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Posted to commits@mahout.apache.org by ra...@apache.org on 2018/09/08 23:35:15 UTC
[11/15] mahout git commit: NO-JIRA Trevors updates
http://git-wip-us.apache.org/repos/asf/mahout/blob/545648f6/core/src/main/java/org/apache/mahout/math/SingularValueDecomposition.java
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diff --git a/core/src/main/java/org/apache/mahout/math/SingularValueDecomposition.java b/core/src/main/java/org/apache/mahout/math/SingularValueDecomposition.java
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+/*
+ * Licensed to the Apache Software Foundation (ASF) under one or more
+ * contributor license agreements. See the NOTICE file distributed with
+ * this work for additional information regarding copyright ownership.
+ * The ASF licenses this file to You 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.
+ *
+ * Copyright 1999 CERN - European Organization for Nuclear Research.
+ * Permission to use, copy, modify, distribute and sell this software and its documentation for any purpose
+ * is hereby granted without fee, provided that the above copyright notice appear in all copies and
+ * that both that copyright notice and this permission notice appear in supporting documentation.
+ * CERN makes no representations about the suitability of this software for any purpose.
+ * It is provided "as is" without expressed or implied warranty.
+ */
+package org.apache.mahout.math;
+
+public class SingularValueDecomposition implements java.io.Serializable {
+
+ /** Arrays for internal storage of U and V. */
+ private final double[][] u;
+ private final double[][] v;
+
+ /** Array for internal storage of singular values. */
+ private final double[] s;
+
+ /** Row and column dimensions. */
+ private final int m;
+ private final int n;
+
+ /**To handle the case where numRows() < numCols() and to use the fact that SVD(A')=VSU'=> SVD(A')'=SVD(A)**/
+ private boolean transpositionNeeded;
+
+ /**
+ * Constructs and returns a new singular value decomposition object; The
+ * decomposed matrices can be retrieved via instance methods of the returned
+ * decomposition object.
+ *
+ * @param arg
+ * A rectangular matrix.
+ */
+ public SingularValueDecomposition(Matrix arg) {
+ if (arg.numRows() < arg.numCols()) {
+ transpositionNeeded = true;
+ }
+
+ // Derived from LINPACK code.
+ // Initialize.
+ double[][] a;
+ if (transpositionNeeded) {
+ //use the transpose Matrix
+ m = arg.numCols();
+ n = arg.numRows();
+ a = new double[m][n];
+ for (int i = 0; i < m; i++) {
+ for (int j = 0; j < n; j++) {
+ a[i][j] = arg.get(j, i);
+ }
+ }
+ } else {
+ m = arg.numRows();
+ n = arg.numCols();
+ a = new double[m][n];
+ for (int i = 0; i < m; i++) {
+ for (int j = 0; j < n; j++) {
+ a[i][j] = arg.get(i, j);
+ }
+ }
+ }
+
+
+ int nu = Math.min(m, n);
+ s = new double[Math.min(m + 1, n)];
+ u = new double[m][nu];
+ v = new double[n][n];
+ double[] e = new double[n];
+ double[] work = new double[m];
+ boolean wantu = true;
+ boolean wantv = true;
+
+ // Reduce A to bidiagonal form, storing the diagonal elements
+ // in s and the super-diagonal elements in e.
+
+ int nct = Math.min(m - 1, n);
+ int nrt = Math.max(0, Math.min(n - 2, m));
+ for (int k = 0; k < Math.max(nct, nrt); k++) {
+ if (k < nct) {
+
+ // Compute the transformation for the k-th column and
+ // place the k-th diagonal in s[k].
+ // Compute 2-norm of k-th column without under/overflow.
+ s[k] = 0;
+ for (int i = k; i < m; i++) {
+ s[k] = Algebra.hypot(s[k], a[i][k]);
+ }
+ if (s[k] != 0.0) {
+ if (a[k][k] < 0.0) {
+ s[k] = -s[k];
+ }
+ for (int i = k; i < m; i++) {
+ a[i][k] /= s[k];
+ }
+ a[k][k] += 1.0;
+ }
+ s[k] = -s[k];
+ }
+ for (int j = k + 1; j < n; j++) {
+ if (k < nct && s[k] != 0.0) {
+
+ // Apply the transformation.
