MaureyOperatorCoveringBounds.java

  1. package org.drip.spaces.cover;

  3. /*
  4.  * -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
  5.  */

  7. /*!
  8.  * Copyright (C) 2020 Lakshmi Krishnamurthy
  9.  * Copyright (C) 2019 Lakshmi Krishnamurthy
  10.  * Copyright (C) 2018 Lakshmi Krishnamurthy
  11.  * Copyright (C) 2017 Lakshmi Krishnamurthy
  12.  * Copyright (C) 2016 Lakshmi Krishnamurthy
  13.  * Copyright (C) 2015 Lakshmi Krishnamurthy
  14.  * 
  15.  *  This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
  16.  *      asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
  17.  *      analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
  18.  *      equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
  19.  *      numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
  20.  *      and computational support.
  21.  *  
  22.  *      https://lakshmidrip.github.io/DROP/
  23.  *  
  24.  *  DROP is composed of three modules:
  25.  *  
  26.  *  - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
  27.  *  - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
  28.  *  - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
  29.  * 
  30.  *  DROP Product Core implements libraries for the following:
  31.  *  - Fixed Income Analytics
  32.  *  - Loan Analytics
  33.  *  - Transaction Cost Analytics
  34.  * 
  35.  *  DROP Portfolio Core implements libraries for the following:
  36.  *  - Asset Allocation Analytics
  37.  *  - Asset Liability Management Analytics
  38.  *  - Capital Estimation Analytics
  39.  *  - Exposure Analytics
  40.  *  - Margin Analytics
  41.  *  - XVA Analytics
  42.  * 
  43.  *  DROP Computational Core implements libraries for the following:
  44.  *  - Algorithm Support
  45.  *  - Computation Support
  46.  *  - Function Analysis
  47.  *  - Model Validation
  48.  *  - Numerical Analysis
  49.  *  - Numerical Optimizer
  50.  *  - Spline Builder
  51.  *  - Statistical Learning
  52.  * 
  53.  *  Documentation for DROP is Spread Over:
  54.  * 
  55.  *  - Main                     => https://lakshmidrip.github.io/DROP/
  56.  *  - Wiki                     => https://github.com/lakshmiDRIP/DROP/wiki
  57.  *  - GitHub                   => https://github.com/lakshmiDRIP/DROP
  58.  *  - Repo Layout Taxonomy     => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
  59.  *  - Javadoc                  => https://lakshmidrip.github.io/DROP/Javadoc/index.html
  60.  *  - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
  61.  *  - Release Versions         => https://lakshmidrip.github.io/DROP/version.html
  62.  *  - Community Credits        => https://lakshmidrip.github.io/DROP/credits.html
  63.  *  - Issues Catalog           => https://github.com/lakshmiDRIP/DROP/issues
  64.  *  - JUnit                    => https://lakshmidrip.github.io/DROP/junit/index.html
  65.  *  - Jacoco                   => https://lakshmidrip.github.io/DROP/jacoco/index.html
  66.  * 
  67.  *  Licensed under the Apache License, Version 2.0 (the "License");
  68.  *      you may not use this file except in compliance with the License.
  69.  *   
  70.  *  You may obtain a copy of the License at
  71.  *      http://www.apache.org/licenses/LICENSE-2.0
  72.  *  
  73.  *  Unless required by applicable law or agreed to in writing, software
  74.  *      distributed under the License is distributed on an "AS IS" BASIS,
  75.  *      WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
  76.  *  
  77.  *  See the License for the specific language governing permissions and
  78.  *      limitations under the License.
  79.  */

