Package org.drip.learning.kernel

Class IntegralOperator

java.lang.Object

org.drip.learning.kernel.IntegralOperator

public abstract class IntegralOperator
extends java.lang.Object

IntegralOperator implements the R^x L₂ To R^x L₂ Mercer Kernel Integral Operator defined by: T_k [f(.)] := Integral Over Input Space {k (., y) * f(y) * d[Prob(y)]}

The References are:

• Ash, R. (1965): Information Theory Inter-science New York
• Konig, H. (1986): Eigenvalue Distribution of Compact Operators Birkhauser Basel, Switzerland
• Smola, A. J., A. Elisseff, B. Scholkopf, and R. C. Williamson (2000): Entropy Numbers for Convex Combinations and mlps, in: Advances in Large Margin Classifiers, A. Smola, P. Bartlett, B. Scholkopf, and D. Schuurmans - editors MIT Press Cambridge, MA

• Module = Computational Core Module
• Library = Statistical Learning
• Project = Agnostic Learning Bounds under Empirical Loss Minimization Schemes
• Package = Statistical Learning Banach Mercer Kernels

Author:

Lakshmi Krishnamurthy

Constructor Summary

Constructors

Constructor	Description
IntegralOperator(SymmetricRdToNormedR1Kernel kernel, RdToR1 funcRdToR1, R1Normed r1OperatorOutput)	IntegralOperator Constructor

Method Summary

Modifier and Type	Method	Description
double	computeOperatorIntegral(double[] adblX)	Compute the Operator's Kernel Integral across the specified X Variate Instance
abstract IntegralOperatorEigenContainer	eigenize()	Eigenize the Kernel Integral Operator
Rd	inputSpaceBorelMeasure()	Retrieve the Input Space Borel Sigma Measure
boolean	isPositiveDefinite(double[] adblX)	Indicate the Kernel Operator Integral's Positive-definiteness across the specified X Variate Instance
SymmetricRdToNormedR1Kernel	kernel()	Retrieve the Symmetric R^d To R^1 Kernel
RdToR1	kernelOperatorFunction()	Retrieve the R^d To R^1 Kernel Operator Function
R1Normed	outputVectorMetricSpace()	Retrieve the Kernel Integral Operator Output Space

Methods inherited from class java.lang.Object

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

Constructor Details

IntegralOperator

public IntegralOperator(SymmetricRdToNormedR1Kernel kernel,
RdToR1 funcRdToR1,
R1Normed r1OperatorOutput)
                 throws java.lang.Exception

IntegralOperator Constructor

Parameters:

kernel - The Symmetric Mercer Kernel - this should be R^x L2 X R^x L2 To R^1

funcRdToR1 - The R^d To R^1 Operator Function

r1OperatorOutput - The Kernel Integral Operator Output Space - this is R^1 L2

Throws:

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

Method Details

kernel

public SymmetricRdToNormedR1Kernel kernel()

Retrieve the Symmetric R^d To R^1 Kernel

Returns:

The Symmetric R^d To R^1 Kernel

kernelOperatorFunction

public RdToR1 kernelOperatorFunction()

Retrieve the R^d To R^1 Kernel Operator Function

Returns:

The R^d To R^1 Kernel Operator Function

inputSpaceBorelMeasure

public Rd inputSpaceBorelMeasure()

Retrieve the Input Space Borel Sigma Measure

Returns:

The Input Space Borel Sigma Measure

outputVectorMetricSpace

public R1Normed outputVectorMetricSpace()

Retrieve the Kernel Integral Operator Output Space

Returns:

The Kernel Integral Operator Output Space

computeOperatorIntegral

public double computeOperatorIntegral(double[] adblX)
                               throws java.lang.Exception

Compute the Operator's Kernel Integral across the specified X Variate Instance

Parameters:

adblX - Validated Vector Instance X

Returns:

The Operator's Kernel Integral across the specified X Variate Instance

Throws:

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

isPositiveDefinite

public boolean isPositiveDefinite(double[] adblX)

Indicate the Kernel Operator Integral's Positive-definiteness across the specified X Variate Instance

Parameters:

adblX - Validated Vector Instance X

Returns:

TRUE - The Kernel Operator Integral is Positive Definite across the specified X Variate Instance

eigenize

public abstract IntegralOperatorEigenContainer eigenize()

Eigenize the Kernel Integral Operator

Returns:

The Eigenization Output