Singular Value Decomposition: Different results wi

Question

I want to perform Singular Value Decomposition on a large (sparse) matrix. In order to choose the best(most accurate) library, I tried replicating the SVD example provided here using different Java and Python libraries. Strangely I am getting different results with each library.

Here's the original example matrix and it's decomposed (U S and VT) matrices:

A =2.0  0.0 8.0 6.0 0.0
   1.0 6.0 0.0 1.0 7.0
   5.0 0.0 7.0 4.0 0.0
   7.0 0.0 8.0 5.0 0.0 
   0.0 10.0 0.0 0.0 7.0

U =-0.54 0.07 0.82 -0.11 0.12
   -0.10 -0.59 -0.11 -0.79 -0.06
   -0.53 0.06 -0.21 0.12 -0.81
   -0.65 0.07 -0.51 0.06 0.56
   -0.06 -0.80 0.09 0.59 0.04

VT =-0.46 0.02 -0.87 -0.00 0.17
    -0.07 -0.76 0.06 0.60 0.23
    -0.74 0.10 0.28 0.22 -0.56
    -0.48 0.03 0.40 -0.33 0.70
    -0.07 -0.64 -0.04 -0.69 -0.32

S (with the top three singular values) = 
   17.92 0 0
   0 15.17 0
   0 0 3.56

I tried using the following Java and Python libraries: Java: PColt, Jama Python: NumPy

Here are the results from each one of them:

Jama:
U = 0.5423  -0.065  -0.8216 0.1057  -0.1245 
    0.1018  0.5935  0.1126  0.7881  0.0603  
    0.525   -0.0594 0.213   -0.1157 0.8137  
    0.6449  -0.0704 0.5087  -0.0599 -0.5628 
    0.0645  0.7969  -0.09   -0.5922 -0.0441 

VT =0.4646  -0.0215 0.8685  8.0E-4  -0.1713 
    0.0701  0.76    -0.0631 -0.6013 -0.2278 
    0.7351  -0.0988 -0.284  -0.2235 0.565   
    0.4844  -0.0254 -0.3989 0.3327  -0.7035 
    0.065   0.6415  0.0443  0.6912  0.3233  

S = 17.9184 0.0 0.0 0.0 0.0 
    0.0 15.1714 0.0 0.0 0.0 
    0.0 0.0 3.564   0.0 0.0 
    0.0 0.0 0.0 1.9842  0.0 
    0.0 0.0 0.0 0.0 0.3496  

PColt:
U = -0.542255   0.0649957  0.821617  0.105747  -0.124490 
    -0.101812  -0.593461 -0.112552  0.788123   0.0602700
    -0.524953   0.0593817 -0.212969 -0.115742   0.813724 
    -0.644870   0.0704063 -0.508744 -0.0599027 -0.562829 
    -0.0644952 -0.796930  0.0900097 -0.592195  -0.0441263

VT =-0.464617   0.0215065 -0.868509   0.000799554 -0.171349
    -0.0700860 -0.759988  0.0630715 -0.601346   -0.227841
    -0.735094   0.0987971  0.284009  -0.223485    0.565040
    -0.484392   0.0254474  0.398866   0.332684   -0.703523
    -0.0649698 -0.641520 -0.0442743  0.691201    0.323284

S = 
(00)    17.91837085874625
(11)    15.17137188041607
(22)    3.5640020352605677
(33)    1.9842281528992616
(44)    0.3495556671751232


Numpy

U = -0.54225536  0.06499573  0.82161708  0.10574661 -0.12448979
    -0.10181247 -0.59346055 -0.11255162  0.78812338  0.06026999
    -0.52495325  0.05938171 -0.21296861 -0.11574223  0.81372354
    -0.64487038  0.07040626 -0.50874368 -0.05990271 -0.56282918
    -0.06449519 -0.79692967  0.09000966 -0.59219473 -0.04412631

VT =-4.64617e-01   2.15065e-02  -8.68508e-01    7.99553e-04  -1.71349e-01
    -7.00859e-02  -7.59987e-01   6.30714e-02   -6.01345e-01  -2.27841e-01
    -7.35093e-01   9.87971e-02   2.84008e-01   -2.23484e-01   5.65040e-01
    -4.84391e-01   2.54473e-02   3.98865e-01    3.32683e-01  -7.03523e-01
    -6.49698e-02  -6.41519e-01  -4.42743e-02    6.91201e-01   3.23283e-01

S = 17.91837086  15.17137188   3.56400204   1.98422815   0.34955567

As can be noticed the sign of each element in the Jama decomposed matrices (u & VT) is opposite to the ones in the original example. Interestingly, for PColt and Numpy only the signs of the elements in the last two columns are inverted. Is there any specific reason behind the inverted signs? Has someone faced similar discrepancies?

