Package org.drip.learning.kernel

Class IntegralOperatorEigenContainer

java.lang.Object

org.drip.learning.kernel.IntegralOperatorEigenContainer

public class IntegralOperatorEigenContainer
extends java.lang.Object

IntegralOperatorEigenContainer holds the Group of Eigen-Components that result from the Eigenization of the R^x L₂ To R^x L₂ Kernel Linear 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
• Carl, B., and I. Stephani (1990): Entropy, Compactness, and Approximation of Operators Cambridge University Press Cambridge UK
• Gordon, Y., H. Konig, and C. Schutt (1987): Geometric and Probabilistic Estimates of Entropy and Approximation Numbers of Operators Journal of Approximation Theory 49 219-237
• 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
IntegralOperatorEigenContainer(IntegralOperatorEigenComponent[] aIOEC)	IntegralOperatorEigenContainer Constructor

Method Summary

Modifier and Type	Method	Description
R1Combinatorial	diagonallyScaledFeatureSpace(DiagonalScalingOperator dso)	Generate the Diagonally Scaled Normed Vector Space of the RKHS Feature Space Bounds that results on applying the Diagonal Scaling Operator
IntegralOperatorEigenComponent[]	eigenComponents()	Retrieve the Array of the Integral Operator Eigen-Components
RdNormed	inputMetricVectorSpace()	Retrieve the Eigen Input Space
R1Normed	outputMetricVectorSpace()	Retrieve the Eigen Output Space
OperatorClassCoveringBounds	scaledCoveringNumberBounds(DiagonalScalingOperator dso)	Generate the Operator Class Covering Number Bounds of the RKHS Feature Space Bounds that result on the Application of the Diagonal Scaling Operator

Methods inherited from class java.lang.Object

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

Constructor Details

IntegralOperatorEigenContainer

public IntegralOperatorEigenContainer(IntegralOperatorEigenComponent[] aIOEC)
                               throws java.lang.Exception

IntegralOperatorEigenContainer Constructor

Parameters:

aIOEC - Array of the Integral Operator Eigen-Components

Throws:

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

Method Details

eigenComponents

public IntegralOperatorEigenComponent[] eigenComponents()

Retrieve the Array of the Integral Operator Eigen-Components

Returns:

The Array of the Integral Operator Eigen-Components

inputMetricVectorSpace

public RdNormed inputMetricVectorSpace()

Retrieve the Eigen Input Space

Returns:

The Eigen Input Space

outputMetricVectorSpace

public R1Normed outputMetricVectorSpace()

Retrieve the Eigen Output Space

Returns:

The Eigen Output Space

diagonallyScaledFeatureSpace

public R1Combinatorial diagonallyScaledFeatureSpace(DiagonalScalingOperator dso)

Generate the Diagonally Scaled Normed Vector Space of the RKHS Feature Space Bounds that results on applying the Diagonal Scaling Operator

Parameters:

dso - The Diagonal Scaling Operator

Returns:

The Diagonally Scaled Normed Vector Space of the RKHS Feature Space

scaledCoveringNumberBounds

public OperatorClassCoveringBounds scaledCoveringNumberBounds(DiagonalScalingOperator dso)

Generate the Operator Class Covering Number Bounds of the RKHS Feature Space Bounds that result on the Application of the Diagonal Scaling Operator

Parameters:

dso - The Diagonal Scaling Operator

Returns:

The Operator Class Covering Number Bounds of the RKHS Feature Space