+
+ double t = 0;
+ for (int i = k; i < m; i++) {
+ t += a[i][k] * a[i][j];
+ }
+ t = -t / a[k][k];
+ for (int i = k; i < m; i++) {
+ a[i][j] += t * a[i][k];
+ }
+ }
+
+ // Place the k-th row of A into e for the
+ // subsequent calculation of the row transformation.
+
+ e[j] = a[k][j];
+ }
+ if (wantu && k < nct) {
+
+ // Place the transformation in U for subsequent back
+ // multiplication.
+
+ for (int i = k; i < m; i++) {
+ u[i][k] = a[i][k];
+ }
+ }
+ if (k < nrt) {
+
+ // Compute the k-th row transformation and place the
+ // k-th super-diagonal in e[k].
+ // Compute 2-norm without under/overflow.
+ e[k] = 0;
+ for (int i = k + 1; i < n; i++) {
+ e[k] = Algebra.hypot(e[k], e[i]);
+ }
+ if (e[k] != 0.0) {
+ if (e[k + 1] < 0.0) {
+ e[k] = -e[k];
+ }
+ for (int i = k + 1; i < n; i++) {
+ e[i] /= e[k];
+ }
+ e[k + 1] += 1.0;
+ }
+ e[k] = -e[k];
+ if (k + 1 < m && e[k] != 0.0) {
+
+ // Apply the transformation.
+
+ for (int i = k + 1; i < m; i++) {
+ work[i] = 0.0;
+ }
+ for (int j = k + 1; j < n; j++) {
+ for (int i = k + 1; i < m; i++) {
+ work[i] += e[j] * a[i][j];
+ }
+ }
+ for (int j = k + 1; j < n; j++) {
+ double t = -e[j] / e[k + 1];
+ for (int i = k + 1; i < m; i++) {
+ a[i][j] += t * work[i];
+ }
+ }
+ }
+ if (wantv) {
+
+ // Place the transformation in V for subsequent
+ // back multiplication.
+
+ for (int i = k + 1; i < n; i++) {
+ v[i][k] = e[i];
+ }
+ }
+ }
+ }
+
+ // Set up the final bidiagonal matrix or order p.
+
+ int p = Math.min(n, m + 1);
+ if (nct < n) {
+ s[nct] = a[nct][nct];
+ }
+ if (m < p) {
+ s[p - 1] = 0.0;
+ }
+ if (nrt + 1 < p) {
+ e[nrt] = a[nrt][p - 1];
+ }
+ e[p - 1] = 0.0;
+
+ // If required, generate U.
+
+ if (wantu) {
+ for (int j = nct; j < nu; j++) {
+ for (int i = 0; i < m; i++) {
+ u[i][j] = 0.0;
+ }
+ u[j][j] = 1.0;
+ }
+ for (int k = nct - 1; k >= 0; k--) {
+ if (s[k] != 0.0) {
+ for (int j = k + 1; j < nu; j++) {
+ double t = 0;
+ for (int i = k; i < m; i++) {
+ t += u[i][k] * u[i][j];
+ }
+ t = -t / u[k][k];
+ for (int i = k; i < m; i++) {
+ u[i][j] += t * u[i][k];
+ }
+ }
+ for (int i = k; i < m; i++) {
+ u[i][k] = -u[i][k];
+ }
+ u[k][k] = 1.0 + u[k][k];
+ for (int i = 0; i < k - 1; i++) {
+ u[i][k] = 0.0;
+ }
+ } else {
+ for (int i = 0; i < m; i++) {
+ u[i][k] = 0.0;
+ }
+ u[k][k] = 1.0;
+ }
+ }
+ }
+
+ // If required, generate V.
+
+ if (wantv) {
+ for (int k = n - 1; k >= 0; k--) {
+ if (k < nrt && e[k] != 0.0) {
+ for (int j = k + 1; j < nu; j++) {
+ double t = 0;
+ for (int i = k + 1; i < n; i++) {
+ t += v[i][k] * v[i][j];
+ }
+ t = -t / v[k + 1][k];
+ for (int i = k + 1; i < n; i++) {
+ v[i][j] += t * v[i][k];
+ }
+ }
+ }
+ for (int i = 0; i < n; i++) {
+ v[i][k] = 0.0;
+ }
+ v[k][k] = 1.0;
+ }
+ }
+
+ // Main iteration loop for the singular values.
