Leonid Gurvits

Bio: Leonid Gurvits is an academic researcher from City College of New York. The author has contributed to research in topics: Very-long-baseline interferometry & Matrix (mathematics). The author has an hindex of 36, co-authored 178 publications receiving 5063 citations. Previous affiliations of Leonid Gurvits include Princeton University & Courant Institute of Mathematical Sciences.

Mobile robot localization using landmarks

Abstract: We describe an efficient method for localizing a mobile robot in an environment with landmarks. We assume that the robot can identify these landmarks and measure their bearings relative to each other. Given such noisy input, the algorithm estimates the robot's position and orientation with respect to the map of the environment. The algorithm makes efficient use of our representation of the landmarks by complex numbers. The algorithm runs in time linear in the number of landmarks. We present results of simulations and propose how to use our method for robot navigation.

Classical deterministic complexity of Edmonds' Problem and quantum entanglement

Abstract: Generalizing a decision problem for bipartite perfect matching, J. Edmonds introduced in [14] the problem (now known as the Edmonds Problem) of deciding if a given linear subspace of M(N) contains a nonsingular matrix, where M(N) stands for the linear space of complex NxN matrices. This problem led to many fundamental developments in matroid theory etc.Classical matching theory can be defined in terms of matrices with nonnegative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with nonnegative entries. (Here operator refers to maps from matrices to matrices.) First, we reformulate the Edmonds Problem in terms of of completely positive operators, or equivalently, in terms of bipartite density matrices. It turns out that one of the most important cases when Edmonds' problem can be solved in polynomial deterministic time, i.e. an intersection of two geometric matroids, corresponds to unentangled (aka separable) bipartite density matrices. We introduce a very general class (or promise) of linear subspaces of M(N) on which there exists a polynomial deterministic time algorithm to solve Edmonds' problem. The algorithm is a thoroughgoing generalization of algorithms in [23], [26], and its analysis benefits from an operator analog of permanents, so called Quantum Permanents. Finally, we prove that the weak membership problem for the convex set of separable normalized bipartite density matrices is NP-HARD.

On the Stability of Switched Positive Linear Systems

Abstract: It was recently conjectured that the Hurwitz stability of the convex hull of a set of Metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching. In this note, we show that (1) this conjecture is true for systems constructed from a pair of second-order Metzler matrices; (2) the conjecture is true for systems constructed from an arbitrary finite number of second-order Metzler matrices; and (3) the conjecture is in general false for higher order systems. The implications of our results, both for the design of switched positive linear systems, and for research directions that arise as a result of our work, are discussed toward the end of the note.

Classical complexity and quantum entanglement

Abstract: Generalizing a decision problem for bipartite perfect matching, Edmonds (J. Res. Natl. Bur. Standards 718(4) (1967) 242) introduced the problem (now known as the Edmonds Problem) of deciding if a given linear subspace of M(N) contains a non-singular matrix, where M(N) stands for the linear space of complex N × N matrices. This problem led to many fundamental developments in matroid theory, etc.Classical matching theory can be defined in terms of matrices with non-negative entries. The notion of Positive operator, central in Quantum Theory, is a natural generalization of matrices with non-negative entries. (Here operator refers to maps from matrices to matrices.) First, we reformulate the Edmonds Problem in terms of completely positive operators, or equivalently, in terms of bipartite density matrices. It turns out that one of the most important cases when Edmonds' problem can be solved in polynomial deterministic time, i.e. an intersection of two geometric matroids, corresponds to unentangled (aka separable) bipartite density matrices. We introduce a very general class (or promise) of linear subspaces of M(N) on which there exists a polynomial deterministic time algorithm to solve Edmonds' problem. The algorithm is a thoroughgoing generalization of algorithms in Linial, Samorodnitsky and Wigderson, Proceedings of the 30th ACM Symposium on Theory of Computing, ACM, New York, 1998; Gurvits and Yianilos, and its analysis benefits from an operator analog of permanents, so-called Quantum Permanents.Finally, we prove that the weak membership problem for the convex set of separable normalized bipartite density matrices is NP-HARD.

