Package org.drip.spaces.cover

Class CarlStephaniProductBounds

java.lang.Object

org.drip.spaces.cover.CarlStephaniProductBounds

public class CarlStephaniProductBounds
extends java.lang.Object

CarlStephaniProductBounds implements the Bounds that result from the Convolution Product Product of 2 Normed R^x To Normed R^x Function Spaces. The 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:

• CarlStephaniProductBounds Constructor
• Retrieve the Function Class A
• Retrieve the Function Class B
• Compute the Upper Bound for the Entropy Number of the Operator Population Metric Covering Number Convolution Product Product across both the Function Classes
• Compute the Upper Bound for the Entropy Number of the Operator Population Supremum Covering Number Convolution Product across both the Function Classes
• Compute the Upper Bound for the Entropy Number of the Operator Sample Metric Covering Number Convolution Product across both the Function Classes
• Compute the Upper Bound for the Entropy Number of the Operator Sample Supremum Covering Number Convolution Product across both the Function Classes
• Compute the Normed Upper Entropy Convolution Product Bound across the Function Classes
• Compute the Population Supremum Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm
• Compute the Population Metric Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm
• Compute the Sample Supremum Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm
• Compute the Sample Metric Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm

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

Constructor Summary

Constructors

Constructor	Description
CarlStephaniProductBounds(NormedRxToNormedRxFinite functionClassRxRxA, NormedRxToNormedRxFinite functionClassRxRxB)	CarlStephaniProductBounds Constructor

Method Summary

Modifier and Type	Method	Description
NormedRxToNormedRxFinite	funcClassA()	Retrieve the Function Class A
NormedRxToNormedRxFinite	funcClassB()	Retrieve the Function Class B
CarlStephaniNormedBounds	normedEntropyUpperBound(MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsA, MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsB, int entropyNumberIndex, boolean useSupremumNorm)	Compute the Normed Upper Entropy Convolution Product Bound across the Function Classes
CarlStephaniNormedBounds	populationMetricEntropyNorm(int entropyNumberIndex, boolean useSupremumNorm)	Compute the Population Metric Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm
double	populationMetricEntropyNumber(int entropyNumberIndexA, int entropyNumberIndexB)	Compute the Upper Bound for the Entropy Number of the Operator Population Metric Covering Number Convolution Product Product across both the Function Classes
CarlStephaniNormedBounds	populationSupremumEntropyNorm(int entropyNumberIndex, boolean useSupremumNorm)	Compute the Population Supremum Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm
double	populationSupremumEntropyNumber(int entropyNumberIndexA, int entropyNumberIndexB)	Compute the Upper Bound for the Entropy Number of the Operator Population Supremum Covering Number Convolution Product across both the Function Classes
CarlStephaniNormedBounds	sampleMetricEntropyNorm(GeneralizedValidatedVector generalizedValidatedVectorA, GeneralizedValidatedVector generalizedValidatedVectorB, int entropyNumberIndex, boolean useSupremumNorm)	Compute the Sample Metric Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm
double	sampleMetricEntropyNumber(GeneralizedValidatedVector generalizedValidatedVectorA, GeneralizedValidatedVector generalizedValidatedVectorB, int entropyNumberIndexA, int entropyNumberIndexB)	Compute the Upper Bound for the Entropy Number of the Operator Sample Metric Covering Number Convolution Product across both the Function Classes
CarlStephaniNormedBounds	sampleSupremumEntropyNorm(GeneralizedValidatedVector generalizedValidatedVectorA, GeneralizedValidatedVector generalizedValidatedVectorB, int entropyNumberIndex, boolean useSupremumNorm)	Compute the Sample Supremum Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm
double	sampleSupremumEntropyNumber(GeneralizedValidatedVector generalizedValidatedVectorA, GeneralizedValidatedVector generalizedValidatedVectorB, int entropyNumberIndexA, int entropyNumberIndexB)	Compute the Upper Bound for the Entropy Number of the Operator Sample Supremum Covering Number Convolution Product across both the Function Classes

Methods inherited from class java.lang.Object

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

Constructor Details

CarlStephaniProductBounds

public CarlStephaniProductBounds(NormedRxToNormedRxFinite functionClassRxRxA,
NormedRxToNormedRxFinite functionClassRxRxB)
                          throws java.lang.Exception

CarlStephaniProductBounds Constructor

Parameters:

functionClassRxRxA - Function Class A

functionClassRxRxB - Function Class B

Throws:

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

Method Details

funcClassA

public NormedRxToNormedRxFinite funcClassA()

Retrieve the Function Class A

Returns:

The Function Class A

funcClassB

public NormedRxToNormedRxFinite funcClassB()

Retrieve the Function Class B

Returns:

The Function Class B

populationMetricEntropyNumber

public double populationMetricEntropyNumber(int entropyNumberIndexA,
int entropyNumberIndexB)
                                     throws java.lang.Exception

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

Parameters:

entropyNumberIndexA - Entropy Number Index for Class A

entropyNumberIndexB - Entropy Number Index for Class B

Returns:

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

Throws:

java.lang.Exception - Thrown if the Convolution Product Product Population Metric Entropy Number cannot be computed

populationSupremumEntropyNumber

public double populationSupremumEntropyNumber(int entropyNumberIndexA,
int entropyNumberIndexB)
                                       throws java.lang.Exception

