Package 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 Rx To Normed Rd 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:
  • Retrieve the Input Vector Space
  • Retrieve the Output Vector Space
  • Compute the Operator Population Metric Norm
  • Compute the Operator Population Supremum Norm
  • Compute the Operator Sample Metric Norm
  • Compute the Operator Sample Supremum Norm
  • Retrieve the Agnostic Covering Number Upper/Lower Bounds for the Function Class
  • Retrieve the Maurey Constant
  • Retrieve the Scale-Sensitive Covering Number Upper/Lower Bounds given the Specified Sample for the Function Class
  • Compute the Output Dimension
  • Compute the Maurey Covering Number Upper Bounds for Operator Population Metric Norm
  • Compute the Maurey Covering Number Upper Bounds for Operator Population Supremum Norm
  • Compute the Maurey Covering Number Upper Bounds for Operator Sample Metric Norm
  • Compute the Maurey Covering Number Upper Bounds for Operator Sample Supremum Norm

Module Computational Core Module
Library Statistical Learning Library
Project R1 and Rd Vector/Tensor Spaces (Validated and/or Normed), and Function Classes
Package Normed Finite Spaces Function Class
Author:
Lakshmi Krishnamurthy

  • Method Details

    • 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 generalizedValidatedVector) throws java.lang.Exception
      Compute the Operator Sample Metric Norm
      Parameters:
      generalizedValidatedVector - 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 generalizedValidatedVector) throws java.lang.Exception
      Compute the Operator Sample Supremum Norm
      Parameters:
      generalizedValidatedVector - 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 generalizedValidatedVector, R1ToR1 r1ToR1FatShatteringFunction)
      Retrieve the Scale-Sensitive Covering Number Upper/Lower Bounds given the Specified Sample for the Function Class
      Parameters:
      generalizedValidatedVector - The Validated Instance Vector Sequence
      r1ToR1FatShatteringFunction - The Cover Fat Shattering Coefficient R1 To R1
      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 generalizedValidatedVector)
      Compute the Maurey Covering Number Upper Bounds for Operator Sample Metric Norm
      Parameters:
      generalizedValidatedVector - 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 generalizedValidatedVector)
      Compute the Maurey Covering Number Upper Bounds for Operator Sample Supremum Norm
      Parameters:
      generalizedValidatedVector - The Validated Vector Space Instance
      Returns:
      The Maurey Operator Covering Number Upper Bounds Instance Corresponding to the Operator Sample Supremum Norm