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* 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
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*
* 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,
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* See the License for the specific language governing permissions and
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package org.apache.mahout.math;
import org.apache.mahout.common.RandomUtils;
import org.junit.Test;
import java.util.Random;
//To launch this test only : mvn test -Dtest=org.apache.mahout.math.TestSingularValueDecomposition
public final class TestSingularValueDecomposition extends MahoutTestCase {
private final double[][] testSquare = {
{ 24.0 / 25.0, 43.0 / 25.0 },
{ 57.0 / 25.0, 24.0 / 25.0 }
};
private final double[][] testNonSquare = {
{ -540.0 / 625.0, 963.0 / 625.0, -216.0 / 625.0 },
{ -1730.0 / 625.0, -744.0 / 625.0, 1008.0 / 625.0 },
{ -720.0 / 625.0, 1284.0 / 625.0, -288.0 / 625.0 },
{ -360.0 / 625.0, 192.0 / 625.0, 1756.0 / 625.0 },
};
private static final double NORM_TOLERANCE = 10.0e-14;
@Test
public void testMoreRows() {
double[] singularValues = { 123.456, 2.3, 1.001, 0.999 };
int rows = singularValues.length + 2;
int columns = singularValues.length;
Random r = RandomUtils.getRandom();
SingularValueDecomposition svd =
new SingularValueDecomposition(createTestMatrix(r, rows, columns, singularValues));
double[] computedSV = svd.getSingularValues();
assertEquals(singularValues.length, computedSV.length);
for (int i = 0; i < singularValues.length; ++i) {
assertEquals(singularValues[i], computedSV[i], 1.0e-10);
}
}
@Test
public void testMoreColumns() {
double[] singularValues = { 123.456, 2.3, 1.001, 0.999 };
int rows = singularValues.length;
int columns = singularValues.length + 2;
Random r = RandomUtils.getRandom();
SingularValueDecomposition svd =
new SingularValueDecomposition(createTestMatrix(r, rows, columns, singularValues));
double[] computedSV = svd.getSingularValues();
assertEquals(singularValues.length, computedSV.length);
for (int i = 0; i < singularValues.length; ++i) {
assertEquals(singularValues[i], computedSV[i], 1.0e-10);
}
}
/** test dimensions */
@Test
public void testDimensions() {
Matrix matrix = new DenseMatrix(testSquare);
int m = matrix.numRows();
int n = matrix.numCols();
SingularValueDecomposition svd = new SingularValueDecomposition(matrix);
assertEquals(m, svd.getU().numRows());
assertEquals(m, svd.getU().numCols());
assertEquals(m, svd.getS().numCols());
assertEquals(n, svd.getS().numCols());
assertEquals(n, svd.getV().numRows());
assertEquals(n, svd.getV().numCols());
}
/** Test based on a dimension 4 Hadamard matrix. */
// getCovariance to be implemented
@Test
public void testHadamard() {
Matrix matrix = new DenseMatrix(new double[][] {
{15.0 / 2.0, 5.0 / 2.0, 9.0 / 2.0, 3.0 / 2.0 },
{ 5.0 / 2.0, 15.0 / 2.0, 3.0 / 2.0, 9.0 / 2.0 },
{ 9.0 / 2.0, 3.0 / 2.0, 15.0 / 2.0, 5.0 / 2.0 },
{ 3.0 / 2.0, 9.0 / 2.0, 5.0 / 2.0, 15.0 / 2.0 }
});
SingularValueDecomposition svd = new SingularValueDecomposition(matrix);
assertEquals(16.0, svd.getSingularValues()[0], 1.0e-14);
assertEquals( 8.0, svd.getSingularValues()[1], 1.0e-14);
assertEquals( 4.0, svd.getSingularValues()[2], 1.0e-14);
assertEquals( 2.0, svd.getSingularValues()[3], 1.0e-14);
Matrix fullCovariance = new DenseMatrix(new double[][] {
{ 85.0 / 1024, -51.0 / 1024, -75.0 / 1024, 45.0 / 1024 },
{ -51.0 / 1024, 85.0 / 1024, 45.0 / 1024, -75.0 / 1024 },
{ -75.0 / 1024, 45.0 / 1024, 85.0 / 1024, -51.0 / 1024 },
{ 45.0 / 1024, -75.0 / 1024, -51.0 / 1024, 85.0 / 1024 }
});
assertEquals(0.0,Algebra.getNorm(fullCovariance.minus(svd.getCovariance(0.0))),1.0e-14);
Matrix halfCovariance = new DenseMatrix(new double[][] {
{ 5.0 / 1024, -3.0 / 1024, 5.0 / 1024, -3.0 / 1024 },
{ -3.0 / 1024, 5.0 / 1024, -3.0 / 1024, 5.0 / 1024 },
{ 5.0 / 1024, -3.0 / 1024, 5.0 / 1024, -3.0 / 1024 },
{ -3.0 / 1024, 5.0 / 1024, -3.0 / 1024, 5.0 / 1024 }
});
assertEquals(0.0,Algebra.getNorm(halfCovariance.minus(svd.getCovariance(6.0))),1.0e-14);
}
/** test A = USVt */
@Test
public void testAEqualUSVt() {
checkAEqualUSVt(new DenseMatrix(testSquare));
