org.drip.spaces.functionclass

Class NormedRxToNormedRdFinite

java.lang.Object

org.drip.spaces.functionclass.NormedRxToNormedRxFinite

org.drip.spaces.functionclass.NormedRxToNormedRdFinite

Direct Known Subclasses:

HilbertRxToSupremumRdFinite

public class NormedRxToNormedRdFinite
extends NormedRxToNormedRxFinite

NormedRxToNormedRdFinite implements the Class F with f E f : Normed R^x To Normed R^d Space of Finite Functions. 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
NormedRxToNormedRdFinite(double dblMaureyConstant, NormedRxToNormedRd[] aNormedRxToNormedRd) NormedRxToNormedRdFinite Constructor

Method Summary

Modifier and Type	Method and Description
FunctionClassCoveringBounds	agnosticCoveringNumberBounds() Retrieve the Agnostic Covering Number Upper/Lower Bounds for the Function Class
NormedRxToNormedRd[]	functionSpaces() Retrieve the Array of Function Spaces in the Class
GeneralizedMetricVectorSpace	inputMetricVectorSpace() Retrieve the Input Vector Space
double	operatorPopulationMetricNorm() Compute the Operator Population Metric Norm
double	operatorPopulationSupremumNorm() Compute the Operator Population Supremum Norm
double	operatorSampleMetricNorm(GeneralizedValidatedVector gvvi) Compute the Operator Sample Metric Norm
double	operatorSampleSupremumNorm(GeneralizedValidatedVector gvvi) Compute the Operator Sample Supremum Norm
RdNormed	outputMetricVectorSpace() Retrieve the Output Vector Space
double[]	populationCoveringNumber(double dblCover) Estimate for the Function Class Population Covering Number Array, one for each dimension
double[]	populationCoveringNumber(double[] adblCover) Estimate for the Function Class Population Covering Number Array, one for each dimension
double[]	populationRdMetricNorm() Compute the Population R^d Metric Norm
double[]	populationRdSupremumNorm() Compute the Population R^d Supremum Norm
double[]	populationSupremumCoveringNumber(double dblCover) Estimate for the Function Class Population Supremum Covering Number Array, one for each dimension
double[]	populationSupremumCoveringNumber(double[] adblCover) Estimate for the Function Class Population Supremum Covering Number Array, one for each dimension
double[]	sampleCoveringNumber(GeneralizedValidatedVector gvvi, double dblCover) Estimate for the Scale-Sensitive Sample Covering Number Array for the specified Cover Size
double[]	sampleCoveringNumber(GeneralizedValidatedVector gvvi, double[] adblCover) Estimate for the Scale-Sensitive Sample Covering Number Array for the specified Cover Size
double[]	sampleRdMetricNorm(GeneralizedValidatedVector gvvi) Compute the Sample R^d Metric Norm
double[]	sampleRdSupremumNorm(GeneralizedValidatedVector gvvi) Compute the Sample R^d Supremum Norm
double[]	sampleSupremumCoveringNumber(GeneralizedValidatedVector gvvi, double dblCover) Estimate for the Scale-Sensitive Sample Supremum Covering Number for the specified Cover Size
double[]	sampleSupremumCoveringNumber(GeneralizedValidatedVector gvvi, double[] adblCover) Estimate for the Scale-Sensitive Sample Supremum Covering Number for the specified Cover Size

Methods inherited from class org.drip.spaces.functionclass.NormedRxToNormedRxFinite

maureyConstant, outputDimension, populationMetricCoveringBounds, populationSupremumCoveringBounds, sampleMetricCoveringBounds, sampleSupremumCoveringBounds, scaleSensitiveCoveringBounds

Methods inherited from class java.lang.Object

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

Constructor Detail

NormedRxToNormedRdFinite

public NormedRxToNormedRdFinite(double dblMaureyConstant,
                                NormedRxToNormedRd[] aNormedRxToNormedRd)
                         throws java.lang.Exception

NormedRxToNormedRdFinite Constructor

Parameters:

dblMaureyConstant - Maurey Constant

aNormedRxToNormedRd - Array of the Normed R^x To Normed R^d Spaces

Throws:

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

Method Detail

agnosticCoveringNumberBounds

public FunctionClassCoveringBounds agnosticCoveringNumberBounds()

Description copied from class: NormedRxToNormedRxFinite

Retrieve the Agnostic Covering Number Upper/Lower Bounds for the Function Class

Specified by:

agnosticCoveringNumberBounds in class NormedRxToNormedRxFinite

Returns:

The Agnostic Covering Number Upper/Lower Bounds for the Function Class

inputMetricVectorSpace

public GeneralizedMetricVectorSpace inputMetricVectorSpace()

Description copied from class: NormedRxToNormedRxFinite

Retrieve the Input Vector Space

Specified by:

inputMetricVectorSpace in class NormedRxToNormedRxFinite

Returns:

The Input Vector Space

outputMetricVectorSpace

public RdNormed outputMetricVectorSpace()

Description copied from class: NormedRxToNormedRxFinite

Retrieve the Output Vector Space

Specified by:

outputMetricVectorSpace in class NormedRxToNormedRxFinite

Returns:

The Output Vector Space

functionSpaces

public NormedRxToNormedRd[] functionSpaces()

Retrieve the Array of Function Spaces in the Class

Returns:

The Array of Function Spaces in the Class

populationCoveringNumber

public double[] populationCoveringNumber(double[] adblCover)

Estimate for the Function Class Population Covering Number Array, one for each dimension

