MaureyOperatorCoveringBounds.java
package org.drip.spaces.cover;
/*
* -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
*/
/*!
* Copyright (C) 2020 Lakshmi Krishnamurthy
* Copyright (C) 2019 Lakshmi Krishnamurthy
* Copyright (C) 2018 Lakshmi Krishnamurthy
* Copyright (C) 2017 Lakshmi Krishnamurthy
* Copyright (C) 2016 Lakshmi Krishnamurthy
* Copyright (C) 2015 Lakshmi Krishnamurthy
*
* This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
* asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
* analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
* equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
* numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
* and computational support.
*
* https://lakshmidrip.github.io/DROP/
*
* DROP is composed of three modules:
*
* - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
* - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
* - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
*
* DROP Product Core implements libraries for the following:
* - Fixed Income Analytics
* - Loan Analytics
* - Transaction Cost Analytics
*
* DROP Portfolio Core implements libraries for the following:
* - Asset Allocation Analytics
* - Asset Liability Management Analytics
* - Capital Estimation Analytics
* - Exposure Analytics
* - Margin Analytics
* - XVA Analytics
*
* DROP Computational Core implements libraries for the following:
* - Algorithm Support
* - Computation Support
* - Function Analysis
* - Model Validation
* - Numerical Analysis
* - Numerical Optimizer
* - Spline Builder
* - Statistical Learning
*
* Documentation for DROP is Spread Over:
*
* - Main => https://lakshmidrip.github.io/DROP/
* - Wiki => https://github.com/lakshmiDRIP/DROP/wiki
* - GitHub => https://github.com/lakshmiDRIP/DROP
* - Repo Layout Taxonomy => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
* - Javadoc => https://lakshmidrip.github.io/DROP/Javadoc/index.html
* - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
* - Release Versions => https://lakshmidrip.github.io/DROP/version.html
* - Community Credits => https://lakshmidrip.github.io/DROP/credits.html
* - Issues Catalog => https://github.com/lakshmiDRIP/DROP/issues
* - JUnit => https://lakshmidrip.github.io/DROP/junit/index.html
* - Jacoco => https://lakshmidrip.github.io/DROP/jacoco/index.html
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
*
* You may obtain a copy of the License at
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
*
* See the License for the specific language governing permissions and
* limitations under the License.
*/
/**
* <i>MaureyOperatorCoveringBounds</i> implements the estimate the Upper Bounds and/or Absolute Values of the
* Covering Number for the Hilbert R<sup>d</sup> To Supremum R<sup>d</sup> Operator Class. The Main
* References are:
*
* <br><br>
* <ul>
* <li>
* Carl, B. (1985): Inequalities of the Bernstein-Jackson type and the Degree of Compactness of
* Operators in Banach Spaces <i>Annals of the Fourier Institute</i> <b>35 (3)</b> 79-118
* </li>
* <li>
* Carl, B., and I. Stephani (1990): <i>Entropy, Compactness, and the Approximation of Operators</i>
* <b>Cambridge University Press</b> Cambridge UK
* </li>
* <li>
* Williamson, R. C., A. J. Smola, and B. Scholkopf (2000): Entropy Numbers of Linear Function
* Classes, in: <i>Proceedings of the 13th Annual Conference on Computational Learning
* Theory</i> <b>ACM</b> New York
* </li>
* </ul>
*
* <br><br>
* <ul>
* <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
* <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/StatisticalLearningLibrary.md">Statistical Learning Library</a></li>
* <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/spaces/README.md">R<sup>1</sup> and R<sup>d</sup> Vector/Tensor Spaces (Validated and/or Normed), and Function Classes</a></li>
* <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/spaces/cover/README.md">Vector Spaces Covering Number Estimator</a></li>
* </ul>
* <br><br>
*
* @author Lakshmi Krishnamurthy
*/
public class MaureyOperatorCoveringBounds {
/**
* Maurey Constant - from the Hilbert - Supremum Identity Map Estimate
*/
