Package org.drip.spaces.cover

Class MaureyOperatorCoveringBounds

java.lang.Object
org.drip.spaces.cover.MaureyOperatorCoveringBounds
public class MaureyOperatorCoveringBounds
extends java.lang.Object
MaureyOperatorCoveringBounds implements the estimate the Upper Bounds and/or Absolute Values of the Covering Number for the Hilbert Rd To Supremum Rd Operator Class. The Main References are:

  • Carl, B. (1985): Inequalities of the Bernstein-Jackson type and the Degree of Compactness of Operators in Banach Spaces Annals of the Fourier Institute 35 (3) 79-118
  • Carl, B., and I. Stephani (1990): Entropy, Compactness, and the Approximation of Operators Cambridge University Press Cambridge UK
  • Williamson, R. C., A. J. Smola, and B. Scholkopf (2000): Entropy Numbers of Linear Function Classes, in: Proceedings of the 13th Annual Conference on Computational Learning Theory ACM New York
It provides the following Functionality:
  • Maurey Constant - from the Hilbert - Supremum Identity Map Estimate
  • Maurey Constant - from the Williamson-Smola-Scholkopf Estimate
  • Construct an Instance Hilbert To Supremum Identity Map based Maurey Operator Covering Bounds
  • Construct an Instance of the Maurey Operator Covering Bounds based upon the Williamson, Smola, and Scholkopf Estimate
  • MaureyOperatorCoveringBounds Constructor
  • Retrieve the Maurey Constant
  • Retrieve the Supremum Dimension
  • Retrieve the Operator Norm of Interest
  • Compute the Upper Bound for the Dyadic Entropy Number
  • Compute the Upper Bound for the Entropy Number

Module Computational Core Module
Library Statistical Learning Library
Project R1 and Rd Vector/Tensor Spaces (Validated and/or Normed), and Function Classes
Package Vector Spaces Covering Number Estimator
Author:
Lakshmi Krishnamurthy

  • Field Summary

    Fields
    Modifier and Type Field Description
    static double HILBERT_SUPREMUM_IDENTITY_CONSTANT
    Maurey Constant - from the Hilbert - Supremum Identity Map Estimate
    static double WILLIAMSON_SMOLA_SCHOLKOPF_CONSTANT
    Maurey Constant - from the Williamson-Smola-Scholkopf Estimate

  • Constructor Summary

    Constructors
    Constructor Description
    MaureyOperatorCoveringBounds​(double maureyConstant, int supremumDimension, double operatorNorm)
    MaureyOperatorCoveringBounds Constructor

  • Method Summary

    Modifier and Type Method Description
    double dyadicEntropyUpperBound​(int entropyNumberIndex)
    Compute the Upper Bound for the Dyadic Entropy Number
    double entropyNumberUpperBound​(int entropyNumberIndex)
    Compute the Upper Bound for the Entropy Number
    static MaureyOperatorCoveringBounds HilbertSupremumIdentityMap​(int supremumDimension, double operatorNorm)
    Construct an Instance Hilbert To Supremum Identity Map based Maurey Operator Covering Bounds
    double maureyConstant()
    Retrieve the Maurey Constant
    double operatorNorm()
    Retrieve the Operator Norm of Interest
    int supremumDimension()
    Retrieve the Supremum Dimension
    static MaureyOperatorCoveringBounds WilliamsonSmolaScholkopfEstimate​(int supremumDimension, double operatorNorm)
    Construct an Instance of the Maurey Operator Covering Bounds based upon the Williamson, Smola, and Scholkopf Estimate

    Methods inherited from class java.lang.Object

    equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

  • Field Details

    • HILBERT_SUPREMUM_IDENTITY_CONSTANT

      public static final double HILBERT_SUPREMUM_IDENTITY_CONSTANT
      Maurey Constant - from the Hilbert - Supremum Identity Map Estimate
      See Also:
      Constant Field Values

    • WILLIAMSON_SMOLA_SCHOLKOPF_CONSTANT

      public static final double WILLIAMSON_SMOLA_SCHOLKOPF_CONSTANT
      Maurey Constant - from the Williamson-Smola-Scholkopf Estimate
      See Also:
      Constant Field Values

  • Constructor Details

    • MaureyOperatorCoveringBounds

      public MaureyOperatorCoveringBounds​(double maureyConstant, int supremumDimension, double operatorNorm) throws java.lang.Exception
      MaureyOperatorCoveringBounds Constructor
      Parameters:
      maureyConstant - The Maurey Constant
      supremumDimension - The Operator Supremum Output Space Dimension
      operatorNorm - The Operator Norm of Interest
      Throws:
      java.lang.Exception - Thrown if the Inputs are Invalid

  • Method Details

    • HilbertSupremumIdentityMap

      public static final MaureyOperatorCoveringBounds HilbertSupremumIdentityMap​(int supremumDimension, double operatorNorm)
      Construct an Instance Hilbert To Supremum Identity Map based Maurey Operator Covering Bounds
      Parameters:
      supremumDimension - The Operator Supremum Output Space Dimension
      operatorNorm - The Operator Norm of Interest
      Returns:
      The Instance Hilbert To Supremum Identity Map based Maurey Operator Covering Bounds

    • WilliamsonSmolaScholkopfEstimate

      public static final MaureyOperatorCoveringBounds WilliamsonSmolaScholkopfEstimate​(int supremumDimension, double operatorNorm)
      Construct an Instance of the Maurey Operator Covering Bounds based upon the Williamson, Smola, and Scholkopf Estimate
      Parameters:
      supremumDimension - The Operator Supremum Output Space Dimension
      operatorNorm - The Operator Norm of Interest
      Returns:
      Maurey Operator Covering Bounds based upon the Williamson, Smola, and Scholkopf Estimate

    • maureyConstant

      public double maureyConstant()
      Retrieve the Maurey Constant
      Returns:
      The Maurey Constant

    • supremumDimension

      public int supremumDimension()
      Retrieve the Supremum Dimension
      Returns:
      The Supremum Dimension

    • operatorNorm

      public double operatorNorm()
      Retrieve the Operator Norm of Interest
      Returns:
      The Operator Norm of Interest

    • dyadicEntropyUpperBound

      public double dyadicEntropyUpperBound​(int entropyNumberIndex) throws java.lang.Exception
      Compute the Upper Bound for the Dyadic Entropy Number
      Parameters:
      entropyNumberIndex - The Entropy Number Index
      Returns:
      The Upper Bound for the DyadicEntropy Number
      Throws:
      java.lang.Exception - Thrown if the Inputs are Invalid

    • entropyNumberUpperBound

      public double entropyNumberUpperBound​(int entropyNumberIndex) throws java.lang.Exception
      Compute the Upper Bound for the Entropy Number
      Parameters:
      entropyNumberIndex - The Entropy Number Index
      Returns:
      The Upper Bound for the Entropy Number
      Throws:
      java.lang.Exception - Thrown if the Inputs are Invalid