+
+ int pp = p - 1;
+ int iter = 0;
+ double eps = Math.pow(2.0, -52.0);
+ double tiny = Math.pow(2.0,-966.0);
+ while (p > 0) {
+ int k;
+
+ // Here is where a test for too many iterations would go.
+
+ // This section of the program inspects for
+ // negligible elements in the s and e arrays. On
+ // completion the variables kase and k are set as follows.
+
+ // kase = 1 if s(p) and e[k-1] are negligible and k<p
+ // kase = 2 if s(k) is negligible and k<p
+ // kase = 3 if e[k-1] is negligible, k<p, and
+ // s(k), ..., s(p) are not negligible (qr step).
+ // kase = 4 if e(p-1) is negligible (convergence).
+
+ for (k = p - 2; k >= -1; k--) {
+ if (k == -1) {
+ break;
+ }
+ if (Math.abs(e[k]) <= tiny +eps * (Math.abs(s[k]) + Math.abs(s[k + 1]))) {
+ e[k] = 0.0;
+ break;
+ }
+ }
+ int kase;
+ if (k == p - 2) {
+ kase = 4;
+ } else {
+ int ks;
+ for (ks = p - 1; ks >= k; ks--) {
+ if (ks == k) {
+ break;
+ }
+ double t =
+ (ks != p ? Math.abs(e[ks]) : 0.) +
+ (ks != k + 1 ? Math.abs(e[ks-1]) : 0.);
+ if (Math.abs(s[ks]) <= tiny + eps * t) {
+ s[ks] = 0.0;
+ break;
+ }
+ }
+ if (ks == k) {
+ kase = 3;
+ } else if (ks == p - 1) {
+ kase = 1;
+ } else {
+ kase = 2;
+ k = ks;
+ }
+ }
+ k++;
+
+ // Perform the task indicated by kase.
+
+ switch (kase) {
+
+ // Deflate negligible s(p).
+
+ case 1: {
+ double f = e[p - 2];
+ e[p - 2] = 0.0;
+ for (int j = p - 2; j >= k; j--) {
+ double t = Algebra.hypot(s[j], f);
+ double cs = s[j] / t;
+ double sn = f / t;
+ s[j] = t;
+ if (j != k) {
+ f = -sn * e[j - 1];
+ e[j - 1] = cs * e[j - 1];
+ }
+ if (wantv) {
+ for (int i = 0; i < n; i++) {
+ t = cs * v[i][j] + sn * v[i][p - 1];
+ v[i][p - 1] = -sn * v[i][j] + cs * v[i][p - 1];
+ v[i][j] = t;
+ }
+ }
+ }
+ }
+ break;
+
+ // Split at negligible s(k).
+
+ case 2: {
+ double f = e[k - 1];
+ e[k - 1] = 0.0;
+ for (int j = k; j < p; j++) {
+ double t = Algebra.hypot(s[j], f);
+ double cs = s[j] / t;
+ double sn = f / t;
+ s[j] = t;
+ f = -sn * e[j];
+ e[j] = cs * e[j];
+ if (wantu) {
+ for (int i = 0; i < m; i++) {
+ t = cs * u[i][j] + sn * u[i][k - 1];
+ u[i][k - 1] = -sn * u[i][j] + cs * u[i][k - 1];
+ u[i][j] = t;
+ }
+ }
+ }
+ }
+ break;
+
+ // Perform one qr step.
+
+ case 3: {
+
+ // Calculate the shift.
+
+ double scale = Math.max(Math.max(Math.max(Math.max(
+ Math.abs(s[p - 1]), Math.abs(s[p - 2])), Math.abs(e[p - 2])),
+ Math.abs(s[k])), Math.abs(e[k]));
+ double sp = s[p - 1] / scale;
+ double spm1 = s[p - 2] / scale;
+ double epm1 = e[p - 2] / scale;
+ double sk = s[k] / scale;
+ double ek = e[k] / scale;
+ double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;
+ double c = sp * epm1 * sp * epm1;
+ double shift = 0.0;
+ if (b != 0.0 || c != 0.0) {
+ shift = Math.sqrt(b * b + c);
+ if (b < 0.0) {
+ shift = -shift;
+ }
+ shift = c / (b + shift);
+ }
+ double f = (sk + sp) * (sk - sp) + shift;
+ double g = sk * ek;
+
+ // Chase zeros.
