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porting LM algorithm to JavaScript
This commit is contained in:
Val Erastov
parent
3d756e6e2c
commit
44b910d512
3 changed files with 989 additions and 0 deletions
136
src/cad/Java2JS.java
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136
src/cad/Java2JS.java
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package cad;
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import java.util.ArrayList;
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import java.util.Collections;
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import java.util.List;
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import java.util.Scanner;
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import java.util.regex.MatchResult;
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import java.util.regex.Matcher;
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import java.util.regex.Pattern;
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public class Java2JS {
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List<MatchResult> comments = new ArrayList<>();
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int initCount = 0;
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public void convert() {
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Scanner scanner = new Scanner(Java2JS.class.getResourceAsStream("lm.in"));
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String text = scanner.useDelimiter("\\A").next();
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text = rplc(text, "(?<![A-Za-z0-9_])(private|public|protected)\\s.+\\s([A-Za-z0-9_]+);", "this.%s = null;", 2);
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text = rplc(text, "\\snew double\\[(.+)\\];", " arr(%s);", 1);
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text = rplc(text, "\\snew int\\[(.+)\\];", " arr(%s);", 1);
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text = rplc(text, "(?<![A-Za-z0-9_])(double|int)[\\[\\]]*\\s", "var ");
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// text = rplc(text, "for\\s+\\((int|double)\\s", "for (var ");
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text = rplc(text, "FastMath", "Math");
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text = rplc(text, "Arrays\\.fill", "Arrays_fill");
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text = rplc(text, "Double\\.NEGATIVE_INFINITY", "Number.NEGATIVE_INFINITY");
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text = rplc(text, "(protected|public|private)\\s[a-zA-Z0-9_]+\\s([a-zA-Z0-9_]+)\\(",
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"this.%s = function(", 2);
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text = rplc(text, "new\\sPointVectorValuePair\\((.+)\\)", "[%s]", 1);
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text = rplc(text, "PointVectorValuePair\\s", "var ");
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text = rplcParams(text);
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text = rplc(text, "final ", "", 0);
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text = rplc(text, "@Override", "", 0);
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System.out.println(text);
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}
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private void buildComment(String text) {
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comments.clear();
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Pattern p = Pattern.compile("(?m)//.+$");
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Matcher matcher = p.matcher(text);
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while (matcher.find()) {
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comments.add(matcher.toMatchResult());
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}
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p = Pattern.compile("(?s)/\\*.+?\\*/");
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matcher = p.matcher(text);
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while (matcher.find()) {
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comments.add(matcher.toMatchResult());
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}
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}
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private String rplc(String text, String pattern, String replacement, int... groups) {
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buildComment(text);
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Pattern p = Pattern.compile(pattern);
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Matcher matcher = p.matcher(text);
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StringBuffer out = new StringBuffer();
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while (matcher.find()) {
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if (isComment(matcher)) {
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continue;
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}
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String[] params = new String[groups.length];
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for (int i = 0; i < groups.length; i++) {
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params[i] = matcher.group(groups[i]);
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}
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matcher.appendReplacement(out, String.format(replacement, params));
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}
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matcher.appendTail(out);
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return out.toString();
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}
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private boolean isComment(MatchResult matcher) {
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for (MatchResult comment : comments) {
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if (matcher.start() >= comment.start() && matcher.start() < comment.end()) {
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return true;
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}
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}
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return false;
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}
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private String rplcParams(String text) {
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buildComment(text);
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Pattern p = Pattern.compile("this\\.[^\\(]+=\\s*function\\s*\\((.+)\\)");
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Matcher matcher = p.matcher(text);
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List<MatchResult> replacements = new ArrayList<>();
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while (matcher.find()) {
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replacements.add(matcher.toMatchResult());
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}
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Collections.reverse(replacements);
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int off = text.length();
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StringBuilder out = new StringBuilder();
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for (MatchResult m : replacements) {
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if (isComment(m)) continue;
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out.insert(0, text.substring(m.end(1), off));
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String sst = m.group();
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out.insert(0, sst.replaceAll("(?<![A-Za-z0-9_])(double|int)[\\[\\]]*\\s", ""));
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off = m.start();
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}
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out.insert(0, text.substring(0, off));
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return out.toString();
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}
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public static void main(String[] args) {
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new Java2JS().convert();
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}
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}
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853
src/cad/lm.in
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853
src/cad/lm.in
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@ -0,0 +1,853 @@
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/**
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* This class solves a least-squares problem using the Levenberg-Marquardt algorithm.
