org.drip.spaces.cover

Class CoveringBoundsHelper

java.lang.Object

org.drip.spaces.cover.CoveringBoundsHelper

public class CoveringBoundsHelper
extends java.lang.Object

CoveringBoundsHelper contains the assortment of Utilities used in the Computation of Upper Bounds for Normed Single Function Spaces and Function Space Products. The 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

Constructor Summary

Constructors

Constructor and Description
CoveringBoundsHelper()

Method Summary

Modifier and Type	Method and Description
static double	CarlStephaniProductBound(MaureyOperatorCoveringBounds mocbA, MaureyOperatorCoveringBounds mocbB, int iEntropyNumberIndexA, int iEntropyNumberIndexB) Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes
static CarlStephaniNormedBounds	CarlStephaniProductNorm(MaureyOperatorCoveringBounds mocbA, MaureyOperatorCoveringBounds mocbB, double dblNormA, double dblNormB, int iEntropyNumberIndex) Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes using the Function Class Norm
static double	DyadicEntropyNumber(double dblLogNEntropyNumber) Compute the Dyadic Entropy Number from the nth Entropy Number

Methods inherited from class java.lang.Object

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

Constructor Detail

CoveringBoundsHelper

public CoveringBoundsHelper()

Method Detail

DyadicEntropyNumber

public static final double DyadicEntropyNumber(double dblLogNEntropyNumber)
                                        throws java.lang.Exception

Compute the Dyadic Entropy Number from the nth Entropy Number

Parameters:

dblLogNEntropyNumber - Log of the nth Entropy Number

Returns:

The Dyadic Entropy Number

Throws:

java.lang.Exception - Thrown if the Dyadic Entropy Number cannot be calculated

CarlStephaniProductBound

public static final double CarlStephaniProductBound(MaureyOperatorCoveringBounds mocbA,
                                                    MaureyOperatorCoveringBounds mocbB,
                                                    int iEntropyNumberIndexA,
                                                    int iEntropyNumberIndexB)
                                             throws java.lang.Exception

Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes

Parameters:

mocbA - The Maurey Operator Covering Bounds for Class A

mocbB - The Maurey Operator Covering Bounds for Class B

iEntropyNumberIndexA - Entropy Number Index for Class A

iEntropyNumberIndexB - Entropy Number Index for Class B

Returns:

The Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes

Throws:

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

CarlStephaniProductNorm

public static final CarlStephaniNormedBounds CarlStephaniProductNorm(MaureyOperatorCoveringBounds mocbA,
                                                                     MaureyOperatorCoveringBounds mocbB,
                                                                     double dblNormA,
                                                                     double dblNormB,
                                                                     int iEntropyNumberIndex)

Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes using the Function Class Norm

Parameters:

mocbA - The Maurey Operator Covering Bounds for Class A

mocbB - The Maurey Operator Covering Bounds for Class B

dblNormA - The Function Class A Norm

dblNormB - The Function Class B Norm

iEntropyNumberIndex - Entropy Number Index for either Class

Returns:

The Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes using the Function Norm