2,356 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

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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

    2023-2024 Catalog

    Get PDF
    The 2023-2024 Governors State University Undergraduate and Graduate Catalog is a comprehensive listing of current information regarding:Degree RequirementsCourse OfferingsUndergraduate and Graduate Rules and Regulation

    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

    Locality and Exceptional Points in Pseudo-Hermitian Physics

    Get PDF
    Pseudo-Hermitian operators generalize the concept of Hermiticity. Included in this class of operators are the quasi-Hermitian operators, which define a generalization of quantum theory with real-valued measurement outcomes and unitary time evolution. This thesis is devoted to the study of locality in quasi-Hermitian theory, the symmetries and conserved quantities associated with non-Hermitian operators, and the perturbative features of pseudo-Hermitian matrices. An implicit assumption of the tensor product model of locality is that the inner product factorizes with the tensor product. Quasi-Hermitian quantum theory generalizes the tensor product model by modifying the Born rule via a metric operator with nontrivial Schmidt rank. Local observable algebras and expectation values are examined in chapter 5. Observable algebras of two one-dimensional fermionic quasi-Hermitian chains are explicitly constructed. Notably, there can be spatial subsystems with no nontrivial observables. Despite devising a new framework for local quantum theory, I show that expectation values of local quasi-Hermitian observables can be equivalently computed as expectation values of Hermitian observables. Thus, quasi-Hermitian theories do not increase the values of nonlocal games set by Hermitian theories. Furthermore, Bell's inequality violations in quasi-Hermitian theories never exceed the Tsirelson bound of Hermitian quantum theory. A perturbative feature present in pseudo-Hermitian curves which has no Hermitian counterpart is the exceptional point, a branch point in the set of eigenvalues. An original finding presented in section 2.6.3 is a correspondence between cusp singularities of algebraic curves and higher-order exceptional points. Eigensystems of one-dimensional lattice models admit closed-form expressions that can be used to explore the new features of non-Hermitian physics. One-dimensional lattice models with a pair of non Hermitian defect potentials with balanced gain and loss, Δ±iγ, are investigated in chapter 3. Conserved quantities and positive-definite metric operators are examined. When the defects are nearest neighbour, the entire spectrum simultaneously becomes complex when γ increases beyond a second-order exceptional point. When the defects are at the edges of the chain and the hopping amplitudes are 2-periodic, as in the Su-Schrieffer-Heeger chain, the PT-phase transition is dictated by the topological phase of the system. In the thermodynamic limit, PT-symmetry spontaneously breaks in the topologically non-trivial phase due to the presence of edge states. Chiral symmetry and representation theory are utilized in chapter 4 to derive large classes of pseudo-Hermitian operators with closed-form intertwining operators. These intertwining operators include positive-definite metric operators in the quasi-Hermitian case. The PT-phase transition is explicitly determined in a special case

    Mechanical characterization, constitutive modeling and applications of ultra-soft magnetorheological elastomers