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*
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* <p>This implementation <em>should</em> work even for over-determined systems
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* (i.e. systems having more point than equations). Over-determined systems
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* are solved by ignoring the point which have the smallest impact according
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* to their jacobian column norm. Only the rank of the matrix and some loop bounds
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* are changed to implement this.</p>
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*
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* <p>The resolution engine is a simple translation of the MINPACK <a
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* href="http://www.netlib.org/minpack/lmder.f">lmder</a> routine with minor
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* changes. The changes include the over-determined resolution, the use of
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* inherited convergence checker and the Q.R. decomposition which has been
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* rewritten following the algorithm described in the
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* P. Lascaux and R. Theodor book <i>Analyse numérique matricielle
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* appliquée à l'art de l'ingénieur</i>, Masson 1986.</p>
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* <p>The authors of the original fortran version are:
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* <ul>
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* <li>Argonne National Laboratory. MINPACK project. March 1980</li>
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* <li>Burton S. Garbow</li>
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* <li>Kenneth E. Hillstrom</li>
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* <li>Jorge J. More</li>
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* </ul>
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* The redistribution policy for MINPACK is available <a
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* href="http://www.netlib.org/minpack/disclaimer">here</a>, for convenience, it
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* is reproduced below.</p>
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*
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* <table border="0" width="80%" cellpadding="10" align="center" bgcolor="#E0E0E0">
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* <tr><td>
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* Minpack Copyright Notice (1999) University of Chicago.
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* All rights reserved
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* </td></tr>
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* <tr><td>
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* Redistribution and use in source and binary forms, with or without
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* modification, are permitted provided that the following conditions
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* are met:
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* <ol>
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* <li>Redistributions of source code must retain the above copyright
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* notice, this list of conditions and the following disclaimer.</li>
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* <li>Redistributions in binary form must reproduce the above
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* copyright notice, this list of conditions and the following
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* disclaimer in the documentation and/or other materials provided
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* with the distribution.</li>
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* <li>The end-user documentation included with the redistribution, if any,
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* must include the following acknowledgment:
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* <code>This product includes software developed by the University of
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* Chicago, as Operator of Argonne National Laboratory.</code>
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* Alternately, this acknowledgment may appear in the software itself,
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* if and wherever such third-party acknowledgments normally appear.</li>
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* <li><strong>WARRANTY DISCLAIMER. THE SOFTWARE IS SUPPLIED "AS IS"
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* WITHOUT WARRANTY OF ANY KIND. THE COPYRIGHT HOLDER, THE
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* UNITED STATES, THE UNITED STATES DEPARTMENT OF ENERGY, AND
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* THEIR EMPLOYEES: (1) DISCLAIM ANY WARRANTIES, EXPRESS OR
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* IMPLIED, INCLUDING BUT NOT LIMITED TO ANY IMPLIED WARRANTIES
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* OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE
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* OR NON-INFRINGEMENT, (2) DO NOT ASSUME ANY LEGAL LIABILITY
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* OR RESPONSIBILITY FOR THE ACCURACY, COMPLETENESS, OR
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* USEFULNESS OF THE SOFTWARE, (3) DO NOT REPRESENT THAT USE OF
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* THE SOFTWARE WOULD NOT INFRINGE PRIVATELY OWNED RIGHTS, (4)
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* DO NOT WARRANT THAT THE SOFTWARE WILL FUNCTION
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* UNINTERRUPTED, THAT IT IS ERROR-FREE OR THAT ANY ERRORS WILL
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* BE CORRECTED.</strong></li>
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* <li><strong>LIMITATION OF LIABILITY. IN NO EVENT WILL THE COPYRIGHT
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* HOLDER, THE UNITED STATES, THE UNITED STATES DEPARTMENT OF
|
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* ENERGY, OR THEIR EMPLOYEES: BE LIABLE FOR ANY INDIRECT,
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* INCIDENTAL, CONSEQUENTIAL, SPECIAL OR PUNITIVE DAMAGES OF
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* ANY KIND OR NATURE, INCLUDING BUT NOT LIMITED TO LOSS OF
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* PROFITS OR LOSS OF DATA, FOR ANY REASON WHATSOEVER, WHETHER
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* SUCH LIABILITY IS ASSERTED ON THE BASIS OF CONTRACT, TORT
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* (INCLUDING NEGLIGENCE OR STRICT LIABILITY), OR OTHERWISE,
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* EVEN IF ANY OF SAID PARTIES HAS BEEN WARNED OF THE
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* POSSIBILITY OF SUCH LOSS OR DAMAGES.</strong></li>
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* <ol></td></tr>
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* </table>
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*
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* @version $Id: LevenbergMarquardtOptimizer.java 1416643 2012-12-03 19:37:14Z tn $
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*/
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function LMOptimizer() {
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/** Number of solved point. */
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private int solvedCols;
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/** Diagonal elements of the R matrix in the Q.R. decomposition. */
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private double[] diagR;
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/** Norms of the columns of the jacobian matrix. */
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private double[] jacNorm;
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/** Coefficients of the Householder transforms vectors. */
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private double[] beta;
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/** Columns permutation array. */
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private int[] permutation;
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/** Rank of the jacobian matrix. */
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private int rank;
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/** Levenberg-Marquardt parameter. */
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private double lmPar;
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/** Parameters evolution direction associated with lmPar. */
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private double[] lmDir;
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/** Positive input variable used in determining the initial step bound. */
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private final double initialStepBoundFactor;
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/** Desired relative error in the sum of squares. */
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private final double costRelativeTolerance;
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/** Desired relative error in the approximate solution parameters. */
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private final double parRelativeTolerance;
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/** Desired max cosine on the orthogonality between the function vector
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* and the columns of the jacobian. */
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private final double orthoTolerance;
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/** Threshold for QR ranking. */
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private final double qrRankingThreshold;
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/** Weighted residuals. */
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private double[] weightedResidual;
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/** Weighted Jacobian. */
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private double[][] weightedJacobian;
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/**
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* Build an optimizer for least squares problems with default values
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* for all the tuning parameters (see the {@link
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* #LevenbergMarquardtOptimizer(double,double,double,double,double)
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* other contructor}.
