Ellipsoid

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M C Campi - One of the best experts on this subject based on the ideXlab platform.

  • brief non asymptotic confidence Ellipsoids for the least squares estimate
    Automatica, 2002
    Co-Authors: Erik Weyer, M C Campi
    Abstract:

    In this paper, we consider the finite sample properties of least-squares system identification, and derive non-asymptotic confidence Ellipsoids for the estimate. The shape of the confidence Ellipsoids is similar to the shape of the Ellipsoids derived using asymptotic theory, but unlike asymptotic theory, they are valid for a finite number of data points. The probability that the estimate belongs to a certain Ellipsoid has a natural dependence on the volume of the Ellipsoid, the data generating mechanism, the model order and the number of data points available.

  • non asymptotic confidence Ellipsoids for the least squares estimate
    Conference on Decision and Control, 2000
    Co-Authors: Erik Weyer, M C Campi
    Abstract:

    In this paper we consider the finite sample properties of least squares system identification, and we derive nonasymptotic confidence Ellipsoids for the estimate. Unlike asymptotic theory, the obtained confidence Ellipsoids are valid for a finite number of data points. The probability that the estimate belongs to a certain Ellipsoid has a natural dependence on the volume of the Ellipsoid, the data generating mechanism, the model order and the number of data points available.

Ge Xiong - One of the best experts on this subject based on the ideXlab platform.

  • The logarithmic John Ellipsoid
    Geometriae Dedicata, 2018
    Co-Authors: Ge Xiong
    Abstract:

    The logarithmic John Ellipsoid of a convex body in \({\mathbb {R}}^{n}\) with its centroid at the origin is introduced by solving a pair of dual optimization problems. Convex bodies with identical John and logarithmic John Ellipsoids are characterized.

  • Orlicz–Legendre Ellipsoids
    The Journal of Geometric Analysis, 2015
    Co-Authors: Du Zou, Ge Xiong
    Abstract:

    The Orlicz–Legendre Ellipsoids, which are in the framework of emerging dual Orlicz Brunn–Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz–John Ellipsoids, and have largely generalized the classical Legendre Ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz–Legendre Ellipsoids and isotropy of measures is demonstrated.

  • Orlicz-Legendre Ellipsoids
    arXiv: Metric Geometry, 2014
    Co-Authors: Du Zou, Ge Xiong
    Abstract:

    The Orlicz-Legendre Ellipsoids, which are in the framework of emerging dual Orlicz Brunn-Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz-John Ellipsoids, and have largely generalized the classical Legendre Ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz-Legendre Ellipsoids and isotropy of measures is demonstrated.

  • Orlicz–John Ellipsoids
    Advances in Mathematics, 2014
    Co-Authors: Du Zou, Ge Xiong
    Abstract:

    Abstract The Orlicz–John Ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John Ellipsoid and the evolved L p John Ellipsoids. The analog of Ball's volume-ratio inequality is established for the new Orlicz–John Ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John Ellipsoids is demonstrated.

Erik Weyer - One of the best experts on this subject based on the ideXlab platform.

  • brief non asymptotic confidence Ellipsoids for the least squares estimate
    Automatica, 2002
    Co-Authors: Erik Weyer, M C Campi
    Abstract:

    In this paper, we consider the finite sample properties of least-squares system identification, and derive non-asymptotic confidence Ellipsoids for the estimate. The shape of the confidence Ellipsoids is similar to the shape of the Ellipsoids derived using asymptotic theory, but unlike asymptotic theory, they are valid for a finite number of data points. The probability that the estimate belongs to a certain Ellipsoid has a natural dependence on the volume of the Ellipsoid, the data generating mechanism, the model order and the number of data points available.

  • non asymptotic confidence Ellipsoids for the least squares estimate
    Conference on Decision and Control, 2000
    Co-Authors: Erik Weyer, M C Campi
    Abstract:

    In this paper we consider the finite sample properties of least squares system identification, and we derive nonasymptotic confidence Ellipsoids for the estimate. Unlike asymptotic theory, the obtained confidence Ellipsoids are valid for a finite number of data points. The probability that the estimate belongs to a certain Ellipsoid has a natural dependence on the volume of the Ellipsoid, the data generating mechanism, the model order and the number of data points available.

Terry Rudolph - One of the best experts on this subject based on the ideXlab platform.

  • Geometric representation of two-qubit entanglement witnesses
    Physical Review A, 2015
    Co-Authors: Antony Milne, David Jennings, Terry Rudolph
    Abstract:

    Any two-qubit state can be represented geometrically by a steering Ellipsoid inside the Blochsphere. We extend this approach to represent any block positive two-qubit operator B. We derive a classification scheme based on the positivity of detBand de t B T B; this shows that any Ellipsoid inside the Bloch sphere must represent either a two-qubit state or a two-qubit entanglement witness. We focus on such witnesses and their corresponding Ellipsoids, finding that properties such as witness optimality are naturally manifest in this geometric representation

  • quantum steering Ellipsoids extremal physical states and monogamy
    New Journal of Physics, 2014
    Co-Authors: Antony Milne, Howard Mark Wiseman, David Jennings, Sania Jevtic, Terry Rudolph
    Abstract:

    Any two-qubit state can be faithfully represented by a steering Ellipsoid inside the Bloch sphere, but not every Ellipsoid inside the Bloch sphere corresponds to a two-qubit state. We give necessary and sufficient conditions for when the geometric data describe a physical state and investigate maximal volume Ellipsoids lying on the physical-unphysical boundary. We derive monogamy relations for steering that are strictly stronger than the Coffman–Kundu–Wootters (CKW) inequality for monogamy of concurrence. The CKW result is thus found to follow from the simple perspective of steering Ellipsoid geometry. Remarkably, we can also use steering Ellipsoids to derive non-trivial results in classical Euclidean geometry, extending Eulerʼs inequality for the circumradius and inradius of a triangle.

Du Zou - One of the best experts on this subject based on the ideXlab platform.

  • Orlicz–Legendre Ellipsoids
    The Journal of Geometric Analysis, 2015
    Co-Authors: Du Zou, Ge Xiong
    Abstract:

    The Orlicz–Legendre Ellipsoids, which are in the framework of emerging dual Orlicz Brunn–Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz–John Ellipsoids, and have largely generalized the classical Legendre Ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz–Legendre Ellipsoids and isotropy of measures is demonstrated.

  • Orlicz-Legendre Ellipsoids
    arXiv: Metric Geometry, 2014
    Co-Authors: Du Zou, Ge Xiong
    Abstract:

    The Orlicz-Legendre Ellipsoids, which are in the framework of emerging dual Orlicz Brunn-Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz-John Ellipsoids, and have largely generalized the classical Legendre Ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz-Legendre Ellipsoids and isotropy of measures is demonstrated.

  • Orlicz–John Ellipsoids
    Advances in Mathematics, 2014
    Co-Authors: Du Zou, Ge Xiong
    Abstract:

    Abstract The Orlicz–John Ellipsoids, which are in the framework of the booming Orlicz Brunn–Minkowski theory, are introduced for the first time. It turns out that they are generalizations of the classical John Ellipsoid and the evolved L p John Ellipsoids. The analog of Ball's volume-ratio inequality is established for the new Orlicz–John Ellipsoids. The connection between the isotropy of measures and the characterization of Orlicz–John Ellipsoids is demonstrated.