org.drip.spaces.functionclass

Class NormedRxToNormedRxFinite

java.lang.Object

org.drip.spaces.functionclass.NormedRxToNormedRxFinite

Direct Known Subclasses:

NormedRxToNormedR1Finite, NormedRxToNormedRdFinite

public abstract class NormedRxToNormedRxFinite
extends java.lang.Object

NormedRxToNormedRxFinite exposes the Space of Functions that are a Transform from the Normed R^x To Normed R^x Spaces. 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

Method Summary

Modifier and Type	Method and Description
abstract FunctionClassCoveringBounds	agnosticCoveringNumberBounds() Retrieve the Agnostic Covering Number Upper/Lower Bounds for the Function Class
abstract GeneralizedMetricVectorSpace	inputMetricVectorSpace() Retrieve the Input Vector Space
double	maureyConstant() Retrieve the Maurey Constant
abstract double	operatorPopulationMetricNorm() Compute the Operator Population Metric Norm
abstract double	operatorPopulationSupremumNorm() Compute the Operator Population Supremum Norm
abstract double	operatorSampleMetricNorm(GeneralizedValidatedVector gvvi) Compute the Operator Sample Metric Norm
abstract double	operatorSampleSupremumNorm(GeneralizedValidatedVector gvvi) Compute the Operator Sample Supremum Norm
int	outputDimension() Compute the Output Dimension
abstract GeneralizedMetricVectorSpace	outputMetricVectorSpace() Retrieve the Output Vector Space
MaureyOperatorCoveringBounds	populationMetricCoveringBounds() Compute the Maurey Covering Number Upper Bounds for Operator Population Metric Norm
MaureyOperatorCoveringBounds	populationSupremumCoveringBounds() Compute the Maurey Covering Number Upper Bounds for Operator Population Supremum Norm
MaureyOperatorCoveringBounds	sampleMetricCoveringBounds(GeneralizedValidatedVector gvvi) Compute the Maurey Covering Number Upper Bounds for Operator Sample Metric Norm
MaureyOperatorCoveringBounds	sampleSupremumCoveringBounds(GeneralizedValidatedVector gvvi) Compute the Maurey Covering Number Upper Bounds for Operator Sample Supremum Norm
FunctionClassCoveringBounds	scaleSensitiveCoveringBounds(GeneralizedValidatedVector gvvi, R1ToR1 funcR1ToR1FatShatter) Retrieve the Scale-Sensitive Covering Number Upper/Lower Bounds given the Specified Sample for the Function Class

Methods inherited from class java.lang.Object

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

Method Detail

inputMetricVectorSpace

public abstract GeneralizedMetricVectorSpace inputMetricVectorSpace()

Retrieve the Input Vector Space

Returns:

The Input Vector Space

outputMetricVectorSpace

public abstract GeneralizedMetricVectorSpace outputMetricVectorSpace()

Retrieve the Output Vector Space

Returns:

The Output Vector Space

operatorPopulationMetricNorm

public abstract double operatorPopulationMetricNorm()
                                             throws java.lang.Exception

Compute the Operator Population Metric Norm

Returns:

The Operator Population Metric Norm

Throws:

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

operatorPopulationSupremumNorm

public abstract double operatorPopulationSupremumNorm()
                                               throws java.lang.Exception

Compute the Operator Population Supremum Norm

Returns:

The Operator Population Supremum Norm

Throws:

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

operatorSampleMetricNorm

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

Compute the Operator Sample Metric Norm

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

Compute the Operator Sample Supremum Norm

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

agnosticCoveringNumberBounds

public abstract FunctionClassCoveringBounds agnosticCoveringNumberBounds()

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

Returns:

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

maureyConstant

public double maureyConstant()

Retrieve the Maurey Constant

Returns:

The Maurey Constant

scaleSensitiveCoveringBounds

public FunctionClassCoveringBounds scaleSensitiveCoveringBounds(GeneralizedValidatedVector gvvi,
                                                                R1ToR1 funcR1ToR1FatShatter)

Retrieve the Scale-Sensitive Covering Number Upper/Lower Bounds given the Specified Sample for the Function Class

Parameters:

gvvi - The Validated Instance Vector Sequence

funcR1ToR1FatShatter - The Cover Fat Shattering Coefficient R^1 To R^1

Returns:

The Scale-Sensitive Covering Number Upper/Lower Bounds given the Specified Sample for the Function Class

outputDimension

public int outputDimension()
                    throws java.lang.Exception

Compute the Output Dimension

Returns:

The Output Dimension

Throws:

java.lang.Exception - Thrown if the Output Dimension is Invalid

populationMetricCoveringBounds

public MaureyOperatorCoveringBounds populationMetricCoveringBounds()

Compute the Maurey Covering Number Upper Bounds for Operator Population Metric Norm

Returns:

The Maurey Operator Covering Number Upper Bounds Instance Corresponding to the Operator Population Metric Norm

populationSupremumCoveringBounds

public MaureyOperatorCoveringBounds populationSupremumCoveringBounds()

Compute the Maurey Covering Number Upper Bounds for Operator Population Supremum Norm

Returns:

The Maurey Operator Covering Number Upper Bounds Instance Corresponding to the Operator Population Supremum Norm

sampleMetricCoveringBounds

public MaureyOperatorCoveringBounds sampleMetricCoveringBounds(GeneralizedValidatedVector gvvi)

Compute the Maurey Covering Number Upper Bounds for Operator Sample Metric Norm

Parameters:

gvvi - The Validated Vector Space Instance

Returns:

The Maurey Operator Covering Number Upper Bounds Instance Corresponding to the Operator Sample Metric Norm

sampleSupremumCoveringBounds

public MaureyOperatorCoveringBounds sampleSupremumCoveringBounds(GeneralizedValidatedVector gvvi)

Compute the Maurey Covering Number Upper Bounds for Operator Sample Supremum Norm

Parameters:

gvvi - The Validated Vector Space Instance

Returns:

The Maurey Operator Covering Number Upper Bounds Instance Corresponding to the Operator Sample Supremum Norm