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* The default values for the algorithm settings are:
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* <ul>
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* <li>Initial step bound factor: 100</li>
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* <li>Cost relative tolerance: 1e-10</li>
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* <li>Parameters relative tolerance: 1e-10</li>
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* <li>Orthogonality tolerance: 1e-10</li>
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* <li>QR ranking threshold: {@link Precision#SAFE_MIN}</li>
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* </ul>
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*/
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public X init() {
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this.init1(100, 1e-10, 1e-10, 1e-10, Precision.SAFE_MIN);
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}
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/**
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* Build an optimizer for least squares problems with default values
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* for some of the tuning parameters (see the {@link
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* #LevenbergMarquardtOptimizer(double,double,double,double,double)
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* other contructor}.
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* The default values for the algorithm settings are:
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* <ul>
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* <li>Initial step bound factor}: 100</li>
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* <li>QR ranking threshold}: {@link Precision#SAFE_MIN}</li>
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* </ul>
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*
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* @param costRelativeTolerance Desired relative error in the sum of
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* squares.
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* @param parRelativeTolerance Desired relative error in the approximate
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* solution parameters.
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* @param orthoTolerance Desired max cosine on the orthogonality between
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* the function vector and the columns of the Jacobian.
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*/
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public X init0(double costRelativeTolerance,
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double parRelativeTolerance,
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double orthoTolerance) {
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this.init1(100,
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costRelativeTolerance, parRelativeTolerance, orthoTolerance,
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Precision.SAFE_MIN);
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}
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/**
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* The arguments control the behaviour of the default convergence checking
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* procedure.
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* Additional criteria can defined through the setting of a {@link
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* ConvergenceChecker}.
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*
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* @param initialStepBoundFactor Positive input variable used in
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* determining the initial step bound. This bound is set to the
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* product of initialStepBoundFactor and the euclidean norm of
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* {@code diag * x} if non-zero, or else to {@code initialStepBoundFactor}
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* itself. In most cases factor should lie in the interval
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* {@code (0.1, 100.0)}. {@code 100} is a generally recommended value.
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* @param costRelativeTolerance Desired relative error in the sum of
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* squares.
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* @param parRelativeTolerance Desired relative error in the approximate
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* solution parameters.
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* @param orthoTolerance Desired max cosine on the orthogonality between
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* the function vector and the columns of the Jacobian.
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* @param threshold Desired threshold for QR ranking. If the squared norm
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* of a column vector is smaller or equal to this threshold during QR
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* decomposition, it is considered to be a zero vector and hence the rank
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* of the matrix is reduced.
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*/
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public X init1(double initialStepBoundFactor,
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double costRelativeTolerance,
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double parRelativeTolerance,
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double orthoTolerance,
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double threshold) {
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this.initialStepBoundFactor = initialStepBoundFactor;
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this.costRelativeTolerance = costRelativeTolerance;
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this.parRelativeTolerance = parRelativeTolerance;
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this.orthoTolerance = orthoTolerance;
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this.qrRankingThreshold = threshold;
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}
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/** {@inheritDoc} */
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@Override
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protected PointVectorValuePair doOptimize() {
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final int nR = getTarget().length; // Number of observed data.
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final double[] currentPoint = getStartPoint();
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final int nC = currentPoint.length; // Number of parameters.
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// arrays shared with the other private methods
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solvedCols = FastMath.min(nR, nC);
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diagR = new double[nC];
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jacNorm = new double[nC];
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beta = new double[nC];
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permutation = new int[nC];
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lmDir = new double[nC];
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// local point
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double delta = 0;
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double xNorm = 0;
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double[] diag = new double[nC];
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double[] oldX = new double[nC];
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double[] oldRes = new double[nR];
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double[] oldObj = new double[nR];
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double[] qtf = new double[nR];
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double[] work1 = new double[nC];
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double[] work2 = new double[nC];
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double[] work3 = new double[nC];
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final RealMatrix weightMatrixSqrt = getWeightSquareRoot();
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// Evaluate the function at the starting point and calculate its norm.
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double[] currentObjective = computeObjectiveValue(currentPoint);
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double[] currentResiduals = computeResiduals(currentObjective);
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PointVectorValuePair current = new PointVectorValuePair(currentPoint, currentObjective);
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double currentCost = computeCost(currentResiduals);
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// Outer loop.