public static final double HILBERT_SUPREMUM_IDENTITY_CONSTANT = 1.86;
/**
* Maurey Constant - from the Williamson-Smola-Scholkopf Estimate
*/
public static final double WILLIAMSON_SMOLA_SCHOLKOPF_CONSTANT = 103.;
private int _iSupremumDimension = -1;
private double _dblOperatorNorm = java.lang.Double.NaN;
private double _dblMaureyConstant = java.lang.Double.NaN;
/**
* Construct an Instance Hilbert To Supremum Identity Map based Maurey Operator Covering Bounds
*
* @param iSupremumDimension The Operator Supremum Output Space Dimension
* @param dblOperatorNorm The Operator Norm of Interest
*
* @return The Instance Hilbert To Supremum Identity Map based Maurey Operator Covering Bounds
*/
public static final MaureyOperatorCoveringBounds HilbertSupremumIdentityMap (
final int iSupremumDimension,
final double dblOperatorNorm)
{
try {
return new MaureyOperatorCoveringBounds (HILBERT_SUPREMUM_IDENTITY_CONSTANT,
iSupremumDimension, dblOperatorNorm);
} catch (java.lang.Exception e) {
e.printStackTrace();
}
return null;
}
/**
* Construct an Instance of the Maurey Operator Covering Bounds based upon the Williamson, Smola, and
* Scholkopf Estimate
*
* @param iSupremumDimension The Operator Supremum Output Space Dimension
* @param dblOperatorNorm The Operator Norm of Interest
*
* @return Maurey Operator Covering Bounds based upon the Williamson, Smola, and Scholkopf Estimate
*/
public static final MaureyOperatorCoveringBounds WilliamsonSmolaScholkopfEstimate (
final int iSupremumDimension,
final double dblOperatorNorm)
{
try {
return new MaureyOperatorCoveringBounds (WILLIAMSON_SMOLA_SCHOLKOPF_CONSTANT,
iSupremumDimension, dblOperatorNorm);
} catch (java.lang.Exception e) {
e.printStackTrace();
}
return null;
}
/**
* MaureyOperatorCoveringBounds Constructor
*
* @param dblMaureyConstant The Maurey Constant
* @param iSupremumDimension The Operator Supremum Output Space Dimension
* @param dblOperatorNorm The Operator Norm of Interest
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public MaureyOperatorCoveringBounds (
final double dblMaureyConstant,
final int iSupremumDimension,
final double dblOperatorNorm)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsValid (_dblMaureyConstant = dblMaureyConstant) || 0 >=
(_iSupremumDimension = iSupremumDimension) || !org.drip.numerical.common.NumberUtil.IsValid
(_dblOperatorNorm = dblOperatorNorm))
throw new java.lang.Exception ("MaureyOperatorCoveringBounds ctr => Invalid Inputs");
}
/**
* Retrieve the Maurey Constant
*
* @return The Maurey Constant
*/
public double maureyConstant()
{
return _dblMaureyConstant;
}
/**
* Retrieve the Supremum Dimension
*
* @return The Supremum Dimension
*/
public int supremumDimension()
{
return _iSupremumDimension;
}
/**
* Retrieve the Operator Norm of Interest
*
* @return The Operator Norm of Interest
*/
public double operatorNorm()
{
return _dblOperatorNorm;
}
/**
* Compute the Upper Bound for the Dyadic Entropy Number
*
* @param iEntropyNumberIndex The Entropy Number Index
*
* @return The Upper Bound for the DyadicEntropy Number
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double dyadicEntropyUpperBound (
final int iEntropyNumberIndex)
throws java.lang.Exception
{
if (0 >= iEntropyNumberIndex)
throw new java.lang.Exception
("MaureyOperatorCoveringBounds::dyadicEntropyUpperBound => Invalid Inputs");
return _dblMaureyConstant * _dblOperatorNorm * java.lang.Math.sqrt ((java.lang.Math.log (1. +
(((double) _iSupremumDimension) / ((double) iEntropyNumberIndex))) / iEntropyNumberIndex));
}
/**
* Compute the Upper Bound for the Entropy Number
*
* @param iEntropyNumberIndex The Entropy Number Index
*
* @return The Upper Bound for the Entropy Number
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public double entropyNumberUpperBound (
final int iEntropyNumberIndex)
throws java.lang.Exception
{
if (0 >= iEntropyNumberIndex)
throw new java.lang.Exception
("MaureyOperatorCoveringBounds::entropyNumberUpperBound => Invalid Inputs");
double dblLogNPlus1 = 1. + java.lang.Math.log (iEntropyNumberIndex);
return _dblMaureyConstant * _dblOperatorNorm * java.lang.Math.sqrt ((1. + (((double)
_iSupremumDimension) / dblLogNPlus1)) / dblLogNPlus1);
}
}