317 research outputs found

    Classical and quantum algorithms for scaling problems

    Get PDF
    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

    Algorithms and complexity for approximately counting hypergraph colourings and related problems

    Get PDF
    The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of LovĂĄsz local lemma when Δ â‰Č qᔏ. In prior, however, fast approximate counting algorithms exist when Δ â‰Č qᔏ/Âł, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows. ‱ When q, k ≄ 4 are evens and Δ ≄ 5·qᔏ/ÂČ, approximating the number of hypergraph colourings is NP-hard. ‱ When the input hypergraph is linear and Δ â‰Č qᔏ/ÂČ, a fast approximate counting algorithm does exist

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Differentially Private Stochastic Convex Optimization in (Non)-Euclidean Space Revisited

    Full text link
    In this paper, we revisit the problem of Differentially Private Stochastic Convex Optimization (DP-SCO) in Euclidean and general ℓpd\ell_p^d spaces. Specifically, we focus on three settings that are still far from well understood: (1) DP-SCO over a constrained and bounded (convex) set in Euclidean space; (2) unconstrained DP-SCO in ℓpd\ell_p^d space; (3) DP-SCO with heavy-tailed data over a constrained and bounded set in ℓpd\ell_p^d space. For problem (1), for both convex and strongly convex loss functions, we propose methods whose outputs could achieve (expected) excess population risks that are only dependent on the Gaussian width of the constraint set rather than the dimension of the space. Moreover, we also show the bound for strongly convex functions is optimal up to a logarithmic factor. For problems (2) and (3), we propose several novel algorithms and provide the first theoretical results for both cases when 1<p<21<p<2 and 2≀p≀∞2\leq p\leq \infty

    Fluidic Nozzles for Automotive Washer Systems: Computational Fluid Dynamics and Experimental Analysis

    Get PDF
    One of the main goals of this project was to cultivate an understanding of fluidic nozzle geometries and characteristic flow. Through this knowledge, three new fluidic nozzle concepts were developed to be used as components in several windscreen washer systems for an automotive part supplier, Kautex Textron CVS Ltd.Accurate and conclusive visualisation of flow through fluidic nozzles was vital in understanding how they can be best utilised for different applications. Over the past century, the specific needs of automotive cleaning systems have greatly developed with new technological discoveries, these advances allow the driver further knowledge of their surroundings. These specialised systems each require a different type of maintenance and cleaning system depending on their usage and the different size and shape of the vehicle. By completing this project, it is hoped to allow manufacturers to accurately identify what sort of fluidic nozzles are best for windscreen cleaning systems for a vehicle and how to design a nozzle to suit their specification. Fluidic nozzles have been researched experimentally and computationally to ensure an accurate comparison of results. By guaranteeing a precise comparison it will negate the need for high volume testing of nozzles in experimental situations, greatly reducing time and resources required to analyse a fluidic nozzle.The fluidic nozzles that are investigated and developed in this project were modelled and examined both experimentally and computationally, this ensured valid and accurate results were achieved by both the computational modelling and experimental testing. The development of the nozzles within this project was conducted using several experimental and computational setups to analyse the spray distribution, angle and oscillatory frequency amongst other parameters significant to the nozzle usage on a vehicle. Through this it was possible to tailor nozzle dimensions to allow for a streamlined design approach, this increased efficiency in fluidic nozzle development for any specification given by a vehicle manufacturing company customer. In addition to this the water flow emitted from the outlet was experimentally tested and modelled with both stationary and high surrounding velocities to examine how external variables affect the flow of the water from the nozzle.iiiThis project has been useful in the design manufacturing process of fluidic nozzles, by utilising computational modelling it has allowed a faster and cheaper method of analysing the effect of design alterations to fluidic nozzles. There is a greatly reduced frequency required for rapid prototyping of an array of fluidic chips with minimal dimensional differences to be used in the experimental stages of design, as once the inlet boundary conditions are established the nozzle can be redesigned completely within reason without the need for additional material wastage. This ensures a more easy and precise method of testing the manufacturing tolerances of a fluidic nozzle with a target of reaching customer specifications are always achieved.Three nozzles were aimed developed to satisfy conditions set by the customers, the vehicle manufacturers at which the new nozzle designs are aimed at are Honda, Nissan and Toyota. The nozzles to be established were designed for use on windscreen washer systems with a varying number of nozzles and with diverse windscreen sizes for different vehicles, resulting in a wide variety of specifications that must be met for each vehicle manufacturer. This meant that a single nozzle could not be utilised for all vehicles, instead a base model of fluidic chip was developed for the Nissan vehicle which was then dimensionally changed to suit the other vehicles.Throughout this project there were design specifications changes and ambiguities from the automotive company customers, leading to redesigns of the fluidic chips designed in this project. This means that although only two of the three fluidic nozzle designs are successfully in production, a much greater understanding of the mechanics of the fluid flow within the fluidic nozzle was achieved