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lmPar = 0;
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boolean firstIteration = true;
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int iter = 0;
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final ConvergenceChecker<PointVectorValuePair> checker = getConvergenceChecker();
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while (true) {
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++iter;
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final PointVectorValuePair previous = current;
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// QR decomposition of the jacobian matrix
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qrDecomposition(computeWeightedJacobian(currentPoint));
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weightedResidual = weightMatrixSqrt.operate(currentResiduals);
|
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for (int i = 0; i < nR; i++) {
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qtf[i] = weightedResidual[i];
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}
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// compute Qt.res
|
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qTy(qtf);
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// now we don't need Q anymore,
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// so let jacobian contain the R matrix with its diagonal elements
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for (int k = 0; k < solvedCols; ++k) {
|
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int pk = permutation[k];
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weightedJacobian[k][pk] = diagR[pk];
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}
|
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|
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if (firstIteration) {
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// scale the point according to the norms of the columns
|
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// of the initial jacobian
|
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xNorm = 0;
|
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for (int k = 0; k < nC; ++k) {
|
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double dk = jacNorm[k];
|
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if (dk == 0) {
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dk = 1.0;
|
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}
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double xk = dk * currentPoint[k];
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xNorm += xk * xk;
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diag[k] = dk;
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}
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xNorm = FastMath.sqrt(xNorm);
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// initialize the step bound delta
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delta = (xNorm == 0) ? initialStepBoundFactor : (initialStepBoundFactor * xNorm);
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}
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// check orthogonality between function vector and jacobian columns
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double maxCosine = 0;
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if (currentCost != 0) {
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for (int j = 0; j < solvedCols; ++j) {
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int pj = permutation[j];
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double s = jacNorm[pj];
|
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if (s != 0) {
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double sum = 0;
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for (int i = 0; i <= j; ++i) {
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sum += weightedJacobian[i][pj] * qtf[i];
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}
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maxCosine = FastMath.max(maxCosine, FastMath.abs(sum) / (s * currentCost));
|
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}
|
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}
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}
|
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if (maxCosine <= orthoTolerance) {
|
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// Convergence has been reached.
|
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setCost(currentCost);
|
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return current;
|
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}
|
||||
|
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// rescale if necessary
|
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for (int j = 0; j < nC; ++j) {
|
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diag[j] = FastMath.max(diag[j], jacNorm[j]);
|
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}
|
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|
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// Inner loop.
|
||||
for (double ratio = 0; ratio < 1.0e-4;) {
|
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|
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// save the state
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
oldX[pj] = currentPoint[pj];
|
||||
}
|
||||
final double previousCost = currentCost;
|
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double[] tmpVec = weightedResidual;
|
||||
weightedResidual = oldRes;
|
||||
oldRes = tmpVec;
|
||||
tmpVec = currentObjective;
|
||||
currentObjective = oldObj;
|
||||
oldObj = tmpVec;
|
||||
|
||||
// determine the Levenberg-Marquardt parameter
|
||||
determineLMParameter(qtf, delta, diag, work1, work2, work3);
|
||||
|
||||
// compute the new point and the norm of the evolution direction
|
||||
double lmNorm = 0;
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
lmDir[pj] = -lmDir[pj];
|
||||
currentPoint[pj] = oldX[pj] + lmDir[pj];
|
||||
double s = diag[pj] * lmDir[pj];
|
||||
lmNorm += s * s;
|
||||
}
|
||||
lmNorm = FastMath.sqrt(lmNorm);
|
||||
// on the first iteration, adjust the initial step bound.
|
||||
if (firstIteration) {
|
||||
delta = FastMath.min(delta, lmNorm);
|
||||
}
|
||||
|
||||
// Evaluate the function at x + p and calculate its norm.
|
||||
currentObjective = computeObjectiveValue(currentPoint);
|
||||
currentResiduals = computeResiduals(currentObjective);
|
||||
current = new PointVectorValuePair(currentPoint, currentObjective);
|
||||
currentCost = computeCost(currentResiduals);
|
||||
|
||||
// compute the scaled actual reduction
|
||||
double actRed = -1.0;
|
||||
if (0.1 * currentCost < previousCost) {
|
||||
double r = currentCost / previousCost;
|
||||
actRed = 1.0 - r * r;
|
||||
}
|
||||
|
||||
// compute the scaled predicted reduction
|
||||
// and the scaled directional derivative
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
double dirJ = lmDir[pj];
|
||||
work1[j] = 0;
|
||||
for (int i = 0; i <= j; ++i) {
|
||||
work1[i] += weightedJacobian[i][pj] * dirJ;
|
||||
}
|
||||
}
|
||||
double coeff1 = 0;
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
coeff1 += work1[j] * work1[j];
|
||||
}
|
||||
double pc2 = previousCost * previousCost;
|
||||
coeff1 = coeff1 / pc2;
|
||||
double coeff2 = lmPar * lmNorm * lmNorm / pc2;
|
||||
double preRed = coeff1 + 2 * coeff2;
|
||||
double dirDer = -(coeff1 + coeff2);
|
||||
|
||||
// ratio of the actual to the predicted reduction
|
||||
ratio = (preRed == 0) ? 0 : (actRed / preRed);
|
||||
|
||||
// update the step bound
|
||||
if (ratio <= 0.25) {
|
||||
double tmp =
|
||||
(actRed < 0) ? (0.5 * dirDer / (dirDer + 0.5 * actRed)) : 0.5;
|
||||
if ((0.1 * currentCost >= previousCost) || (tmp < 0.1)) {
|
||||
tmp = 0.1;
|
||||
}
|
||||
delta = tmp * FastMath.min(delta, 10.0 * lmNorm);
|
||||
lmPar /= tmp;
|
||||
} else if ((lmPar == 0) || (ratio >= 0.75)) {
|
||||
delta = 2 * lmNorm;
|
||||
lmPar *= 0.5;
|
||||
}
|
||||
|
||||
// test for successful iteration.
