DecisionFunctionOperatorBounds.java
package org.drip.learning.svm;
/*
* -*- 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>DecisionFunctionOperatorBounds</i> implements the Dot Product Entropy Number Upper Bounds for the
* Product of Kernel Feature Map Function and the Scaling Diagonal Operator.
*
* <br><br>
* The References are:
* <br><br>
* <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>
* </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/svm">Kernel SVM Decision Function Operator</a></li>
* </ul>
*
* @author Lakshmi Krishnamurthy
*/
public class DecisionFunctionOperatorBounds {
private int _iFeatureSpaceDimension = -1;
private double _dblInverseMarginNormBound = java.lang.Double.NaN;
private double _dblFeatureSpaceMaureyConstant = java.lang.Double.NaN;
private org.drip.learning.kernel.DiagonalScalingOperator _dsoFactorizer = null;
/**
* DecisionFunctionOperatorBounds Constructor
*
* @param dsoFactorizer The Factorizing Diagonal Scaling Operator
* @param dblInverseMarginNormBound The Decision Function Inverse Margin Norm Bound
* @param dblFeatureSpaceMaureyConstant The Kernel Feature Space Function Maurey Constant
* @param iFeatureSpaceDimension The Kernel Feature Space Dimension
*
* @throws java.lang.Exception Thrown if the Inputs are Invalid
*/
public DecisionFunctionOperatorBounds (
final org.drip.learning.kernel.DiagonalScalingOperator dsoFactorizer,
final double dblInverseMarginNormBound,
final double dblFeatureSpaceMaureyConstant,
final int iFeatureSpaceDimension)
throws java.lang.Exception
{
if (null == (_dsoFactorizer = dsoFactorizer) || !org.drip.numerical.common.NumberUtil.IsValid
(_dblInverseMarginNormBound = dblInverseMarginNormBound) ||
!org.drip.numerical.common.NumberUtil.IsValid (_dblFeatureSpaceMaureyConstant =
dblFeatureSpaceMaureyConstant) || 0 >= (_iFeatureSpaceDimension =
iFeatureSpaceDimension))
throw new java.lang.Exception ("DecisionFunctionOperatorBounds ctr => Invalid Inputs");
}
/**
* Retrieve the Factorizing Diagonal Scaling Operator Instance
*
* @return The Factorizing Diagonal Scaling Operator Instance
*/
public org.drip.learning.kernel.DiagonalScalingOperator factorizingOperator()
{
return _dsoFactorizer;
}
/**
* Retrieve the Norm Upper Bound of the Inverse Margin
*
* @return The Norm Upper Bound of the Inverse Margin
*/
public double inverseMarginNormBound()
{
return _dblInverseMarginNormBound;
}
/**
* Retrieve the Feature Space Maurey Constant
*
* @return The Feature Space Maurey Constant
*/
public double featureSpaceMaureyConstant()
{
return _dblFeatureSpaceMaureyConstant;
}
/**
* Retrieve the Feature Space Dimension
*
* @return The Feature Space Dimension
*/
public double featureSpaceDimension()
{
return _iFeatureSpaceDimension;
}
/**
* Compute the Feature Space's Maurey Bound for the Entropy Number given the specified Entropy Number
*
* @param iFeatureSpaceEntropyNumber The Feature Space Entropy Number
*
* @return The Feature Space's Maurey Bound for the specified Entropy Number
*
* @throws java.lang.Exception The Feature Space's Maurey Bound cannot be computed
*/
public double featureSpaceMaureyBound (
final int iFeatureSpaceEntropyNumber)
throws java.lang.Exception
{
if (0 >= iFeatureSpaceEntropyNumber)
throw new java.lang.Exception
("DecisionFunctionOperatorBounds::featureSpaceMaureyBound => Invalid Inputs");
return java.lang.Math.sqrt (1. / (iFeatureSpaceEntropyNumber * java.lang.Math.sqrt
(java.lang.Math.log (1. + (((double) _iFeatureSpaceDimension) / java.lang.Math.log
(iFeatureSpaceEntropyNumber))))));
}
/**
* Compute the Decision Function Entropy Number Upper Bound using the Product of the Feature Space's
