I couldn’t sleep one recent evening. So I figured I’d implement a matrix pseudo-inverse function using C#, via a recent implementation of singular value decomposition function using the Householder + QR technique. Bottom line: My implementation didn’t work as well as my older SVD using the Jacobi method.
In ordinary arithmetic, the inverse of 4 is 0.25 because 4 * 0.25 = 1. In matrix arithmetic, the inverse of a square n-by-n matrix A is an n-by-n matrix Ainv where A * Ainv = I. (Also Ainv * A = I). The * is matrix multiplication, the I is the n-by-n identity matrix which has 1s on the upper-left to lower-right diagonal, 0s elsewhere.
But for matrices that aren’t square, such as a 3-by-5 or a 4-by-2 matrix, there’s no inverse. But it’s sometimes useful to define a pseudo-inverse. There are several different types of pseudo-inverse, but the most common is called the Moore-Penrose pseudo-inverse.
For a matrix A, the MP pseudo-inverse is a matrix Apinv where A * Apinv * A = A. Computing a pseudo-inverse is tricky. One technique I sometimes prefer is the singular value decomposition (SVD) method. SVD breaks a matrix A down into three components: a matrix U, a matrix Vh, and a vector s. If you compute U * S * Vh you get the original matrix A. Here S is a square matrix with the elements of vector s on the diagonal, 0s elsewhere. The values in the s vector are called the singular values of the matrix.
Computing an SVD decomposition is very difficult. Based on my experience (but no hard data), the two most common techniques are the Jacobi method and the QR decomposition method. My recent SVD implementation uses the QR technique. (There are three common versions of QR — Householder, modified Gram-Schmidt, Givens. I used Householder which is theoretically the most stable of the three).
If you can compute the U, Vh, and S from a source matrix A, then the pseudo-inverse of A is V * Sinv * Uh. Here V is the inverse of Vh, which fortunately is just the transpose (rows and columns switched) of Vh. The Uh is the transpose of U, which again, fortunately is just the transpose of U. The Sinv is the S matrix but each value on the diagonal is 1/s instead of s.
To summarize, in order to find the MP pseudo-inverse of a matrix A, perform SVD decomposition to get U, Vh, and s. Compute V = transpose(Vh), Uh = transpose(U), Sinv (as described above), and then compute Apinv = V * Sinv * Uh.
This code shows how easy it is if you have an SVD decomposition function:
public static double[][] MatPinv(double[][] A)
{
double[][] U; double[] s; double[][] Vh;
MatDecompSVD(A, out U, out s, out Vh);
double[][] Uh = MatTranspose(U);
double[][] V = MatTranspose(Vh);
double[][] S = MatMake(s.Length, s.Length);
for (int i = 0; i < s.Length; ++i)
S[i][i] = 1.0 / (s[i] + 1.0e-12); // avoid div by 0
double[][] result = MatProduct(V, MatProduct(S, Uh));
return result;
}
Unfortunately, computing the SVD decomposition is very, very difficult. Luckily, as I mentioned above, I had previously implemented SVD using the Householder + QR technique and so all the hard work was done.
Here’s the output of my demo program:
Begin pseudo-inverse via SVD (QR) using C# Source (tall) matrix: 1.0 2.0 3.0 4.0 5.0 0.0 -3.0 5.0 -7.0 9.0 2.0 0.0 -2.0 0.0 -2.0 4.0 -1.0 5.0 6.0 1.0 3.0 6.0 8.0 2.0 2.0 5.0 -2.0 4.0 -4.0 3.0 Computing Moore-Penrose pseudo-inverse Done Pseudo-inverse: 0.091074 -0.056097 0.165008 -0.025042 -0.014424 0.144469 0.092124 -0.041739 0.075457 -0.137851 0.093893 0.005337 -0.161758 0.043008 -0.142053 0.104065 0.078951 -0.041449 0.066366 -0.025042 -0.014465 0.075336 -0.037227 -0.045420 0.180763 0.037573 0.062447 -0.054091 -0.047030 0.010359 ============================ Source (wide) matrix: 3.0 1.0 1.0 0.0 5.0 -1.0 3.0 1.0 -2.0 4.0 0.0 2.0 2.0 1.0 -3.0 Computing Moore-Penrose pseudo-inverse Done Pseudo-inverse: 0.190177 -0.148152 0.066835 0.001620 0.125468 0.153924 0.064810 0.018734 0.156962 0.084962 -0.108253 0.071392 0.072608 0.060051 -0.102278 End demo
I validated the results using the NumPy library np.linalg.pinv() function. Note: There are subtle differences between computing a true mathematically valid Moore-Penrose pseudo-inverse and computing a "sort-of Moore-Penrose pseudo-inverse that's good enough for machine learning scenarios". I ran some tests using this pseudo-inverse code and they mostly passed, but there are still bugs, so this version is not good enough for anything other than experimentation.