|
||||
if (ratio >= 1.0e-4) {
|
||||
// successful iteration, update the norm
|
||||
firstIteration = false;
|
||||
xNorm = 0;
|
||||
for (int k = 0; k < nC; ++k) {
|
||||
double xK = diag[k] * currentPoint[k];
|
||||
xNorm += xK * xK;
|
||||
}
|
||||
xNorm = FastMath.sqrt(xNorm);
|
||||
|
||||
// tests for convergence.
|
||||
if (checker != null) {
|
||||
// we use the vectorial convergence checker
|
||||
if (checker.converged(iter, previous, current)) {
|
||||
setCost(currentCost);
|
||||
return current;
|
||||
}
|
||||
}
|
||||
} else {
|
||||
// failed iteration, reset the previous values
|
||||
currentCost = previousCost;
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
currentPoint[pj] = oldX[pj];
|
||||
}
|
||||
tmpVec = weightedResidual;
|
||||
weightedResidual = oldRes;
|
||||
oldRes = tmpVec;
|
||||
tmpVec = currentObjective;
|
||||
currentObjective = oldObj;
|
||||
oldObj = tmpVec;
|
||||
// Reset "current" to previous values.
|
||||
current = new PointVectorValuePair(currentPoint, currentObjective);
|
||||
}
|
||||
|
||||
// Default convergence criteria.
|
||||
if ((FastMath.abs(actRed) <= costRelativeTolerance &&
|
||||
preRed <= costRelativeTolerance &&
|
||||
ratio <= 2.0) ||
|
||||
delta <= parRelativeTolerance * xNorm) {
|
||||
setCost(currentCost);
|
||||
return current;
|
||||
}
|
||||
|
||||
// tests for termination and stringent tolerances
|
||||
// (2.2204e-16 is the machine epsilon for IEEE754)
|
||||
if ((FastMath.abs(actRed) <= 2.2204e-16) && (preRed <= 2.2204e-16) && (ratio <= 2.0)) {
|
||||
throw "TOO_SMALL_COST_RELATIVE_TOLERANCE: " + costRelativeTolerance;
|
||||
} else if (delta <= 2.2204e-16 * xNorm) {
|
||||
throw "TOO_SMALL_PARAMETERS_RELATIVE_TOLERANCE: " + parRelativeTolerance;
|
||||
} else if (maxCosine <= 2.2204e-16) {
|
||||
throw "TOO_SMALL_ORTHOGONALITY_TOLERANCE: " + orthoTolerance;
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
/**
|
||||
* Determine the Levenberg-Marquardt parameter.
|
||||
* <p>This implementation is a translation in Java of the MINPACK
|
||||
* <a href="http://www.netlib.org/minpack/lmpar.f">lmpar</a>
|
||||
* routine.</p>
|
||||
* <p>This method sets the lmPar and lmDir attributes.</p>
|
||||
* <p>The authors of the original fortran function are:</p>
|
||||
* <ul>
|
||||
* <li>Argonne National Laboratory. MINPACK project. March 1980</li>
|
||||
* <li>Burton S. Garbow</li>
|
||||
* <li>Kenneth E. Hillstrom</li>
|
||||
* <li>Jorge J. More</li>
|
||||
* </ul>
|
||||
* <p>Luc Maisonobe did the Java translation.</p>
|
||||
*
|
||||
* @param qy array containing qTy
|
||||
* @param delta upper bound on the euclidean norm of diagR * lmDir
|
||||
* @param diag diagonal matrix
|
||||
* @param work1 work array
|
||||
* @param work2 work array
|
||||
* @param work3 work array
|
||||
*/
|
||||
private void determineLMParameter(double[] qy, double delta, double[] diag,
|
||||
double[] work1, double[] work2, double[] work3) {
|
||||
final int nC = weightedJacobian[0].length;
|
||||
|
||||
// compute and store in x the gauss-newton direction, if the
|
||||
// jacobian is rank-deficient, obtain a least squares solution
|
||||
for (int j = 0; j < rank; ++j) {
|
||||
lmDir[permutation[j]] = qy[j];
|
||||
}
|
||||
for (int j = rank; j < nC; ++j) {
|
||||
lmDir[permutation[j]] = 0;
|
||||
}
|
||||
for (int k = rank - 1; k >= 0; --k) {
|
||||
int pk = permutation[k];
|
||||
double ypk = lmDir[pk] / diagR[pk];
|
||||
for (int i = 0; i < k; ++i) {
|
||||
lmDir[permutation[i]] -= ypk * weightedJacobian[i][pk];
|
||||
}
|
||||
lmDir[pk] = ypk;
|
||||
}
|
||||
|
||||
// evaluate the function at the origin, and test
|
||||
// for acceptance of the Gauss-Newton direction
|
||||
double dxNorm = 0;