+
+ for (int j = k; j < p - 1; j++) {
+ double t = Algebra.hypot(f, g);
+ double cs = f / t;
+ double sn = g / t;
+ if (j != k) {
+ e[j - 1] = t;
+ }
+ f = cs * s[j] + sn * e[j];
+ e[j] = cs * e[j] - sn * s[j];
+ g = sn * s[j + 1];
+ s[j + 1] = cs * s[j + 1];
+ if (wantv) {
+ for (int i = 0; i < n; i++) {
+ t = cs * v[i][j] + sn * v[i][j + 1];
+ v[i][j + 1] = -sn * v[i][j] + cs * v[i][j + 1];
+ v[i][j] = t;
+ }
+ }
+ t = Algebra.hypot(f, g);
+ cs = f / t;
+ sn = g / t;
+ s[j] = t;
+ f = cs * e[j] + sn * s[j + 1];
+ s[j + 1] = -sn * e[j] + cs * s[j + 1];
+ g = sn * e[j + 1];
+ e[j + 1] = cs * e[j + 1];
+ if (wantu && j < m - 1) {
+ for (int i = 0; i < m; i++) {
+ t = cs * u[i][j] + sn * u[i][j + 1];
+ u[i][j + 1] = -sn * u[i][j] + cs * u[i][j + 1];
+ u[i][j] = t;
+ }
+ }
+ }
+ e[p - 2] = f;
+ iter = iter + 1;
+ }
+ break;
+
+ // Convergence.
+
+ case 4: {
+
+ // Make the singular values positive.
+
+ if (s[k] <= 0.0) {
+ s[k] = s[k] < 0.0 ? -s[k] : 0.0;
+ if (wantv) {
+ for (int i = 0; i <= pp; i++) {
+ v[i][k] = -v[i][k];
+ }
+ }
+ }
+
+ // Order the singular values.
+
+ while (k < pp) {
+ if (s[k] >= s[k + 1]) {
+ break;
+ }
+ double t = s[k];
+ s[k] = s[k + 1];
+ s[k + 1] = t;
+ if (wantv && k < n - 1) {
+ for (int i = 0; i < n; i++) {
+ t = v[i][k + 1];
+ v[i][k + 1] = v[i][k];
+ v[i][k] = t;
+ }
+ }
+ if (wantu && k < m - 1) {
+ for (int i = 0; i < m; i++) {
+ t = u[i][k + 1];
+ u[i][k + 1] = u[i][k];
+ u[i][k] = t;
+ }
+ }
+ k++;
+ }
+ iter = 0;
+ p--;
+ }
+ break;
+ default:
+ throw new IllegalStateException();
+ }
+ }
+ }
+
+ /**
+ * Returns the two norm condition number, which is <tt>max(S) / min(S)</tt>.
+ */
+ public double cond() {
+ return s[0] / s[Math.min(m, n) - 1];
+ }
+
+ /**
+ * @return the diagonal matrix of singular values.
+ */
+ public Matrix getS() {
+ double[][] s = new double[n][n];
+ for (int i = 0; i < n; i++) {
+ for (int j = 0; j < n; j++) {
+ s[i][j] = 0.0;
+ }
+ s[i][i] = this.s[i];
+ }
+
+ return new DenseMatrix(s);
+ }
+
+ /**
+ * Returns the diagonal of <tt>S</tt>, which is a one-dimensional array of
+ * singular values
+ *
+ * @return diagonal of <tt>S</tt>.
+ */
+ public double[] getSingularValues() {
+ return s;
+ }
+
+ /**
+ * Returns the left singular vectors <tt>U</tt>.
