158 research outputs found

    Classical and quantum algorithms for scaling problems

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    This thesis is concerned with scaling problems, which have a plethora of connections to different areas of mathematics, physics and computer science. Although many structural aspects of these problems are understood by now, we only know how to solve them efficiently in special cases.We give new algorithms for non-commutative scaling problems with complexity guarantees that match the prior state of the art. To this end, we extend the well-known (self-concordance based) interior-point method (IPM) framework to Riemannian manifolds, motivated by its success in the commutative setting. Moreover, the IPM framework does not obviously suffer from the same obstructions to efficiency as previous methods. It also yields the first high-precision algorithms for other natural geometric problems in non-positive curvature.For the (commutative) problems of matrix scaling and balancing, we show that quantum algorithms can outperform the (already very efficient) state-of-the-art classical algorithms. Their time complexity can be sublinear in the input size; in certain parameter regimes they are also optimal, whereas in others we show no quantum speedup over the classical methods is possible. Along the way, we provide improvements over the long-standing state of the art for searching for all marked elements in a list, and computing the sum of a list of numbers.We identify a new application in the context of tensor networks for quantum many-body physics. We define a computable canonical form for uniform projected entangled pair states (as the solution to a scaling problem), circumventing previously known undecidability results. We also show, by characterizing the invariant polynomials, that the canonical form is determined by evaluating the tensor network contractions on networks of bounded size

    Approximate Methods for Marginal Likelihood Estimation

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    We consider the estimation of the marginal likelihood in Bayesian statistics, a essential and important task known to be computationally expensive when the dimension of the parameter space is large. We propose a general algorithm with numerous extensions that can be widely applied to a variety of problem settings and excels particularly when dealing with near log-concave posteriors. Our method hinges on a novel idea that uses MCMC samples to partition the parameter space and forms local approximations over these partition sets as a means of estimating the marginal likelihood. In this dissertation, we provide both the motivation and the groundwork for developing what we call the Hybrid estimator. Our numerical experiments show the versatility and accuracy of the proposed estimator, even as the parameter space becomes increasingly high-dimensional and complicated

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    RankSEG: A Consistent Ranking-based Framework for Segmentation

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    Segmentation has emerged as a fundamental field of computer vision and natural language processing, which assigns a label to every pixel/feature to extract regions of interest from an image/text. To evaluate the performance of segmentation, the Dice and IoU metrics are used to measure the degree of overlap between the ground truth and the predicted segmentation. In this paper, we establish a theoretical foundation of segmentation with respect to the Dice/IoU metrics, including the Bayes rule and Dice-/IoU-calibration, analogous to classification-calibration or Fisher consistency in classification. We prove that the existing thresholding-based framework with most operating losses are not consistent with respect to the Dice/IoU metrics, and thus may lead to a suboptimal solution. To address this pitfall, we propose a novel consistent ranking-based framework, namely RankDice/RankIoU, inspired by plug-in rules of the Bayes segmentation rule. Three numerical algorithms with GPU parallel execution are developed to implement the proposed framework in large-scale and high-dimensional segmentation. We study statistical properties of the proposed framework. We show it is Dice-/IoU-calibrated, and its excess risk bounds and the rate of convergence are also provided. The numerical effectiveness of RankDice/mRankDice is demonstrated in various simulated examples and Fine-annotated CityScapes, Pascal VOC and Kvasir-SEG datasets with state-of-the-art deep learning architectures.Comment: 50 page

    LIPIcs, Volume 274, ESA 2023, Complete Volume

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    LIPIcs, Volume 274, ESA 2023, Complete Volum