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
double s = diag[pj] * lmDir[pj];
|
||||
work1[pj] = s;
|
||||
dxNorm += s * s;
|
||||
}
|
||||
dxNorm = FastMath.sqrt(dxNorm);
|
||||
double fp = dxNorm - delta;
|
||||
if (fp <= 0.1 * delta) {
|
||||
lmPar = 0;
|
||||
return;
|
||||
}
|
||||
|
||||
// if the jacobian is not rank deficient, the Newton step provides
|
||||
// a lower bound, parl, for the zero of the function,
|
||||
// otherwise set this bound to zero
|
||||
double sum2;
|
||||
double parl = 0;
|
||||
if (rank == solvedCols) {
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
work1[pj] *= diag[pj] / dxNorm;
|
||||
}
|
||||
sum2 = 0;
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
double sum = 0;
|
||||
for (int i = 0; i < j; ++i) {
|
||||
sum += weightedJacobian[i][pj] * work1[permutation[i]];
|
||||
}
|
||||
double s = (work1[pj] - sum) / diagR[pj];
|
||||
work1[pj] = s;
|
||||
sum2 += s * s;
|
||||
}
|
||||
parl = fp / (delta * sum2);
|
||||
}
|
||||
|
||||
// calculate an upper bound, paru, for the zero of the function
|
||||
sum2 = 0;
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
double sum = 0;
|
||||
for (int i = 0; i <= j; ++i) {
|
||||
sum += weightedJacobian[i][pj] * qy[i];
|
||||
}
|
||||
sum /= diag[pj];
|
||||
sum2 += sum * sum;
|
||||
}
|
||||
double gNorm = FastMath.sqrt(sum2);
|
||||
double paru = gNorm / delta;
|
||||
if (paru == 0) {
|
||||
// 2.2251e-308 is the smallest positive real for IEE754
|
||||
paru = 2.2251e-308 / FastMath.min(delta, 0.1);
|
||||
}
|
||||
|
||||
// if the input par lies outside of the interval (parl,paru),
|
||||
// set par to the closer endpoint
|
||||
lmPar = FastMath.min(paru, FastMath.max(lmPar, parl));
|
||||
if (lmPar == 0) {
|
||||
lmPar = gNorm / dxNorm;
|
||||
}
|
||||
|
||||
for (int countdown = 10; countdown >= 0; --countdown) {
|
||||
|
||||
// evaluate the function at the current value of lmPar
|
||||
if (lmPar == 0) {
|
||||
lmPar = FastMath.max(2.2251e-308, 0.001 * paru);
|
||||
}
|
||||
double sPar = FastMath.sqrt(lmPar);
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
work1[pj] = sPar * diag[pj];
|
||||
}
|
||||
determineLMDirection(qy, work1, work2, work3);
|
||||
|
||||
dxNorm = 0;
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
double s = diag[pj] * lmDir[pj];
|
||||
work3[pj] = s;
|
||||
dxNorm += s * s;
|
||||
}
|
||||
dxNorm = FastMath.sqrt(dxNorm);
|
||||
double previousFP = fp;
|
||||
fp = dxNorm - delta;
|
||||
|
||||
// if the function is small enough, accept the current value
|
||||
// of lmPar, also test for the exceptional cases where parl is zero
|
||||
if ((FastMath.abs(fp) <= 0.1 * delta) ||
|
||||
((parl == 0) && (fp <= previousFP) && (previousFP < 0))) {
|
||||
return;
|
||||
}
|
||||
|
||||
// compute the Newton correction
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
work1[pj] = work3[pj] * diag[pj] / dxNorm;
|
||||
}
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
work1[pj] /= work2[j];
|
||||
double tmp = work1[pj];
|
||||
for (int i = j + 1; i < solvedCols; ++i) {
|
||||
work1[permutation[i]] -= weightedJacobian[i][pj] * tmp;
|
||||
}
|
||||
}
|
||||
sum2 = 0;
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
double s = work1[permutation[j]];
|
||||
sum2 += s * s;
|
||||
}
|
||||
double correction = fp / (delta * sum2);
|
||||
|
||||
// depending on the sign of the function, update parl or paru.
|
||||
if (fp > 0) {
|
||||
parl = FastMath.max(parl, lmPar);
|
||||
} else if (fp < 0) {
|
||||
paru = FastMath.min(paru, lmPar);
|
||||
}
|
||||
|
||||
// compute an improved estimate for lmPar
|
||||
lmPar = FastMath.max(parl, lmPar + correction);
|
||||
|
||||
}
|
||||
}
|
||||
|
||||
/**
|
||||
* Solve a*x = b and d*x = 0 in the least squares sense.