An interesting way to spend time during a sleepless evening.
"The Pike" was an amusement zone in Long Beach, California, from 1902 to 1979. I went there several times as a young man. There was an interesting game called "Looff's Lite-a-Line" at the Pike. There were 15 seats for 15 players. The goal was to shoot a pinball multiple times to get five-in-a-row on a bingo-like score card before any other player did so. I think experiences like playing this game influenced my love of mathematics, which led to my love of computer science, which led to this blog site.
Demo program. Very long, very complex. Replace "lt" (less than), "gt", "lte", "gte" with Boolean operator symbols (my blog editor often chokes on symbols).
using System;
using System.IO;
using System.Collections.Generic;
// no significant .NET dependencies
namespace MatrixPseudoInverseSingularValueDecompQR
{
internal class PinvSVDProgram
{
static void Main(string[] args)
{
Console.WriteLine("\nBegin pseudo-inverse" +
" via SVD (QR) using C# ");
double[][] A = new double[6][];
A[0] = new double[] { 1, 2, 3, 4, 5 };
A[1] = new double[] { 0, -3, 5, -7, 9 };
A[2] = new double[] { 2, 0, -2, 0, -2 };
A[3] = new double[] { 4, -1, 5, 6, 1 };
A[4] = new double[] { 3, 6, 8, 2, 2 };
A[5] = new double[] { 5, -2, 4, -4, 3 };
Console.WriteLine("\nSource (tall) matrix: ");
MatShow(A, 1, 6);
Console.WriteLine("\nComputing Moore-Penrose" +
" pseudo-inverse ");
double[][] Apinv = SVD.MatPinv(A);
Console.WriteLine("Done ");
Console.WriteLine("\nPseudo-inverse: ");
MatShow(Apinv, 6, 11);
Console.WriteLine("\n============================ ");
double[][] B = new double[3][];
B[0] = new double[] { 3, 1, 1, 0, 5 };
B[1] = new double[] { -1, 3, 1, -2, 4 };
B[2] = new double[] { 0, 2, 2, 1, -3 };
Console.WriteLine("\nSource (wide) matrix: ");
MatShow(B, 1, 6);
Console.WriteLine("\nComputing Moore-Penrose" +
" pseudo-inverse ");
double[][] Bpinv = SVD.MatPinv(B);
Console.WriteLine("Done ");
Console.WriteLine("\nPseudo-inverse: ");
MatShow(Bpinv, 6, 11);
Console.WriteLine("\nEnd demo ");
Console.ReadLine();
} // Main
// helpers for Main()
// ------------------------------------------------------
static void MatShow(double[][] M, int dec, int wid)
{
for (int i = 0; i "lt" M.Length; ++i)
{
for (int j = 0; j "lt" M[0].Length; ++j)
{
double v = M[i][j];
Console.Write(v.ToString("F" + dec).