Parameters:

adblCover - The Size of the Cover Array

Returns:

Function Class Population Covering Number Estimate Array, one for each dimension

populationCoveringNumber

public double[] populationCoveringNumber(double dblCover)

Estimate for the Function Class Population Covering Number Array, one for each dimension

Parameters:

dblCover - The Cover

Returns:

Function Class Population Covering Number Estimate Array, one for each dimension

populationSupremumCoveringNumber

public double[] populationSupremumCoveringNumber(double[] adblCover)

Estimate for the Function Class Population Supremum Covering Number Array, one for each dimension

Parameters:

adblCover - The Size of the Cover Array

Returns:

Function Class Population Supremum Covering Number Estimate Array, one for each dimension

populationSupremumCoveringNumber

public double[] populationSupremumCoveringNumber(double dblCover)

Estimate for the Function Class Population Supremum Covering Number Array, one for each dimension

Parameters:

dblCover - The Cover

Returns:

Function Class Population Covering Supremum Number Estimate Array, one for each dimension

sampleCoveringNumber

public double[] sampleCoveringNumber(GeneralizedValidatedVector gvvi,
                                     double[] adblCover)

Estimate for the Scale-Sensitive Sample Covering Number Array for the specified Cover Size

Parameters:

gvvi - The Validated Instance Vector Sequence

adblCover - The Size of the Cover Array

Returns:

The Scale-Sensitive Sample Covering Number Array for the specified Cover Size

sampleCoveringNumber

public double[] sampleCoveringNumber(GeneralizedValidatedVector gvvi,
                                     double dblCover)

Estimate for the Scale-Sensitive Sample Covering Number Array for the specified Cover Size

Parameters:

gvvi - The Validated Instance Vector Sequence

dblCover - The Size of the Cover Array

Returns:

The Scale-Sensitive Sample Covering Number Array for the specified Cover Size

sampleSupremumCoveringNumber

public double[] sampleSupremumCoveringNumber(GeneralizedValidatedVector gvvi,
                                             double[] adblCover)

Estimate for the Scale-Sensitive Sample Supremum Covering Number for the specified Cover Size

Parameters:

gvvi - The Validated Instance Vector Sequence

adblCover - The Size of the Cover Array

Returns:

The Scale-Sensitive Sample Supremum Covering Number for the specified Cover Size

sampleSupremumCoveringNumber

public double[] sampleSupremumCoveringNumber(GeneralizedValidatedVector gvvi,
                                             double dblCover)

Estimate for the Scale-Sensitive Sample Supremum Covering Number for the specified Cover Size

Parameters:

gvvi - The Validated Instance Vector Sequence

dblCover - The Cover

Returns:

The Scale-Sensitive Sample Supremum Covering Number for the specified Cover Size

populationRdMetricNorm

public double[] populationRdMetricNorm()

Compute the Population R^d Metric Norm

Returns:

The Population R^d Metric Norm

populationRdSupremumNorm

public double[] populationRdSupremumNorm()

Compute the Population R^d Supremum Norm

Returns:

The Population R^d Supremum Norm

sampleRdMetricNorm

public double[] sampleRdMetricNorm(GeneralizedValidatedVector gvvi)

Compute the Sample R^d Metric Norm

Parameters:

gvvi - The Validated Vector Space Instance

Returns:

The Sample R^d Metric Norm

sampleRdSupremumNorm

public double[] sampleRdSupremumNorm(GeneralizedValidatedVector gvvi)

Compute the Sample R^d Supremum Norm

Parameters:

gvvi - The Validated Vector Space Instance

Returns:

The Sample R^d Supremum Norm

operatorPopulationMetricNorm

public double operatorPopulationMetricNorm()
                                    throws java.lang.Exception

Description copied from class: NormedRxToNormedRxFinite

Compute the Operator Population Metric Norm

Specified by:

operatorPopulationMetricNorm in class NormedRxToNormedRxFinite

Returns:

The Operator Population Metric Norm

Throws:

java.lang.Exception - Thrown if the Operator Norm cannot be computed

operatorPopulationSupremumNorm

public double operatorPopulationSupremumNorm()
                                      throws java.lang.Exception

Description copied from class: NormedRxToNormedRxFinite

Compute the Operator Population Supremum Norm

Specified by:

operatorPopulationSupremumNorm in class NormedRxToNormedRxFinite

Returns:

The Operator Population Supremum Norm

Throws:

java.lang.Exception - Thrown if the Operator Population Supremum Norm cannot be computed

operatorSampleMetricNorm

public double operatorSampleMetricNorm(GeneralizedValidatedVector gvvi)
                                throws java.lang.Exception

Description copied from class: NormedRxToNormedRxFinite

Compute the Operator Sample Metric Norm

Specified by:

operatorSampleMetricNorm in class NormedRxToNormedRxFinite

Parameters:

gvvi - The Validated Vector Space Instance

Returns:

The Operator Sample Metric Norm

Throws:

java.lang.Exception - Thrown if the Operator Norm cannot be computed

operatorSampleSupremumNorm

public double operatorSampleSupremumNorm(GeneralizedValidatedVector gvvi)
                                  throws java.lang.Exception

Description copied from class: NormedRxToNormedRxFinite

Compute the Operator Sample Supremum Norm

Specified by:

operatorSampleSupremumNorm in class NormedRxToNormedRxFinite

Parameters:

gvvi - The Validated Vector Space Instance

Returns:

The Operator Sample Supremum Norm

Throws:

java.lang.Exception - Thrown if the Operator Sample Supremum Norm cannot be computed