DiagonalOperatorCoveringBound.java

package org.drip.learning.bound;

/*
 * -*- 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>DiagonalOperatorCoveringBound</i> implements the Behavior of the Bound on the Covering Number of the
 * Diagonal Scaling Operator. The Asymptote is set as either
 * <br><br>
 *              log [e_n(A)] ~ O {(1/log n)^alpha}
 * <br><br>
 *      - OR -
 * <br><br>
 *                    e_n(A) ~ O {(1/log n)^alpha}
 * <br><br>
 *  The References are:
 * <ul>
 *  <li>
 *      Ash, R. (1965): <i>Information Theory</i> <b>Inter-science</b> New York
 *  </li>
 *  <li>
 *      Carl, B., and I. Stephani (1990): <i>Entropy, Compactness, and Approximation of Operators</i>
 *          <b>Cambridge University Press</b> Cambridge UK
 *  </li>
 *  <li>
 *      Gordon, Y., H. Konig, and C. Schutt (1987): Geometric and Probabilistic Estimates of Entropy and
 *          Approximation Numbers of Operators <i>Journal of Approximation Theory</i> <b>49</b> 219-237
 *  </li>
 *  <li>
 *      Konig, H. (1986): <i>Eigenvalue Distribution of Compact Operators</i> <b>Birkhauser</b> Basel,
 *          Switzerland
 *  </li>
 *  <li>
 *      Smola, A. J., A. Elisseff, B. Scholkopf, and R. C. Williamson (2000): Entropy Numbers for Convex
 *          Combinations and mlps, in: <i>Advances in Large Margin Classifiers, A. Smola, P. Bartlett, B.
 *          Scholkopf, and D. Schuurmans - editors</i> <b>MIT Press</b> Cambridge, MA
 *  </li>
 *  <li>
 *      Williamson, R. C., A. J. Smola, and B. Scholkopf (2001): Generalization Performance of Regularization
 *          Networks and Support Vector Machines via Entropy Numbers of Compact Operators <i>IEEE
 *          Transactions on Information Theory</i> <b>47 (6)</b> 2516-2532
 *  </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</a></li>
 *      <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/learning">Agnostic Learning Bounds under Empirical Loss Minimization Schemes</a></li>
 *      <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/learning/bound">Covering Numbers, Concentration, Lipschitz Bounds</a></li>
 *  </ul>
 *
 * @author Lakshmi Krishnamurthy
 */

public class DiagonalOperatorCoveringBound {

    /**
     * Asymptote on the Base Diagonal Operator Entropy Number
     */

    public static final int BASE_DIAGONAL_ENTROPY_ASYMPTOTE_EXPONENT = 1;

    /**
     * Asymptote on the Log of the Diagonal Operator Entropy Number
     */

    public static final int LOG_DIAGONAL_ENTROPY_ASYMPTOTE_EXPONENT = 2;

    private double _dblOperatorEntropyAsymptoteExponent = java.lang.Double.NaN;
    private int _iOperatorEntropyAsymptoteBase = BASE_DIAGONAL_ENTROPY_ASYMPTOTE_EXPONENT;

    /**
     * DiagonalOperatorCoveringBound Constructor
     * 
     * @param iOperatorEntropyAsymptoteBase Indicate the Asymptote is on the Base Value or Log Value
     * @param dblOperatorEntropyAsymptoteExponent The Entropy Number Asymptote Exponent
     * 
     * @throws java.lang.Exception Throws if the Inputs are Invalid
     */

    public DiagonalOperatorCoveringBound (
        final int iOperatorEntropyAsymptoteBase,
        final double dblOperatorEntropyAsymptoteExponent)
        throws java.lang.Exception
    {
        if ((BASE_DIAGONAL_ENTROPY_ASYMPTOTE_EXPONENT != (_iOperatorEntropyAsymptoteBase =
            iOperatorEntropyAsymptoteBase) && LOG_DIAGONAL_ENTROPY_ASYMPTOTE_EXPONENT !=
                _iOperatorEntropyAsymptoteBase) || !org.drip.numerical.common.NumberUtil.IsValid
                    (_dblOperatorEntropyAsymptoteExponent = dblOperatorEntropyAsymptoteExponent))
            throw new java.lang.Exception ("DiagonalOperatorCoveringBound ctr => Invalid Inputs");
    }

    /**
     * Retrieve the Entropy Number Asymptote Type
     * 
     * @return The Entropy Number Asymptote Type
     */

    public int entropyNumberAsymptoteType()
    {
        return _iOperatorEntropyAsymptoteBase;
    }

    /**
     * Retrieve the Entropy Number Asymptote Exponent
     * 
     * @return The Entropy Number Asymptote Exponent
     */

    public double entropyNumberAsymptoteExponent()
    {
        return _dblOperatorEntropyAsymptoteExponent;
    }
}