arising from structural constraints in MAS(Chapter 3), combinatorial generalization results in cooperative MAS(Chapter 5), generalization results on observation shifts(Chapter 7), learning deterministic policies in a probabilistic RL framework(Chapter 6). Our algorithms exhibit provably enhanced performance and sample efficiency along with better scalability. Additionally, we also shed light on generalization aspects of the agents under different frameworks. These properties have been been driven by the use of several advanced tools (e.g. statistical machine learning, state abstraction, variational inference, tensor theory). In summary, the contributions in this thesis significantly advance progress towards making RL agents ready for large scale, real world applications

    Hilbert Spaces Without Countable AC

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    This article examines Hilbert spaces constructed from sets whose existence is incompatible with the Countable Axiom of Choice (CC). Our point of view is twofold: (1) We examine what can and cannot be said about Hilbert spaces and operators on them in ZF set theory without any assumptions of Choice axioms, even the CC. (2) We view Hilbert spaces as ``quantized'' sets and obtain some set-theoretic results from associated Hilbert spaces.Comment: 51 page

    Ill-posedness of time-dependent inverse problems in Lebesgue-Bochner spaces

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    We consider time-dependent inverse problems in a mathematical setting using Lebesgue-Bochner spaces. Such problems arise when one aims to recover parameters from given observations where the parameters or the data depend on time. There are various important applications being subject of current research that belong to this class of problems. Typically inverse problems are ill-posed in the sense that already small noise in the data causes tremendous errors in the solution. In this article we present two different concepts of ill-posedness: temporally (pointwise) ill-posedness and uniform ill-posedness with respect to the Lebesgue-Bochner setting. We investigate the two concepts by means of a typical setting consisting of a time-depending observation operator composed by a compact operator. Furthermore we develop regularization methods that are adapted to the respective class of ill-posedness.Comment: 21 pages, no figure

    Boundedness of Operators on Local Hardy Spaces and Periodic Solutions of Stochastic Partial Differential Equations with Regime-Switching

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    In the first part of the thesis, we discuss the boundedness of inhomogeneous singular integral operators suitable for local Hardy spaces as well as their commutators. First, we consider the equivalence of different localizations of a given convolution operator by giving minimal conditions on the localizing functions; in the case of the Riesz transforms this results in equivalent characterizations of h1h^1. Then, we provide weaker integral conditions on the kernel of the operator and sufficient and necessary cancellation conditions to ensure the boundedness on local Hardy spaces for all values of p. Finally, we introduce a new class of atoms and use them to establish the boundedness of the commutators of inhomogeneous singular integral operators with bmo function. In the second part of the thesis, we investigate periodic solutions of a class of stochastic partial differential equations driven by degenerate noises with regime-switching. First, we consider the existence and uniqueness of solutions to the equations. Then, we discuss the existence and uniqueness of periodic measures for the equations. In particular, we establish the uniqueness of periodic measures by proving the strong Feller property and irreducibility of semigroups associated with the equations. Finally, we use the stochastic fractional porous medium equation as an example to illustrate the main results

    University of Windsor Graduate Calendar 2023 Spring

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    https://scholar.uwindsor.ca/universitywindsorgraduatecalendars/1027/thumbnail.jp
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