+ *
+ * @return <tt>U</tt>
+ */
+ public Matrix getU() {
+ if (transpositionNeeded) { //case numRows() < numCols()
+ return new DenseMatrix(v);
+ } else {
+ int numCols = Math.min(m + 1, n);
+ Matrix r = new DenseMatrix(m, numCols);
+ for (int i = 0; i < m; i++) {
+ for (int j = 0; j < numCols; j++) {
+ r.set(i, j, u[i][j]);
+ }
+ }
+
+ return r;
+ }
+ }
+
+ /**
+ * Returns the right singular vectors <tt>V</tt>.
+ *
+ * @return <tt>V</tt>
+ */
+ public Matrix getV() {
+ if (transpositionNeeded) { //case numRows() < numCols()
+ int numCols = Math.min(m + 1, n);
+ Matrix r = new DenseMatrix(m, numCols);
+ for (int i = 0; i < m; i++) {
+ for (int j = 0; j < numCols; j++) {
+ r.set(i, j, u[i][j]);
+ }
+ }
+
+ return r;
+ } else {
+ return new DenseMatrix(v);
+ }
+ }
+
+ /** Returns the two norm, which is <tt>max(S)</tt>. */
+ public double norm2() {
+ return s[0];
+ }
+
+ /**
+ * Returns the effective numerical matrix rank, which is the number of
+ * nonnegligible singular values.
+ */
+ public int rank() {
+ double eps = Math.pow(2.0, -52.0);
+ double tol = Math.max(m, n) * s[0] * eps;
+ int r = 0;
+ for (double value : s) {
+ if (value > tol) {
+ r++;
+ }
+ }
+ return r;
+ }
+
+ /**
+ * @param minSingularValue
+ * minSingularValue - value below which singular values are ignored (a 0 or negative
+ * value implies all singular value will be used)
+ * @return Returns the n × n covariance matrix.
+ * The covariance matrix is V × J × Vt where J is the diagonal matrix of the inverse
+ * of the squares of the singular values.
+ */
+ Matrix getCovariance(double minSingularValue) {
+ Matrix j = new DenseMatrix(s.length,s.length);
+ Matrix vMat = new DenseMatrix(this.v);
+ for (int i = 0; i < s.length; i++) {
+ j.set(i, i, s[i] >= minSingularValue ? 1 / (s[i] * s[i]) : 0.0);
+ }
+ return vMat.times(j).times(vMat.transpose());
+ }
+
+ /**
+ * Returns a String with (propertyName, propertyValue) pairs. Useful for
+ * debugging or to quickly get the rough picture. For example,
+ *
+ * <pre>
+ * rank : 3
+ * trace : 0
+ * </pre>
+ */
+ @Override
+ public String toString() {
+ StringBuilder buf = new StringBuilder();
+ buf.append("---------------------------------------------------------------------\n");
+ buf.append("SingularValueDecomposition(A) --> cond(A), rank(A), norm2(A), U, S, V\n");
+ buf.append("---------------------------------------------------------------------\n");
+
+ buf.append("cond = ");
+ String unknown = "Illegal operation or error: ";
+ try {
+ buf.append(String.valueOf(this.cond()));
+ } catch (IllegalArgumentException exc) {
+ buf.append(unknown).append(exc.getMessage());
+ }
+
+ buf.append("\nrank = ");
+ try {
+ buf.append(String.valueOf(this.rank()));
+ } catch (IllegalArgumentException exc) {
+ buf.append(unknown).append(exc.getMessage());
+ }
+
+ buf.append("\nnorm2 = ");
+ try {
+ buf.append(String.valueOf(this.norm2()));
+ } catch (IllegalArgumentException exc) {
+ buf.append(unknown).append(exc.getMessage());
+ }
+
+ buf.append("\n\nU = ");
+ try {
+ buf.append(String.valueOf(this.getU()));
+ } catch (IllegalArgumentException exc) {
+ buf.append(unknown).append(exc.getMessage());
+ }
+
+ buf.append("\n\nS = ");
+ try {
+ buf.append(String.valueOf(this.getS()));
+ } catch (IllegalArgumentException exc) {
+ buf.append(unknown).append(exc.getMessage());
+ }
+
+ buf.append("\n\nV = ");
+ try {
+ buf.append(String.valueOf(this.getV()));
+ } catch (IllegalArgumentException exc) {
+ buf.append(unknown).append(exc.getMessage());
+ }
+
+ return buf.toString();
+ }
+}