  81. /**
  82.  * <i>MaureyOperatorCoveringBounds</i> implements the estimate the Upper Bounds and/or Absolute Values of the
  83.  * Covering Number for the Hilbert R<sup>d</sup> To Supremum R<sup>d</sup> Operator Class. The Main
  84.  * References are:
  85.  *
  86.  * <br><br>
  87.  *  <ul>
  88.  *      <li>
  89.  *          Carl, B. (1985): Inequalities of the Bernstein-Jackson type and the Degree of Compactness of
  90.  *              Operators in Banach Spaces <i>Annals of the Fourier Institute</i> <b>35 (3)</b> 79-118
  91.  *      </li>
  92.  *      <li>
  93.  *          Carl, B., and I. Stephani (1990): <i>Entropy, Compactness, and the Approximation of Operators</i>
  94.  *              <b>Cambridge University Press</b> Cambridge UK 
  95.  *      </li>
  96.  *      <li>
  97.  *          Williamson, R. C., A. J. Smola, and B. Scholkopf (2000): Entropy Numbers of Linear Function
  98.  *              Classes, in: <i>Proceedings of the 13th Annual Conference on Computational Learning
  99.  *                  Theory</i> <b>ACM</b> New York
  100.  *      </li>
  101.  *  </ul>
  102.  *
  103.  * <br><br>
  104.  *  <ul>
  105.  *      <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
  106.  *      <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/StatisticalLearningLibrary.md">Statistical Learning Library</a></li>
  107.  *      <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>
  108.  *      <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>
  109.  *  </ul>
  110.  * <br><br>
  111.  *
  112.  * @author Lakshmi Krishnamurthy
  113.  */

  115. public class MaureyOperatorCoveringBounds {

  117.     /**
  118.      * Maurey Constant - from the Hilbert - Supremum Identity Map Estimate
  119.      */

  121.     public static final double HILBERT_SUPREMUM_IDENTITY_CONSTANT = 1.86;

  123.     /**
  124.      * Maurey Constant - from the Williamson-Smola-Scholkopf Estimate
  125.      */

  127.     public static final double WILLIAMSON_SMOLA_SCHOLKOPF_CONSTANT = 103.;

  129.     private int _iSupremumDimension = -1;
  130.     private double _dblOperatorNorm = java.lang.Double.NaN;
  131.     private double _dblMaureyConstant = java.lang.Double.NaN;

  133.     /**
  134.      * Construct an Instance Hilbert To Supremum Identity Map based Maurey Operator Covering Bounds
  135.      * 
  136.      * @param iSupremumDimension The Operator Supremum Output Space Dimension
  137.      * @param dblOperatorNorm The Operator Norm of Interest
  138.      * 
  139.      * @return The Instance Hilbert To Supremum Identity Map based Maurey Operator Covering Bounds
  140.      */

  142.     public static final MaureyOperatorCoveringBounds HilbertSupremumIdentityMap (
  143.         final int iSupremumDimension,
  144.         final double dblOperatorNorm)
  145.     {
  146.         try {
  147.             return new MaureyOperatorCoveringBounds (HILBERT_SUPREMUM_IDENTITY_CONSTANT,
  148.                 iSupremumDimension, dblOperatorNorm);
  149.         } catch (java.lang.Exception e) {
  150.             e.printStackTrace();
  151.         }

  153.         return null;
  154.     }

  156.     /**
  157.      * Construct an Instance of the Maurey Operator Covering Bounds based upon the Williamson, Smola, and
  158.      *  Scholkopf Estimate
  159.      * 
  160.      * @param iSupremumDimension The Operator Supremum Output Space Dimension
  161.      * @param dblOperatorNorm The Operator Norm of Interest
  162.      * 
  163.      * @return Maurey Operator Covering Bounds based upon the Williamson, Smola, and Scholkopf Estimate
  164.      */

  166.     public static final MaureyOperatorCoveringBounds WilliamsonSmolaScholkopfEstimate (
  167.         final int iSupremumDimension,
  168.         final double dblOperatorNorm)
  169.     {
  170.         try {
  171.             return new MaureyOperatorCoveringBounds (WILLIAMSON_SMOLA_SCHOLKOPF_CONSTANT,
  172.                 iSupremumDimension, dblOperatorNorm);
  173.         } catch (java.lang.Exception e) {
  174.             e.printStackTrace();
  175.         }

