Package org.drip.spaces.cover

Class CoveringBoundsHelper

java.lang.Object
org.drip.spaces.cover.CoveringBoundsHelper
public class CoveringBoundsHelper
extends java.lang.Object
CoveringBoundsHelper contains the assortment of Utilities used in the Computation of Upper Bounds for Normed Single Function Spaces and Function Space Products. 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:
  • Compute the Dyadic Entropy Number from the nth Entropy Number
  • Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes
  • Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes using the Function Class Norm

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

  • Constructor Summary

    Constructors
    Constructor Description
    CoveringBoundsHelper()  

  • Method Summary

    Modifier and Type Method Description
    static double CarlStephaniProductBound​(MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsA, MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsB, int entropyNumberIndexA, int entropyNumberIndexB)
    Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes
    static CarlStephaniNormedBounds CarlStephaniProductNorm​(MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsA, MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsB, double normA, double normB, int entropyNumberIndex)
    Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes using the Function Class Norm
    static double DyadicEntropyNumber​(double logNEntropyNumber)
    Compute the Dyadic Entropy Number from the nth Entropy Number

    Methods inherited from class java.lang.Object

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

  • Constructor Details

    • CoveringBoundsHelper

      public CoveringBoundsHelper()

  • Method Details

    • DyadicEntropyNumber

      public static final double DyadicEntropyNumber​(double logNEntropyNumber) throws java.lang.Exception
      Compute the Dyadic Entropy Number from the nth Entropy Number
      Parameters:
      logNEntropyNumber - Log of the nth Entropy Number
      Returns:
      The Dyadic Entropy Number
      Throws:
      java.lang.Exception - Thrown if the Dyadic Entropy Number cannot be calculated

    • CarlStephaniProductBound

      public static final double CarlStephaniProductBound​(MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsA, MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsB, int entropyNumberIndexA, int entropyNumberIndexB) throws java.lang.Exception
      Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes
      Parameters:
      maureyOperatorCoveringBoundsA - The Maurey Operator Covering Bounds for Class A
      maureyOperatorCoveringBoundsB - The Maurey Operator Covering Bounds for Class B
      entropyNumberIndexA - Entropy Number Index for Class A
      entropyNumberIndexB - Entropy Number Index for Class B
      Returns:
      The Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes
      Throws:
      java.lang.Exception - Thrown if the Inputs are Invalid

    • CarlStephaniProductNorm

      public static final CarlStephaniNormedBounds CarlStephaniProductNorm​(MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsA, MaureyOperatorCoveringBounds maureyOperatorCoveringBoundsB, double normA, double normB, int entropyNumberIndex)
      Compute the Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes using the Function Class Norm
      Parameters:
      maureyOperatorCoveringBoundsA - The Maurey Operator Covering Bounds for Class A
      maureyOperatorCoveringBoundsB - The Maurey Operator Covering Bounds for Class B
      normA - The Function Class A Norm
      normB - The Function Class B Norm
      entropyNumberIndex - Entropy Number Index for either Class
      Returns:
      The Upper Bound for the Entropy Number of the Operator Custom Covering Number Metric Product across both the Function Classes using the Function Norm