|
||||
* <p>This implementation is a translation in Java of the MINPACK
|
||||
* <a href="http://www.netlib.org/minpack/qrsolv.f">qrsolv</a>
|
||||
* routine.</p>
|
||||
* <p>This method sets the lmDir and lmDiag attributes.</p>
|
||||
* <p>The authors of the original fortran function are:</p>
|
||||
* <ul>
|
||||
* <li>Argonne National Laboratory. MINPACK project. March 1980</li>
|
||||
* <li>Burton S. Garbow</li>
|
||||
* <li>Kenneth E. Hillstrom</li>
|
||||
* <li>Jorge J. More</li>
|
||||
* </ul>
|
||||
* <p>Luc Maisonobe did the Java translation.</p>
|
||||
*
|
||||
* @param qy array containing qTy
|
||||
* @param diag diagonal matrix
|
||||
* @param lmDiag diagonal elements associated with lmDir
|
||||
* @param work work array
|
||||
*/
|
||||
private void determineLMDirection(double[] qy, double[] diag,
|
||||
double[] lmDiag, double[] work) {
|
||||
|
||||
// copy R and Qty to preserve input and initialize s
|
||||
// in particular, save the diagonal elements of R in lmDir
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
int pj = permutation[j];
|
||||
for (int i = j + 1; i < solvedCols; ++i) {
|
||||
weightedJacobian[i][pj] = weightedJacobian[j][permutation[i]];
|
||||
}
|
||||
lmDir[j] = diagR[pj];
|
||||
work[j] = qy[j];
|
||||
}
|
||||
|
||||
// eliminate the diagonal matrix d using a Givens rotation
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
|
||||
// prepare the row of d to be eliminated, locating the
|
||||
// diagonal element using p from the Q.R. factorization
|
||||
int pj = permutation[j];
|
||||
double dpj = diag[pj];
|
||||
if (dpj != 0) {
|
||||
Arrays.fill(lmDiag, j + 1, lmDiag.length, 0);
|
||||
}
|
||||
lmDiag[j] = dpj;
|
||||
|
||||
// the transformations to eliminate the row of d
|
||||
// modify only a single element of Qty
|
||||
// beyond the first n, which is initially zero.
|
||||
double qtbpj = 0;
|
||||
for (int k = j; k < solvedCols; ++k) {
|
||||
int pk = permutation[k];
|
||||
|
||||
// determine a Givens rotation which eliminates the
|
||||
// appropriate element in the current row of d
|
||||
if (lmDiag[k] != 0) {
|
||||
|
||||
final double sin;
|
||||
final double cos;
|
||||
double rkk = weightedJacobian[k][pk];
|
||||
if (FastMath.abs(rkk) < FastMath.abs(lmDiag[k])) {
|
||||
final double cotan = rkk / lmDiag[k];
|
||||
sin = 1.0 / FastMath.sqrt(1.0 + cotan * cotan);
|
||||
cos = sin * cotan;
|
||||
} else {
|
||||
final double tan = lmDiag[k] / rkk;
|
||||
cos = 1.0 / FastMath.sqrt(1.0 + tan * tan);
|
||||
sin = cos * tan;
|
||||
}
|
||||
|
||||
// compute the modified diagonal element of R and
|
||||
// the modified element of (Qty,0)
|
||||
weightedJacobian[k][pk] = cos * rkk + sin * lmDiag[k];
|
||||
final double temp = cos * work[k] + sin * qtbpj;
|
||||
qtbpj = -sin * work[k] + cos * qtbpj;
|
||||
work[k] = temp;
|
||||
|
||||
// accumulate the tranformation in the row of s
|
||||
for (int i = k + 1; i < solvedCols; ++i) {
|
||||
double rik = weightedJacobian[i][pk];
|
||||
final double temp2 = cos * rik + sin * lmDiag[i];
|
||||
lmDiag[i] = -sin * rik + cos * lmDiag[i];
|
||||
weightedJacobian[i][pk] = temp2;
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
// store the diagonal element of s and restore
|
||||
// the corresponding diagonal element of R
|
||||
lmDiag[j] = weightedJacobian[j][permutation[j]];
|
||||
weightedJacobian[j][permutation[j]] = lmDir[j];
|
||||
}
|
||||
|
||||
// solve the triangular system for z, if the system is
|
||||
// singular, then obtain a least squares solution
|
||||
int nSing = solvedCols;
|
||||
for (int j = 0; j < solvedCols; ++j) {
|
||||
if ((lmDiag[j] == 0) && (nSing == solvedCols)) {
|
||||
nSing = j;
|
||||
}
|
||||
if (nSing < solvedCols) {
|
||||
work[j] = 0;
|
||||
}
|
||||
}
|
||||
if (nSing > 0) {
|
||||
for (int j = nSing - 1; j >= 0; --j) {
|
||||
int pj = permutation[j];
|
||||
double sum = 0;
|
||||
for (int i = j + 1; i < nSing; ++i) {
|
||||
sum += weightedJacobian[i][pj] * work[i];
|
||||
}
|
||||
work[j] = (work[j] - sum) / lmDiag[j];
|
||||
}
|
||||
}
|
||||
|
||||
// permute the components of z back to components of lmDir
|
||||
for (int j = 0; j < lmDir.length; ++j) {
|
||||
lmDir[permutation[j]] = work[j];
|
||||
}
|
||||
}
|
||||
|
||||
/**
|
||||
* Decompose a matrix A as A.P = Q.R using Householder transforms.
|
||||
* <p>As suggested in the P. Lascaux and R. Theodor book
|
||||
* <i>Analyse numérique matricielle appliquée à
|
||||
* l'art de l'ingénieur</i> (Masson, 1986), instead of representing
|
||||
* the Householder transforms with u<sub>k</sub> unit vectors such that:
|
||||
* <pre>
|
||||
* H<sub>k</sub> = I - 2u<sub>k</sub>.u<sub>k</sub><sup>t</sup>
|
||||
* </pre>
|
||||
* we use <sub>k</sub> non-unit vectors such that:
|
||||
* <pre>
|
||||
* H<sub>k</sub> = I - beta<sub>k</sub>v<sub>k</sub>.v<sub>k</sub><sup>t</sup>
|
||||
* </pre>
|
||||
* where v<sub>k</sub> = a<sub>k</sub> - alpha<sub>k</sub> e<sub>k</sub>.