    Get PDF
    Mención Internacional en el título de doctorSmart materials are bringing sweeping changes in the way humans interact with engineering devices. A myriad of state-of-the-art applications are based on novel ways to actuate on structures that respond under different types of stimuli. Among them, materials that respond to magnetic fields allow to remotely modify their mechanical properties and macroscopic shape. Ultra-soft magnetorheological elastomers (MREs) are composed of a highly stretchable soft elastomeric matrix in the order of 1 kPa and magnetic particles embedded in it. This combination allows large deformations with small external actuations. The type of the magnetic particles plays a crucial role as it defines the reversibility or remanence of the material magnetization. According to the fillers used, MREs are referred to as soft-magnetic magnetorheological elastomers (sMREs) and hard-magnetic magnetorheological elastomers (hMREs). sMREs exhibit strong changes in their mechanical properties when an external magnetic field is applied, whereas hMREs allow sustained magnetic effects along time and complex shape-morphing capabilities. In this regard, end-of-pipe applications of MREs in the literature are based on two major characteristics: the modification of their mechanical properties and macrostructural shape changes. For instance, smart actuators, sensors and soft robots for bioengineering applications are remotely actuated to perform functional deformations and autonomous locomotion. In addition, hMREs have been used for industrial applications, such as damping systems and electrical machines. From the analysis of the current state of the art, we identified some impediments to advance in certain research fields that may be overcome with new solutions based on ultrasoft MREs. On the mechanobiology area, we found no available experimental methodologies to transmit complex and dynamic heterogeneous strain patterns to biological systems in a reversible manner. To remedy this shortcoming, this doctoral research proposes a new mechanobiology experimental setup based on responsive ultra-soft MRE biological substrates. Such an endeavor requires deeper insights into the magneto-viscoelastic and microstructural mechanisms of ultra-soft MREs. In addition, there is still a lack of guidance for the selection of the magnetic fillers to be used for MREs and the final properties provided to the structure. Eventually, the great advances on both sMREs and hMREs to date pose a timely question on whether the combination of both types of particles in a hybrid MRE may optimize the multifunctional response of these active structures. To overcome these roadblocks, this thesis provides an extensive and comprehensive experimental characterization of ultra-soft sMREs, hMREs and hybrid MREs. The experimental methodology uncovers magneto-mechanical rate dependences under numerous loading and manufacturing conditions. Then, a set of modeling frameworks allows to delve into such mechanisms and develop three ground-breaking applications. Therefore, the thesis has lead to three main contributions. First and motivated on mechanobiology research, a computational framework guides a sMRE substrate to transmit complex strain patterns in vitro to biological systems. Second, we demonstrate the ability of remanent magnetic fields in hMREs to arrest cracks propagations and improve fracture toughness. Finally, the combination of soft- and hard-magnetic particles is proved to enhance the magnetorheological and magnetostrictive effects, providing promising results for soft robotics.Los materiales inteligentes están generando cambios radicales en la forma que los humanos interactúan con dispositivos ingenieriles. Distintas aplicaciones punteras se basan en formas novedosas de actuar sobre materiales que responden a diferentes estímulos. Entre ellos, las estructuras que responden a campos magnéticos permiten la modificación de manera remota tanto de sus propiedades mecánicas como de su forma. Los elastómeros magnetorreológicos (MREs) ultra blandos están compuestos por una matriz elastomérica con gran ductilidad y una rigidez en torno a 1 kPa, reforzada con partículas magnéticas. Esta combinación permite inducir grandes deformaciones en el material mediante la aplicación de campos magnéticos pequeños. La naturaleza de las partículas magnéticas define la reversibilidad o remanencia de la magnetización del material compuesto. De esta manera, según el tipo de partículas que contengan, los MREs pueden presentar magnetización débil (sMRE) o magnetización fuerte (hMRE). Los sMREs experimentan grandes cambios en sus propiedades mecánicas al aplicar un campo magnético externo, mientras que los hMREs permiten efectos magneto-mecánicos sostenidos a lo largo del tiempo, así como programar cambios de forma complejos. En este sentido, las aplicaciones de los MREs se basan en dos características principales: la modificación de sus propiedades mecánicas y los cambios de forma macroestructurales. Por ejemplo, los campos magnéticos pueden emplearse para inducir deformaciones funcionales en actuadores y sensores inteligentes, o en robótica blanda para bioingeniería. Los hMREs también se han aplicado en el ámbito industrial en sistemas de amortiguación y máquinas eléctricas. A partir del análisis del estado del arte, se identifican algunas limitaciones que impiden el avance en ciertos campos de investigación y que podrían resolverse con nuevas soluciones basadas en MREs ultra blandos. En el área de la mecanobiología, no existen metodologías experimentales para transmitir patrones de deformación complejos y dinámicos a sistemas biológicos de manera reversible. En esta investigación doctoral se propone una configuración experimental novedosa basada en sustratos biológicos fabricados con MREs ultra blandos. Dicha solución requiere la identificación de los mecanismos magneto-viscoelásticos y microestructurales de estos materiales, según el tipo de partículas magnéticas, y las consiguientes propiedades macroscópicas del material. Además, investigaciones recientes en sMREs y hMREs plantean la pregunta sobre si la combinación de distintos tipos de partículas magnéticas en un MRE híbrido puede optimizar su respuesta multifuncional. Para superar estos obstáculos, la presente tesis proporciona una caracterización experimental completa de sMREs, hMREs y MREs híbridos ultra blandos. Estos resultados muestran las dependencias del comportamiento multifuncional del material con la velocidad de aplicación de cargas magneto-mecánicas. El desarrollo de un conjunto de modelos teórico-computacionales permite profundizar en dichos mecanismos y desarrollar aplicaciones innovadoras. De este modo, la tesis doctoral ha dado lugar a tres bloques de aportaciones principales. En primer lugar, este trabajo proporciona un marco computacional para guiar el diseño de sustratos basados en sMREs para transmitir patrones de deformación complejos in vitro a sistemas biológicos. En segundo lugar, se demuestra la capacidad de los campos magnéticos remanentes en los hMRE para detener la propagación de grietas y mejorar la tenacidad a la fractura. Finalmente, se establece que la combinación de partículas magnéticas de magnetización débil y fuerte mejora el efecto magnetorreológico y magnetoestrictivo, abriendo nuevas posibilidades para el diseño de robots blandos.I want to acknowledge the support from the Ministerio de Ciencia, Innovación y Universidades, Spain (FPU19/03874), and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 947723, project: 4D-BIOMAP).Programa de Doctorado en Ingeniería Mecánica y de Organización Industrial por la Universidad Carlos III de MadridPresidente: Ramón Eulalio Zaera Polo.- Secretario: Abdón Pena Francesch.- Vocal: Laura de Lorenzi