Quantum entanglement

Abstract: All our former experience with application of quantum theory seems to say: {\it what is predicted by quantum formalism must occur in laboratory} But the essence of quantum formalism - entanglement, recognized by Einstein, Podolsky, Rosen and Schr\"odinger - waited over 70 years to enter to laboratories as a new resource as real as energy This holistic property of compound quantum systems, which involves nonclassical correlations between subsystems, is a potential for many quantum processes, including ``canonical'' ones: quantum cryptography, quantum teleportation and dense coding However, it appeared that this new resource is very complex and difficult to detect Being usually fragile to environment, it is robust against conceptual and mathematical tools, the task of which is to decipher its rich structure This article reviews basic aspects of entanglement including its characterization, detection, distillation and quantifying In particular, the authors discuss various manifestations of entanglement via Bell inequalities, entropic inequalities, entanglement witnesses, quantum cryptography and point out some interrelations They also discuss a basic role of entanglement in quantum communication within distant labs paradigm and stress some peculiarities such as irreversibility of entanglement manipulations including its extremal form - bound entanglement phenomenon A basic role of entanglement witnesses in detection of entanglement is emphasized

A Mathematical Introduction to Robotic Manipulation

Abstract: INTRODUCTION: Brief History. Multifingered Hands and Dextrous Manipulation. Outline of the Book. Bibliography. RIGID BODY MOTION: Rigid Body Transformations. Rotational Motion in R3. Rigid Motion in R3. Velocity of a Rigid Body. Wrenches and Reciprocal Screws. MANIPULATOR KINEMATICS: Introduction. Forward Kinematics. Inverse Kinematics. The Manipulator Jacobian. Redundant and Parallel Manipulators. ROBOT DYNAMICS AND CONTROL: Introduction. Lagrange's Equations. Dynamics of Open-Chain Manipulators. Lyapunov Stability Theory. Position Control and Trajectory Tracking. Control of Constrained Manipulators. MULTIFINGERED HAND KINEMATICS: Introduction to Grasping. Grasp Statics. Force-Closure. Grasp Planning. Grasp Constraints. Rolling Contact Kinematics. HAND DYNAMICS AND CONTROL: Lagrange's Equations with Constraints. Robot Hand Dynamics. Redundant and Nonmanipulable Robot Systems. Kinematics and Statics of Tendon Actuation. Control of Robot Hands. NONHOLONOMIC BEHAVIOR IN ROBOTIC SYSTEMS: Introduction. Controllability and Frobenius' Theorem. Examples of Nonholonomic Systems. Structure of Nonholonomic Systems. NONHOLONOMIC MOTION PLANNING: Introduction. Steering Model Control Systems Using Sinusoids. General Methods for Steering. Dynamic Finger Repositioning. FUTURE PROSPECTS: Robots in Hazardous Environments. Medical Applications for Multifingered Hands. Robots on a Small Scale: Microrobotics. APPENDICES: Lie Groups and Robot Kinematics. A Mathematica Package for Screw Calculus. Bibliography. Index Each chapter also includes a Summary, Bibliography, and Exercises

Planning Algorithms: Introductory Material

Abstract: Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Advances in kernel methods: support vector learning

Abstract: Introduction to support vector learning roadmap. Part 1 Theory: three remarks on the support vector method of function estimation, Vladimir Vapnik generalization performance of support vector machines and other pattern classifiers, Peter Bartlett and John Shawe-Taylor Bayesian voting schemes and large margin classifiers, Nello Cristianini and John Shawe-Taylor support vector machines, reproducing kernel Hilbert spaces, and randomized GACV, Grace Wahba geometry and invariance in kernel based methods, Christopher J.C. Burges on the annealed VC entropy for margin classifiers - a statistical mechanics study, Manfred Opper entropy numbers, operators and support vector kernels, Robert C. Williamson et al. Part 2 Implementations: solving the quadratic programming problem arising in support vector classification, Linda Kaufman making large-scale support vector machine learning practical, Thorsten Joachims fast training of support vector machines using sequential minimal optimization, John C. Platt. Part 3 Applications: support vector machines for dynamic reconstruction of a chaotic system, Davide Mattera and Simon Haykin using support vector machines for time series prediction, Klaus-Robert Muller et al pairwise classification and support vector machines, Ulrich Kressel. Part 4 Extensions of the algorithm: reducing the run-time complexity in support vector machines, Edgar E. Osuna and Federico Girosi support vector regression with ANOVA decomposition kernels, Mark O. Stitson et al support vector density estimation, Jason Weston et al combining support vector and mathematical programming methods for classification, Bernhard Scholkopf et al.