    Semantic Security with Infinite Dimensional Quantum Eavesdropping Channel

    Full text link
    We propose a new proof method for direct coding theorems for wiretap channels where the eavesdropper has access to a quantum version of the transmitted signal on an infinite-dimensional Hilbert space and the legitimate parties communicate through a classical channel or a classical input, quantum output (cq) channel. The transmitter input can be subject to an additive cost constraint, which specializes to the case of an average energy constraint. This method yields errors that decay exponentially with increasing block lengths. Moreover, it provides a guarantee of a quantum version of semantic security, which is an established concept in classical cryptography and physical layer security. Therefore, it complements existing works which either do not prove the exponential error decay or use weaker notions of security. The main part of this proof method is a direct coding result on channel resolvability which states that there is only a doubly exponentially small probability that a standard random codebook does not solve the channel resolvability problem for the cq channel. Semantic security has strong operational implications meaning essentially that the eavesdropper cannot use its quantum observation to gather any meaningful information about the transmitted signal. We also discuss the connections between semantic security and various other established notions of secrecy

    Learning and Control of Dynamical Systems

    Get PDF
    Despite the remarkable success of machine learning in various domains in recent years, our understanding of its fundamental limitations remains incomplete. This knowledge gap poses a grand challenge when deploying machine learning methods in critical decision-making tasks, where incorrect decisions can have catastrophic consequences. To effectively utilize these learning-based methods in such contexts, it is crucial to explicitly characterize their performance. Over the years, significant research efforts have been dedicated to learning and control of dynamical systems where the underlying dynamics are unknown or only partially known a priori, and must be inferred from collected data. However, much of these classical results have focused on asymptotic guarantees, providing limited insights into the amount of data required to achieve desired control performance while satisfying operational constraints such as safety and stability, especially in the presence of statistical noise. In this thesis, we study the statistical complexity of learning and control of unknown dynamical systems. By utilizing recent advances in statistical learning theory, high-dimensional statistics, and control theoretic tools, we aim to establish a fundamental understanding of the number of samples required to achieve desired (i) accuracy in learning the unknown dynamics, (ii) performance in the control of the underlying system, and (iii) satisfaction of the operational constraints such as safety and stability. We provide finite-sample guarantees for these objectives and propose efficient learning and control algorithms that achieve the desired performance at these statistical limits in various dynamical systems. Our investigation covers a broad range of dynamical systems, starting from fully observable linear dynamical systems to partially observable linear dynamical systems, and ultimately, nonlinear systems. We deploy our learning and control algorithms in various adaptive control tasks in real-world control systems and demonstrate their strong empirical performance along with their learning, robustness, and stability guarantees. In particular, we implement one of our proposed methods, Fourier Adaptive Learning and Control (FALCON), on an experimental aerodynamic testbed under extreme turbulent flow dynamics in a wind tunnel. The results show that FALCON achieves state-of-the-art stabilization performance and consistently outperforms conventional and other learning-based methods by at least 37%, despite using 8 times less data. The superior performance of FALCON arises from its physically and theoretically accurate modeling of the underlying nonlinear turbulent dynamics, which yields rigorous finite-sample learning and performance guarantees. These findings underscore the importance of characterizing the statistical complexity of learning and control of unknown dynamical systems.</p

    When Deep Learning Meets Polyhedral Theory: A Survey

    Full text link
    In the past decade, deep learning became the prevalent methodology for predictive modeling thanks to the remarkable accuracy of deep neural networks in tasks such as computer vision and natural language processing. Meanwhile, the structure of neural networks converged back to simpler representations based on piecewise constant and piecewise linear functions such as the Rectified Linear Unit (ReLU), which became the most commonly used type of activation function in neural networks. That made certain types of network structure \unicode{x2014}such as the typical fully-connected feedforward neural network\unicode{x2014} amenable to analysis through polyhedral theory and to the application of methodologies such as Linear Programming (LP) and Mixed-Integer Linear Programming (MILP) for a variety of purposes. In this paper, we survey the main topics emerging from this fast-paced area of work, which bring a fresh perspective to understanding neural networks in more detail as well as to applying linear optimization techniques to train, verify, and reduce the size of such networks

    Asymptotics of stochastic learning in structured networks

    Get PDF
    • 

    corecore