  177.         return null;
  178.     }

  180.     /**
  181.      * MaureyOperatorCoveringBounds Constructor
  182.      * 
  183.      * @param dblMaureyConstant The Maurey Constant
  184.      * @param iSupremumDimension The Operator Supremum Output Space Dimension
  185.      * @param dblOperatorNorm The Operator Norm of Interest
  186.      * 
  187.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  188.      */

  190.     public MaureyOperatorCoveringBounds (
  191.         final double dblMaureyConstant,
  192.         final int iSupremumDimension,
  193.         final double dblOperatorNorm)
  194.         throws java.lang.Exception
  195.     {
  196.         if (!org.drip.numerical.common.NumberUtil.IsValid (_dblMaureyConstant = dblMaureyConstant) || 0 >=
  197.             (_iSupremumDimension = iSupremumDimension) || !org.drip.numerical.common.NumberUtil.IsValid
  198.                 (_dblOperatorNorm = dblOperatorNorm))
  199.             throw new java.lang.Exception ("MaureyOperatorCoveringBounds ctr => Invalid Inputs");
  200.     }

  202.     /**
  203.      * Retrieve the Maurey Constant
  204.      * 
  205.      * @return The Maurey Constant
  206.      */

  208.     public double maureyConstant()
  209.     {
  210.         return _dblMaureyConstant;
  211.     }

  213.     /**
  214.      * Retrieve the Supremum Dimension
  215.      * 
  216.      * @return The Supremum Dimension
  217.      */

  219.     public int supremumDimension()
  220.     {
  221.         return _iSupremumDimension;
  222.     }

  224.     /**
  225.      * Retrieve the Operator Norm of Interest
  226.      * 
  227.      * @return The Operator Norm of Interest
  228.      */

  230.     public double operatorNorm()
  231.     {
  232.         return _dblOperatorNorm;
  233.     }

  235.     /**
  236.      * Compute the Upper Bound for the Dyadic Entropy Number
  237.      * 
  238.      * @param iEntropyNumberIndex The Entropy Number Index
  239.      * 
  240.      * @return The Upper Bound for the DyadicEntropy Number
  241.      * 
  242.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  243.      */

  245.     public double dyadicEntropyUpperBound (
  246.         final int iEntropyNumberIndex)
  247.         throws java.lang.Exception
  248.     {
  249.         if (0 >= iEntropyNumberIndex)
  250.             throw new java.lang.Exception
  251.                 ("MaureyOperatorCoveringBounds::dyadicEntropyUpperBound => Invalid Inputs");

  253.         return _dblMaureyConstant * _dblOperatorNorm * java.lang.Math.sqrt ((java.lang.Math.log (1. +
  254.             (((double) _iSupremumDimension) / ((double) iEntropyNumberIndex))) / iEntropyNumberIndex));
  255.     }

  257.     /**
  258.      * Compute the Upper Bound for the Entropy Number
  259.      * 
  260.      * @param iEntropyNumberIndex The Entropy Number Index
  261.      * 
  262.      * @return The Upper Bound for the Entropy Number
  263.      * 
  264.      * @throws java.lang.Exception Thrown if the Inputs are Invalid
  265.      */

  267.     public double entropyNumberUpperBound (
  268.         final int iEntropyNumberIndex)
  269.         throws java.lang.Exception
  270.     {
  271.         if (0 >= iEntropyNumberIndex)
  272.             throw new java.lang.Exception
  273.                 ("MaureyOperatorCoveringBounds::entropyNumberUpperBound => Invalid Inputs");

  275.         double dblLogNPlus1 = 1. + java.lang.Math.log (iEntropyNumberIndex);

  277.         return _dblMaureyConstant * _dblOperatorNorm * java.lang.Math.sqrt ((1. + (((double)
  278.             _iSupremumDimension) / dblLogNPlus1)) / dblLogNPlus1);
  279.     }
  280. }
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