Package org.drip.spaces.cover

Class ScaleSensitiveCoveringBounds

java.lang.Object
org.drip.spaces.cover.ScaleSensitiveCoveringBounds
All Implemented Interfaces:
FunctionClassCoveringBounds
public class ScaleSensitiveCoveringBounds
extends java.lang.Object
implements FunctionClassCoveringBounds
ScaleSensitiveCoveringBounds implements the Lower/Upper Bounds for the General Class of Functions in terms of their scale-sensitive dimensions (i.e., the fat shattering coefficients). The References are:

  • N. Alon, S. Ben-David, N. Cesa-Bianchi, and D. Haussler (1993): Scale-sensitive Dimensions, Uniform-Convergence, and Learnability Proceedings of the ACM Symposium on the Foundations of Computer Science
  • P. L. Bartlett, S. R. Kulkarni, and S. E. Posner (1997): Covering Numbers for Real-valued Function Classes IEEE Transactions on Information Theory 43 (5) 1721-1724
  • D. Pollard (1984): Convergence of Stochastic Processes Springer New York
It provides the following Functionality:
  • ScaleSensitiveCoveringBounds Constructor
  • Retrieve the Fat Shattering Coefficient Function
  • Retrieve the Sample Size
  • Compute the Minimum Sample Size required to Estimate the Cardinality corresponding to the Specified Cover
  • Compute the Cardinality for the Subset T (|x) that possesses the Specified Cover for the Restriction of the Input Function Class Family F (|x)
  • Compute the Log of the Weight Loading Coefficient for the Maximum Cover Term

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
    ScaleSensitiveCoveringBounds​(R1ToR1 r1r1FatShatter, int iSampleSize)
    ScaleSensitiveCoveringBounds Constructor

  • Method Summary

    Modifier and Type Method Description
    R1ToR1 fatShatteringFunction()
    Retrieve the Fat Shattering Coefficient Function
    double logLowerBound​(double cover)
    Log of the Lower Bound of the Function Covering Number
    double logUpperBound​(double cover)
    Log of the Upper Bound of the Function Covering Number
    double restrictedSubsetCardinality​(double cover)
    Compute the Cardinality for the Subset T (|x) that possesses the Specified Cover for the Restriction of the Input Function Class Family F (|x).
    int sampleSize()
    Retrieve the Sample Size
    double sampleSizeLowerBound​(double cover)
    Compute the Minimum Sample Size required to Estimate the Cardinality corresponding to the Specified Cover
    double upperProbabilityBoundWeight​(double cover)
    Compute the Log of the Weight Loading Coefficient for the Maximum Cover Term in: {Probability that the Empirical Error .gt.

    Methods inherited from class java.lang.Object

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

  • Constructor Details

    • ScaleSensitiveCoveringBounds

      public ScaleSensitiveCoveringBounds​(R1ToR1 r1r1FatShatter, int iSampleSize) throws java.lang.Exception
      ScaleSensitiveCoveringBounds Constructor
      Parameters:
      r1r1FatShatter - The Cover Fat Shattering Coefficient Function
      iSampleSize - Sample Size
      Throws:
      java.lang.Exception - Thrown if the Inputs are Invalid

  • Method Details

    • fatShatteringFunction

      public R1ToR1 fatShatteringFunction()
      Retrieve the Fat Shattering Coefficient Function
      Returns:
      The Fat Shattering Coefficient Function

    • sampleSize

      public int sampleSize()
      Retrieve the Sample Size
      Returns:
      The Sample Size

    • sampleSizeLowerBound

      public double sampleSizeLowerBound​(double cover) throws java.lang.Exception
      Compute the Minimum Sample Size required to Estimate the Cardinality corresponding to the Specified Cover
      Parameters:
      cover - The Cover
      Returns:
      The Minimum Sample Size
      Throws:
      java.lang.Exception - Thrown if the Minimum Sample Size Cannot be computed

    • restrictedSubsetCardinality

      public double restrictedSubsetCardinality​(double cover) throws java.lang.Exception
      Compute the Cardinality for the Subset T (|x) that possesses the Specified Cover for the Restriction of the Input Function Class Family F (|x).
      Parameters:
      cover - The Specified Cover
      Returns:
      The Restricted Subset Cardinality
      Throws:
      java.lang.Exception - Thrown if the Restricted Subset Cardinality cannot be computed

    • upperProbabilityBoundWeight

      public double upperProbabilityBoundWeight​(double cover) throws java.lang.Exception
      Compute the Log of the Weight Loading Coefficient for the Maximum Cover Term in: {Probability that the Empirical Error .gt. Cover} .lte. 4 * exp (-m * Cover^2 / 128) * [[Max Covering Number Over the Specified Sample]] Reference is: - D. Haussler (1995): Sphere Packing Numbers for Subsets of the Boolean n-Cube with Bounded Vapnik-Chervonenkis Dimension, Journal of the COmbinatorial Theory A 69 (2) 217.
      Parameters:
      cover - The Specified Cover
      Returns:
      Log of the Weight Loading Coefficient for the Maximum Cover Term
      Throws:
      java.lang.Exception - Thrown if the Log of the Weight Loading Coefficient cannot be computed

    • logLowerBound

      public double logLowerBound​(double cover) throws java.lang.Exception
      Description copied from interface: FunctionClassCoveringBounds
      Log of the Lower Bound of the Function Covering Number
      Specified by:
      logLowerBound in interface FunctionClassCoveringBounds
      Parameters:
      cover - The Size of the Cover
      Returns:
      Log of the Lower Bound of the Function Covering Number
      Throws:
      java.lang.Exception - Thrown if the Inputs are Invalid

    • logUpperBound

      public double logUpperBound​(double cover) throws java.lang.Exception
      Description copied from interface: FunctionClassCoveringBounds
      Log of the Upper Bound of the Function Covering Number
      Specified by:
      logUpperBound in interface FunctionClassCoveringBounds
      Parameters:
      cover - The Size of the Cover
      Returns:
      Log of the Upper Bound of the Function Covering Number
      Throws:
      java.lang.Exception - Thrown if the Inputs are Invalid