    On the Computation of Multi-Scalar Multiplication for Pairing-Based zkSNARKs

    Get PDF
    Multi-scalar multiplication refers to the operation of computing multiple scalar multiplications in an elliptic curve group and then adding them together. It is an essential operation for proof generation and verification in pairing-based trusted setup zero-knowledge succinct non-interactive argument of knowledge (zkSNARK) schemes, which enable privacy-preserving features in many blockchain applications. Pairing-based trusted setup zkSNARKs usually follow a common paradigm. A public string composed of a list of fixed points in an elliptic curve group called common reference string is generated in a trusted setup and accessible to all parties involved. The prover generates a zkSNARK proof by computing multi-scalar multiplications over the points in the common reference string and performing other operations. The verifier verifies the proof by computing multi-scalar multiplications and elliptic curve bilinear pairings. Multi-scalar multiplication in pairing-based trusted setup zkSNARKs has two characteristics. First, all the points are fixed once the common reference string is generated. Second, the number of points n is typically large, with the thesis targeting at n = 2^e (10 ≤ e ≤ 21). Our goal in this thesis is to propose and implement efficient algorithms for computing multi-scalar multiplication in order to enable efficient zkSNARKs. This thesis primarily includes three aspects. First, the background knowledge is introduced and the classical multi-scalar multiplication algorithms are reviewed. Second, two frameworks for computing multi-scalar multiplications over fixed points and five corresponding auxiliary set pairs are proposed. Finally, the theoretical analysis, software implementation, and experimental tests on the representative instantiations of the proposed frameworks are presented

    Asymptotics of stochastic learning in structured networks

    Get PDF

    Implicit Loss of Surjectivity and Facial Reduction: Theory and Applications

    Get PDF
    Facial reduction, pioneered by Borwein and Wolkowicz, is a preprocessing method that is commonly used to obtain strict feasibility in the reformulated, reduced constraint system. The importance of strict feasibility is often addressed in the context of the convergence results for interior point methods. Beyond the theoretical properties that the facial reduction conveys, we show that facial reduction, not only limited to interior point methods, leads to strong numerical performances in different classes of algorithms. In this thesis we study various consequences and the broad applicability of facial reduction. The thesis is organized in two parts. In the first part, we show the instabilities accompanied by the absence of strict feasibility through the lens of facially reduced systems. In particular, we exploit the implicit redundancies, revealed by each nontrivial facial reduction step, resulting in the implicit loss of surjectivity. This leads to the two-step facial reduction and two novel related notions of singularity. For the area of semidefinite programming, we use these singularities to strengthen a known bound on the solution rank, the Barvinok-Pataki bound. For the area of linear programming, we reveal degeneracies caused by the implicit redundancies. Furthermore, we propose a preprocessing tool that uses the simplex method. In the second part of this thesis, we continue with the semidefinite programs that do not have strictly feasible points. We focus on the doubly-nonnegative relaxation of the binary quadratic program and a semidefinite program with a nonlinear objective function. We closely work with two classes of algorithms, the splitting method and the Gauss-Newton interior point method. We elaborate on the advantages in building models from facial reduction. Moreover, we develop algorithms for real-world problems including the quadratic assignment problem, the protein side-chain positioning problem, and the key rate computation for quantum key distribution. Facial reduction continues to play an important role for providing robust reformulated models in both the theoretical and the practical aspects, resulting in successful numerical performances
    corecore