checkAEqualUSVt(new DenseMatrix(testNonSquare));
checkAEqualUSVt(new DenseMatrix(testNonSquare).transpose());
}
public static void checkAEqualUSVt(Matrix matrix) {
SingularValueDecomposition svd = new SingularValueDecomposition(matrix);
Matrix u = svd.getU();
Matrix s = svd.getS();
Matrix v = svd.getV();
//pad with 0, to be able to check some properties if some singular values are equal to 0
if(s.numRows()<matrix.numRows()) {
Matrix sp = new DenseMatrix(s.numRows()+1,s.numCols());
Matrix up = new DenseMatrix(u.numRows(),u.numCols()+1);
for (int i = 0; i < u.numRows(); i++) {
for (int j = 0; j < u.numCols(); j++) {
up.set(i, j, u.get(i, j));
}
}
for (int i = 0; i < s.numRows(); i++) {
for (int j = 0; j < s.numCols(); j++) {
sp.set(i, j, s.get(i, j));
}
}
u = up;
s = sp;
}
double norm = Algebra.getNorm(u.times(s).times(v.transpose()).minus(matrix));
assertEquals(0, norm, NORM_TOLERANCE);
}
/** test that U is orthogonal */
@Test
public void testUOrthogonal() {
checkOrthogonal(new SingularValueDecomposition(new DenseMatrix(testSquare)).getU());
checkOrthogonal(new SingularValueDecomposition(new DenseMatrix(testNonSquare)).getU());
checkOrthogonal(new SingularValueDecomposition(new DenseMatrix(testNonSquare).transpose()).getU());
}
/** test that V is orthogonal */
@Test
public void testVOrthogonal() {
checkOrthogonal(new SingularValueDecomposition(new DenseMatrix(testSquare)).getV());
checkOrthogonal(new SingularValueDecomposition(new DenseMatrix(testNonSquare)).getV());
checkOrthogonal(new SingularValueDecomposition(new DenseMatrix(testNonSquare).transpose()).getV());
}
public static void checkOrthogonal(Matrix m) {
Matrix mTm = m.transpose().times(m);
Matrix id = new DenseMatrix(mTm.numRows(),mTm.numRows());
for (int i = 0; i < mTm.numRows(); i++) {
id.set(i, i, 1);
}
assertEquals(0, Algebra.getNorm(mTm.minus(id)), NORM_TOLERANCE);
}
/** test matrices values */
@Test
public void testMatricesValues1() {
SingularValueDecomposition svd =
new SingularValueDecomposition(new DenseMatrix(testSquare));
Matrix uRef = new DenseMatrix(new double[][] {
{ 3.0 / 5.0, 4.0 / 5.0 },
{ 4.0 / 5.0, -3.0 / 5.0 }
});
Matrix sRef = new DenseMatrix(new double[][] {
{ 3.0, 0.0 },
{ 0.0, 1.0 }
});
Matrix vRef = new DenseMatrix(new double[][] {
{ 4.0 / 5.0, -3.0 / 5.0 },
{ 3.0 / 5.0, 4.0 / 5.0 }
});
// check values against known references
Matrix u = svd.getU();
assertEquals(0, Algebra.getNorm(u.minus(uRef)), NORM_TOLERANCE);
Matrix s = svd.getS();
assertEquals(0, Algebra.getNorm(s.minus(sRef)), NORM_TOLERANCE);
Matrix v = svd.getV();
assertEquals(0, Algebra.getNorm(v.minus(vRef)), NORM_TOLERANCE);
}
/** test condition number */
@Test
public void testConditionNumber() {
SingularValueDecomposition svd =
new SingularValueDecomposition(new DenseMatrix(testSquare));
// replace 1.0e-15 with 1.5e-15
assertEquals(3.0, svd.cond(), 1.5e-15);
}
private static Matrix createTestMatrix(Random r, int rows, int columns, double[] singularValues) {
Matrix u = createOrthogonalMatrix(r, rows);
Matrix d = createDiagonalMatrix(singularValues, rows, columns);
Matrix v = createOrthogonalMatrix(r, columns);
return u.times(d).times(v);
}
public static Matrix createOrthogonalMatrix(Random r, int size) {
double[][] data = new double[size][size];
for (int i = 0; i < size; ++i) {
double[] dataI = data[i];
double norm2;
do {
// generate randomly row I
for (int j = 0; j < size; ++j) {
dataI[j] = 2 * r.nextDouble() - 1;
}
// project the row in the subspace orthogonal to previous rows
for (int k = 0; k < i; ++k) {
double[] dataK = data[k];
double dotProduct = 0;
for (int j = 0; j < size; ++j) {
dotProduct += dataI[j] * dataK[j];
}
for (int j = 0; j < size; ++j) {
dataI[j] -= dotProduct * dataK[j];
}
}
// normalize the row
norm2 = 0;
for (double dataIJ : dataI) {
norm2 += dataIJ * dataIJ;
}
double inv = 1.0 / Math.sqrt(norm2);
for (int j = 0; j < size; ++j) {
dataI[j] *= inv;
}
} while (norm2 * size < 0.01);
}
return new DenseMatrix(data);
}
public static Matrix createDiagonalMatrix(double[] diagonal, int rows, int columns) {
double[][] dData = new double[rows][columns];
for (int i = 0; i < Math.min(rows, columns); ++i) {
dData[i][i] = diagonal[i];
}
return new DenseMatrix(dData);
}
}