Here are the pieces of code which I used: Java

import java.text.DecimalFormat;
import cern.colt.matrix.tdouble.DoubleMatrix2D;
import cern.colt.matrix.tdouble.algo.DenseDoubleAlgebra;
import cern.colt.matrix.tdouble.algo.decomposition.DenseDoubleSingularValueDecomposition;
import cern.colt.matrix.tdouble.impl.DenseDoubleMatrix2D;
import Jama.Matrix;
import Jama.SingularValueDecomposition;
public class SVD_Test implements java.io.Serializable{

    public static void main(String[] args)
    {   

        double[][] data2 = new double[][]
                {{ 2.0, 0.0, 8.0, 6.0, 0.0},
                { 1.0, 6.0, 0.0, 1.0, 7.0},
                { 5.0, 0.0, 7.0, 4.0, 0.0},
                { 7.0, 0.0, 8.0, 5.0, 0.0},
                { 0.0, 10.0, 0.0, 0.0, 7.0}};

        DoubleMatrix2D pColt_matrix = new DenseDoubleMatrix2D(5,5);
        pColt_matrix.assign(data2);
        Matrix j = new Matrix(data2);

        SingularValueDecomposition svd_jama = j.svd();

        DenseDoubleSingularValueDecomposition svd_pColt = new DenseDoubleSingularValueDecomposition(pColt_matrix, true, true);
        System.out.println("U:");
        System.out.println("pColt:");
        System.out.println(svd_pColt.getU());
        printJamaMatrix(svd_jama.getU());
        System.out.println("S:");
        System.out.println("pColt:");
        System.out.println(svd_pColt.getS());
        printJamaMatrix(svd_jama.getS());
        System.out.println("V:");
        System.out.println("pColt:");
        System.out.println(svd_pColt.getV());
        printJamaMatrix(svd_jama.getV());

    }

    public static void printJamaMatrix(Matrix inp){
        System.out.println("Jama: ");
        System.out.println(String.valueOf(inp.getRowDimension())+" X "+String.valueOf(inp.getColumnDimension()));
        DecimalFormat twoDForm = new DecimalFormat("#.####");
        StringBuffer sb = new StringBuffer();
        for (int r = 0; r < inp.getRowDimension(); r++) {
            for (int c = 0; c < inp.getColumnDimension(); c++)
                sb.append(Double.valueOf(twoDForm.format(inp.get(r, c)))).append("\t");
            sb.append("\n");
        }
        System.out.println(sb.toString());      
    }   
}

Python:

>>> import numpy
>>> numpy_matrix = numpy.array([[ 2.0, 0.0, 8.0, 6.0, 0.0], 
                [1.0, 6.0, 0.0, 1.0, 7.0], 
                [5.0, 0.0, 7.0, 4.0, 0.0], 
                [7.0, 0.0, 8.0, 5.0, 0.0], 
                [0.0, 10.0, 0.0, 0.0, 7.0]])
>>> u,s,v = numpy.linalg.svd(numpy_matrix, full_matrices=True)

Is there something wrong with the code? .

Answer 1

Nothing wrong: the s.v.d. is not unique up to a sign change of the columns of U and V. (i.e. if you change the sign of i-th column of U and the i-th column of V, you still have a valid s.v.d: A = U*S*V^T). Different implementations of the svd will give slightly different results: to check for a correct svd you have to compute norm(A-U*S*V^T) / norm(A) and verify that it is a small number.

Answer 2

There is nothing wrong. The SVD resolves the column space and the row space of the target matrix into orthonormal bases in such a fashion as to align these two spaces and account for the dilations along the eigenvectors. The alignment angles may be unique, a discrete set, or a continuum as below.

For example, given two angles t and p and the target matrix (see footnote)

A = ( (1, -1), (2, 2) )

The general decomposition is

U = ( (0, exp[ i p ]), (-exp[ i t ], 0) )

S = sqrt(2) ( (2,0), (0,1) )

V* = ( 1 / sqrt( 2 ) ) ( (exp[ i t ], exp[ i t ]), (exp[ i p ], -exp[ i p ]) )

To recover the target matrix use A = U S V*

A quick test of the quality of the answer is to verify the unit length of each column vector in both U and V.

Footnote: Matrices are in row major format. That is, the first row vector in the matrix A is (1, -1).

• Finally I have enough points to post an image file.