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 R^d To Supremum R^d Operator Class. The Main References are: 1) 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. 2) Carl, B., and I. Stephani (1990): Entropy, Compactness, and the Approximation of Operators, Cambridge University Press, Cambridge UK. 3) 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.

Author:

Lakshmi Krishnamurthy

Field Summary

Fields

Modifier and Type	Field and 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 and Description
MaureyOperatorCoveringBounds(double dblMaureyConstant, int iSupremumDimension, double dblOperatorNorm) MaureyOperatorCoveringBounds Constructor

Method Summary

Modifier and Type	Method and Description
double	dyadicEntropyUpperBound(int iEntropyNumberIndex) Compute the Upper Bound for the Dyadic Entropy Number
double	entropyNumberUpperBound(int iEntropyNumberIndex) Compute the Upper Bound for the Entropy Number
static MaureyOperatorCoveringBounds	HilbertSupremumIdentityMap(int iSupremumDimension, double dblOperatorNorm) 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 iSupremumDimension, double dblOperatorNorm) 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 Detail

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 Detail

MaureyOperatorCoveringBounds

public MaureyOperatorCoveringBounds(double dblMaureyConstant,
                                    int iSupremumDimension,
                                    double dblOperatorNorm)
                             throws java.lang.Exception

MaureyOperatorCoveringBounds Constructor

Parameters:

dblMaureyConstant - The Maurey Constant

iSupremumDimension - The Operator Supremum Output Space Dimension

dblOperatorNorm - The Operator Norm of Interest

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid

Method Detail

HilbertSupremumIdentityMap

public static final MaureyOperatorCoveringBounds HilbertSupremumIdentityMap(int iSupremumDimension,
                                                                            double dblOperatorNorm)

Construct an Instance Hilbert To Supremum Identity Map based Maurey Operator Covering Bounds

Parameters:

iSupremumDimension - The Operator Supremum Output Space Dimension

dblOperatorNorm - 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 iSupremumDimension,
                                                                                  double dblOperatorNorm)

Construct an Instance of the Maurey Operator Covering Bounds based upon the Williamson, Smola, and Scholkopf Estimate

Parameters:

iSupremumDimension - The Operator Supremum Output Space Dimension

dblOperatorNorm - 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 iEntropyNumberIndex)
                               throws java.lang.Exception

Compute the Upper Bound for the Dyadic Entropy Number

Parameters:

iEntropyNumberIndex - 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 iEntropyNumberIndex)
                               throws java.lang.Exception

Compute the Upper Bound for the Entropy Number

Parameters:

iEntropyNumberIndex - The Entropy Number Index

Returns:

The Upper Bound for the Entropy Number

Throws:

java.lang.Exception - Thrown if the Inputs are Invalid