Compute the Upper Bound for the Entropy Number of the Operator Population Supremum Covering Number Convolution Product across both the Function Classes

Parameters:

entropyNumberIndexA - Entropy Number Index for Class A

entropyNumberIndexB - Entropy Number Index for Class B

Returns:

The Upper Bound for the Entropy Number of the Operator Population Supremum Covering Number Convolution Product across both the Function Classes

Throws:

java.lang.Exception - Thrown if the Convolution Product Population Supremum Dyadic Entropy cannot be Computed

sampleMetricEntropyNumber

public double sampleMetricEntropyNumber(GeneralizedValidatedVector generalizedValidatedVectorA,
GeneralizedValidatedVector generalizedValidatedVectorB,
int entropyNumberIndexA,
int entropyNumberIndexB)
                                 throws java.lang.Exception

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

Parameters:

generalizedValidatedVectorA - The Validated Input Vector Space Instance for Class A

generalizedValidatedVectorB - The Validated Input Vector Space Instance for Class B

entropyNumberIndexA - Entropy Number Index for Class A

entropyNumberIndexB - Entropy Number Index for Class B

Returns:

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

Throws:

java.lang.Exception - Thrown if the Convolution Product Sample Metric Entropy Number cannot be Computed

sampleSupremumEntropyNumber

public double sampleSupremumEntropyNumber(GeneralizedValidatedVector generalizedValidatedVectorA,
GeneralizedValidatedVector generalizedValidatedVectorB,
int entropyNumberIndexA,
int entropyNumberIndexB)
                                   throws java.lang.Exception

Compute the Upper Bound for the Entropy Number of the Operator Sample Supremum Covering Number Convolution Product across both the Function Classes

Parameters:

generalizedValidatedVectorA - The Validated Input Vector Space Instance for Class A

generalizedValidatedVectorB - The Validated Input Vector Space Instance for Class B

entropyNumberIndexA - Entropy Number Index for Class A

entropyNumberIndexB - Entropy Number Index for Class B

Returns:

The Upper Bound for the Entropy Number of the Operator Sample Supremum Covering Number Convolution Product across both the Function Classes

Throws:

java.lang.Exception - Thrown if the Convolution Product Sample Supremum Entropy Number cannot be Computed

normedEntropyUpperBound

public CarlStephaniNormedBounds normedEntropyUpperBound(MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsA,
MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsB,
int entropyNumberIndex,
boolean useSupremumNorm)

Compute the Normed Upper Entropy Convolution Product Bound across the Function Classes

Parameters:

maureyOperatorCoveringBoundsA - The Maurey Operator Covering Bounds for Class A

maureyOperatorCoveringBoundsB - The Maurey Operator Covering Bounds for Class B

entropyNumberIndex - Entropy Number Index for either Class

useSupremumNorm - TRUE/FALSE - Use the Supremum/Metric Bound as the Operator Function Class

Returns:

The Normed Upper Entropy Convolution Product Bound across the Function Classes

populationSupremumEntropyNorm

public CarlStephaniNormedBounds populationSupremumEntropyNorm(int entropyNumberIndex,
boolean useSupremumNorm)

Compute the Population Supremum Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm

Parameters:

entropyNumberIndex - Entropy Number Index for either Class

useSupremumNorm - TRUE/FALSE - Use the Supremum/Metric Bound as the Operator Function Class

Returns:

The Population Supremum Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm

populationMetricEntropyNorm

public CarlStephaniNormedBounds populationMetricEntropyNorm(int entropyNumberIndex,
boolean useSupremumNorm)

Compute the Population Metric Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm

Parameters:

entropyNumberIndex - Entropy Number Index for either Class

useSupremumNorm - TRUE/FALSE - Use the Supremum/Metric Bound as the Operator Function Class

Returns:

The Population Metric Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm

sampleSupremumEntropyNorm

public CarlStephaniNormedBounds sampleSupremumEntropyNorm(GeneralizedValidatedVector generalizedValidatedVectorA,
GeneralizedValidatedVector generalizedValidatedVectorB,
int entropyNumberIndex,
boolean useSupremumNorm)

Compute the Sample Supremum Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm

Parameters:

generalizedValidatedVectorA - The Validated Input Vector Space Instance for Class A

generalizedValidatedVectorB - The Validated Input Vector Space Instance for Class B

entropyNumberIndex - Entropy Number Index for either Class

useSupremumNorm - TRUE/FALSE - Use the Supremum/Metric Bound as the Operator Function Class

Returns:

The Sample Supremum Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm

sampleMetricEntropyNorm

public CarlStephaniNormedBounds sampleMetricEntropyNorm(GeneralizedValidatedVector generalizedValidatedVectorA,
GeneralizedValidatedVector generalizedValidatedVectorB,
int entropyNumberIndex,
boolean useSupremumNorm)

Compute the Sample Metric Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm

Parameters:

generalizedValidatedVectorA - The Validated Input Vector Space Instance for Class A

generalizedValidatedVectorB - The Validated Input Vector Space Instance for Class B

entropyNumberIndex - Entropy Number Index for either Class

useSupremumNorm - TRUE/FALSE - Use the Supremum/Metric Bound as the Operator Function Class

Returns:

The Sample Metric Carl-Stephani Entropy Number Upper Bound using either the Metric/Supremum Population Norm