|
||||
* The beta<sub>k</sub> coefficients are provided upon exit as recomputing
|
||||
* them from the v<sub>k</sub> vectors would be costly.</p>
|
||||
* <p>This decomposition handles rank deficient cases since the tranformations
|
||||
* are performed in non-increasing columns norms order thanks to columns
|
||||
* pivoting. The diagonal elements of the R matrix are therefore also in
|
||||
* non-increasing absolute values order.</p>
|
||||
*
|
||||
* @param jacobian Weighted Jacobian matrix at the current point.
|
||||
* @exception ConvergenceException if the decomposition cannot be performed
|
||||
*/
|
||||
private void qrDecomposition(RealMatrix jacobian) {
|
||||
// Code in this class assumes that the weighted Jacobian is -(W^(1/2) J),
|
||||
// hence the multiplication by -1.
|
||||
weightedJacobian = jacobian.scalarMultiply(-1).getData();
|
||||
|
||||
final int nR = weightedJacobian.length;
|
||||
final int nC = weightedJacobian[0].length;
|
||||
|
||||
// initializations
|
||||
for (int k = 0; k < nC; ++k) {
|
||||
permutation[k] = k;
|
||||
double norm2 = 0;
|
||||
for (int i = 0; i < nR; ++i) {
|
||||
double akk = weightedJacobian[i][k];
|
||||
norm2 += akk * akk;
|
||||
}
|
||||
jacNorm[k] = FastMath.sqrt(norm2);
|
||||
}
|
||||
|
||||
// transform the matrix column after column
|
||||
for (int k = 0; k < nC; ++k) {
|
||||
|
||||
// select the column with the greatest norm on active components
|
||||
int nextColumn = -1;
|
||||
double ak2 = Double.NEGATIVE_INFINITY;
|
||||
for (int i = k; i < nC; ++i) {
|
||||
double norm2 = 0;
|
||||
for (int j = k; j < nR; ++j) {
|
||||
double aki = weightedJacobian[j][permutation[i]];
|
||||
norm2 += aki * aki;
|
||||
}
|
||||
if (!isFinite(norm2)) {
|
||||
throw "UNABLE_TO_PERFORM_QR_DECOMPOSITION_ON_JACOBIAN";
|
||||
}
|
||||
if (norm2 > ak2) {
|
||||
nextColumn = i;
|
||||
ak2 = norm2;
|
||||
}
|
||||
}
|
||||
if (ak2 <= qrRankingThreshold) {
|
||||
rank = k;
|
||||
return;
|
||||
}
|
||||
int pk = permutation[nextColumn];
|
||||
permutation[nextColumn] = permutation[k];
|
||||
permutation[k] = pk;
|
||||
|
||||
// choose alpha such that Hk.u = alpha ek
|
||||
double akk = weightedJacobian[k][pk];
|
||||
double alpha = (akk > 0) ? -FastMath.sqrt(ak2) : FastMath.sqrt(ak2);
|
||||
double betak = 1.0 / (ak2 - akk * alpha);
|
||||
beta[pk] = betak;
|
||||
|
||||
// transform the current column
|
||||
diagR[pk] = alpha;
|
||||
weightedJacobian[k][pk] -= alpha;
|
||||
|
||||
// transform the remaining columns
|
||||
for (int dk = nC - 1 - k; dk > 0; --dk) {
|
||||
double gamma = 0;
|
||||
for (int j = k; j < nR; ++j) {
|
||||
gamma += weightedJacobian[j][pk] * weightedJacobian[j][permutation[k + dk]];
|
||||
}
|
||||
gamma *= betak;
|
||||
for (int j = k; j < nR; ++j) {
|
||||
weightedJacobian[j][permutation[k + dk]] -= gamma * weightedJacobian[j][pk];
|
||||
}
|
||||
}
|
||||
}
|
||||
rank = solvedCols;
|
||||
}
|
||||
|
||||
/**
|
||||
* Compute the product Qt.y for some Q.R. decomposition.
|
||||
*
|
||||
* @param y vector to multiply (will be overwritten with the result)
|
||||
*/
|
||||
private void qTy(double[] y) {
|
||||
final int nR = weightedJacobian.length;
|
||||
final int nC = weightedJacobian[0].length;
|
||||
|
||||
for (int k = 0; k < nC; ++k) {
|
||||
int pk = permutation[k];
|
||||
double gamma = 0;
|
||||
for (int i = k; i < nR; ++i) {
|
||||
gamma += weightedJacobian[i][pk] * y[i];
|
||||
}
|
||||
gamma *= beta[pk];
|
||||
for (int i = k; i < nR; ++i) {
|
||||
y[i] -= gamma * weightedJacobian[i][pk];
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
0
web/app/optim/lm.js
Normal file
0
web/app/optim/lm.js
Normal file
Loading…
Reference in a new issue