* Maurey Upper Bound for the Entropy for the specified Entropy Number and the Scaling Operator Entropy
* Number Upper Bound
*
* @param iFeatureSpaceEntropyNumber The Feature Space Entropy Number
*
* @return The Feature Space's Operator Entropy for the specified Entropy Number
*
* @throws java.lang.Exception The Feature Space's Operator Entropy cannot be computed
*/
public double featureMaureyOperatorEntropy (
final int iFeatureSpaceEntropyNumber)
throws java.lang.Exception
{
return _dblInverseMarginNormBound * _dsoFactorizer.entropyNumberUpperBound() *
featureSpaceMaureyBound (iFeatureSpaceEntropyNumber);
}
/**
* Compute the Decision Function Entropy Number Upper Bound using the Product of the Feature Space's
* Maurey Upper Bound for the Entropy for the specified Entropy Number and the Scaling Operator Norm
*
* @param iFeatureSpaceEntropyNumber The Feature Space Entropy Number
*
* @return The Feature Space's Operator Norm for the specified Entropy Number
*
* @throws java.lang.Exception The Feature Space's Operator Norm cannot be computed
*/
public double featureMaureyOperatorNorm (
final int iFeatureSpaceEntropyNumber)
throws java.lang.Exception
{
return _dblInverseMarginNormBound * _dsoFactorizer.norm() * featureSpaceMaureyBound
(iFeatureSpaceEntropyNumber);
}
/**
* Compute the Decision Function Entropy Number Upper Bound using the Product of the Feature Space's
* Norm for the Upper Bound of the Entropy Number and the Scaling Operator Norm
*
* @return The Entropy Number Upper Bound using the Product Norm
*
* @throws java.lang.Exception The Entropy Number Upper Bound cannot be computed
*/
public double productFeatureOperatorNorm()
throws java.lang.Exception
{
return _dblInverseMarginNormBound * _dsoFactorizer.norm();
}
/**
* Compute the Decision Function Entropy Number Upper Bound using the Product of the Feature Space's
* Norm for the Upper Bound of the Entropy Number and the Scaling Operator Entropy Number Upper Bound
*
* @return The Entropy Number Upper Bound using the Product Norm
*
* @throws java.lang.Exception The Entropy Number Upper Bound cannot be computed
*/
public double featureNormOperatorEntropy()
throws java.lang.Exception
{
return _dblInverseMarginNormBound * _dsoFactorizer.entropyNumberUpperBound();
}
/**
* Compute the Infimum of the Decision Function Operator Upper Bound across all the Product Bounds for
* the specified Feature Space Entropy Number
*
* @param iFeatureSpaceEntropyNumber The specified Feature Space Entropy Number
*
* @return Infimum of the Decision Function Operator Upper Bound
*
* @throws java.lang.Exception Thrown if the Infimum of the Decision Function Operator Upper Bound cannot
* be calculated
*/
public double infimumUpperBound (
final int iFeatureSpaceEntropyNumber)
throws java.lang.Exception
{
double dblFactorizerNorm = _dsoFactorizer.norm();
double dblFactorizerEntropyUpperBound = _dsoFactorizer.entropyNumberUpperBound();
double dblFeatureSpaceMaureyBound = featureSpaceMaureyBound (iFeatureSpaceEntropyNumber);
double dblProductFeatureOperatorNorm = _dblInverseMarginNormBound * dblFactorizerNorm;
double dblFeatureMaureyOperatorNorm = dblProductFeatureOperatorNorm * dblFeatureSpaceMaureyBound;
double dblFeatureNormOperatorEntropy = _dblInverseMarginNormBound * dblFactorizerEntropyUpperBound;
double dblInfimumUpperBound = dblFeatureNormOperatorEntropy * dblFeatureSpaceMaureyBound;
if (dblInfimumUpperBound > dblFeatureMaureyOperatorNorm)
dblInfimumUpperBound = dblFeatureMaureyOperatorNorm;
if (dblInfimumUpperBound > dblProductFeatureOperatorNorm)
dblInfimumUpperBound = dblProductFeatureOperatorNorm;
return dblInfimumUpperBound > dblFeatureNormOperatorEntropy ? dblInfimumUpperBound :
dblFeatureNormOperatorEntropy;
}
}