    Dissipative ground state preparation and the Dissipative Quantum Eigensolver

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    For any local Hamiltonian H, I construct a local CPT map and stopping condition which converges to the ground state subspace of H. Like any ground state preparation algorithm, this algorithm necessarily has exponential run-time in general (otherwise BQP=QMA), even for gapped, frustration-free Hamiltonians (otherwise BQP is in NP). However, this dissipative quantum eigensolver has a number of interesting characteristics, which give advantages over previous ground state preparation algorithms. - The entire algorithm consists simply of iterating the same set of local measurements repeatedly. - The expected overlap with the ground state subspace increases monotonically with the length of time this process is allowed to run. - It converges to the ground state subspace unconditionally, without any assumptions on or prior information about the Hamiltonian. - The algorithm does not require any variational optimisation over parameters. - It is often able to find the ground state in low circuit depth in practice. - It has a simple implementation on certain types of quantum hardware, in particular photonic quantum computers. - The process is immune to errors in the initial state. - It is inherently error- and noise-resilient, i.e. to errors during execution of the algorithm and also to faulty implementation of the algorithm itself, without incurring any computational overhead: the overlap of the output with the ground state subspace degrades smoothly with the error rate, independent of the algorithm's run-time. I give rigorous proofs of the above claims, and benchmark the algorithm on some concrete examples numerically.Comment: 58 pages, 6 tables+figures, 58 theorems etc. v2: Small generalisations and clarifications of results; 63 pages, 5 tables+figures, 62 theorems et

    Graph polynomials associated with Dyson-Schwinger equations

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    Quantum motions are encoded by a particular family of recursive Hochschild equations in the renormalization Hopf algebra which represent Dyson-Schwinger equations, combinatorially. Feynman graphons, which topologically complete the space of Feynman diagrams of a gauge field theory, are considered to formulate some random graph representations for solutions of quantum motions. This framework leads us to explain the structures of Tutte and Kirchhoff-Symanzik polynomials associated with solutions of Dyson-Schwinger equations. These new graph polynomials are applied to formulate a new parametric representation for large Feynman diagrams and their corresponding Feynman rules

    Around stability for functional inequalities

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    Les inégalités fonctionnelles sont des inégalités qui encodent beaucoup d'information, tant de nature probabiliste (concentration de la mesure), qu'analytique (théorie spectrale des opérateurs) ou encore géométrique (profil isopérimétrique). L'inégalité de Poincaré en est un exemple fondamental. Dans cette thèse, nous obtenons des résultats de stabilité dans le cadre d'hypothèses de normalisation de moments, ainsi que dans le cadre de conditions de courbure-dimension. Un résultat de stabilité est une façon de quantifier la différence entre deux situations dans lesquelles les mêmes inégalités fonctionnelles sont presque vérifiées. Les résultats de stabilité obtenus dans cette thèse sont en particulier basés sur la méthode de Stein, qui est une méthode en plein développement ces dernières années, provenant du domaine des statistiques et permettant d'établir des estimations quantitatives sur des résultats de convergence. Par ailleurs, une partie de cette thèse est consacrée à l'étude des constantes optimales des inégalités de Bobkov, qui sont des inégalités fonctionnelles à caractère isopérimétrique.Functional inequalities are inequalities that encode a lot of information, both of a probabilistic (the concentration of measure phenomenon), analytical (the spectral theory of operators) and geometric (isoperimetric profile) nature. The Poincaré inequality is a fundamental example. In this thesis, we obtain stability results under moment normalisation assumptions, as well as under curvature-dimension conditions. A stability result is a way to quantify the difference between two situations where almost the same functional inequalities are verified. The stability results obtained in this thesis are in particular based on the Stein method, which is a method in full development in recent years, coming from the field of statistics and allowing to establish quantitative estimates on convergence results. In addition, a part of this thesis is devoted to the study of the optimal constants of Bobkov inequalities, which are functional inequalities of isoperimetric character

    Modern Cryptography Volume 1

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    This open access book systematically explores the statistical characteristics of cryptographic systems, the computational complexity theory of cryptographic algorithms and the mathematical principles behind various encryption and decryption algorithms. The theory stems from technology. Based on Shannon's information theory, this book systematically introduces the information theory, statistical characteristics and computational complexity theory of public key cryptography, focusing on the three main algorithms of public key cryptography, RSA, discrete logarithm and elliptic curve cryptosystem. It aims to indicate what it is and why it is. It systematically simplifies and combs the theory and technology of lattice cryptography, which is the greatest feature of this book. It requires a good knowledge in algebra, number theory and probability statistics for readers to read this book. The senior students majoring in mathematics, compulsory for cryptography and science and engineering postgraduates will find this book helpful. It can also be used as the main reference book for researchers in cryptography and cryptographic engineering areas

    LIPIcs, Volume 244, ESA 2022, Complete Volume

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    LIPIcs, Volume 244, ESA 2022, Complete Volum
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