PadLeft(wid));
}
Console.WriteLine("");
}
}
// ------------------------------------------------------
static double[][] MatLoad(string fn, int[] usecols,
char sep, string comment)
{
// ex: double[][] A =
// MatLoad("data.txt", new int[] { 0,1,2,3 }, ',', "#");
List"lt"double[]"gt" result =
new List"lt"double[]"gt"();
string line = "";
FileStream ifs = new FileStream(fn, FileMode.Open);
StreamReader sr = new StreamReader(ifs);
while ((line = sr.ReadLine()) != null)
{
if (line.StartsWith(comment) == true)
continue;
string[] tokens = line.Split(sep);
List"lt"double"gt" lst = new List"lt"double"gt"();
for (int j = 0; j "lt" usecols.Length; ++j)
lst.Add(double.Parse(tokens[usecols[j]]));
double[] row = lst.ToArray();
result.Add(row);
}
sr.Close(); ifs.Close();
return result.ToArray();
}
} // class Program
// --------------------------------------------------------
public class SVD
{
public static double[][] MatPinv(double[][] A)
{
double[][] U; double[] s; double[][] Vh;
MatDecompSVD(A, out U, out s, out Vh);
double[][] Uh = MatTranspose(U);
double[][] V = MatTranspose(Vh);
double[][] S = MatMake(s.Length, s.Length);
for (int i = 0; i "lt" s.Length; ++i)
S[i][i] = 1.0 / (s[i] + 1.0e-12); // avoid div by 0
double[][] result = MatProduct(V, MatProduct(S, Uh));
return result;
}
// ------------------------------------------------------
public static void MatDecompSVD(double[][] A,
out double[][] U, out double[] s, out double[][] Vt,
double tol = 1.0e-12)
{
int m = A.Length; int n = A[0].Length;
if (m "gte" n)
{
double[][] Q; double[][] R;
MatDecompQR(A, out Q, out R);
double[][] RtR = MatProduct(MatTranspose(R), R);
double[] evals; double[][] V;
MatDecompEigen(RtR, out evals, out V);
int[] sortIdxs = VecReversed(ArgSort(evals));
evals = VecSortedUsingIdxs(evals, sortIdxs);
V = MatSortedUsingIdxs(V, sortIdxs);
double[] ss = VecSqrt(VecClip(evals, 0.0));
int[] idxs = VecIndicesWhereNonNegative(ss, tol);
double[] ssNonZero = VecRemoveZeros(ss, idxs);
double[][] VNonZero = MatRemoveCols(V, idxs);
double[][] tmp1 = MatProduct(R, VNonZero);
double[][] tmp2 = MatDivVector(tmp1, ssNonZero);
double[][] UU = MatProduct(Q, tmp2);
U = UU;
s = ssNonZero;
Vt = MatTranspose(VNonZero);
} // m "gte" n
else
{
// use the transpose trick
double[][] dummyU;
double[] dummyS;
double[][] dummyVt;
MatDecompSVD(MatTranspose(A),
out dummyU, out dummyS, out dummyVt);
U = MatTranspose(dummyVt);
s = dummyS;
Vt = MatTranspose(dummyU);
} // m "lt" n
} // MatDecomp()
// ------------------------------------------------------
// primary helpers: MatDecompQR() and MatDecompEigen()
// ------------------------------------------------------
private static void MatDecompQR(double[][] A,
out double[][] Q, out double[][] R)
{
// reduced QR decomp using Householder reflections
int m = A.Length; int n = A[0].Length;
double[][] QQ = MatIdentity(m); // working Q
double[][] RR = MatCopy(A); // working R
for (int k = 0; k "lt" n; ++k)
{
double[] x = MatColToEnd(RR, k); // RR[k:, k]
double normx = VecNorm(x);
if (Math.Abs(normx) "lt" 1.0e-12) continue;
double[] v = VecCopy(x);
double sign;
if (x[0] "gte" 0.0) sign = 1.0; else sign = -1.0;
v[0] += sign * normx;
double normv = VecNorm(v);
for (int i = 0; i "lt" v.Length; ++i)
v[i] /= normv;
// apply reflection to R
double[][] tmp1 =
MatRowColToEnd(RR, k); // RR[k:, k:]
double[] tmp2 = VecMatProduct(v, tmp1);
double[][] tmp3 = VecOuter(v, tmp2);
double[][] B = MatScalarMult(2.0, tmp3);
MatSubtractCorner(RR, k, B); // R[k:, k:] -= B
// accumulate Q
double[][] tmp4 =
MatExtractAllRowsColToEnd(QQ, k); // QQ[:, k:]
double[] tmp5 = MatVecProduct(tmp4, v);
double[][] tmp6 = VecOuter(tmp5, v);
double[][] C = MatScalarMult(2.0, tmp6);
MatSubtractRows(QQ, k, C); // QQ[:, k:] -= C
} // for k
// return parts of QQ and RR
Q = MatExtractFirstCols(QQ, n); // Q[:, :n]
R = MatExtractFirstRows(RR, n); // R[:n, :]
} // MatDecompQR()
// ------------------------------------------------------
private static void MatDecompEigen(double[][] A,
out double[] evals, out double[][] Evecs,
double tol = 1.0e-12, int maxIter = 5000)
{
// eigen-decomp of symmetric matrix using QR
int m = A.Length; int n = A[0].Length;
double[][] AA = MatCopy(A); // don't destroy A
double[][] V = MatIdentity(n);
int iter = 0;
while (iter "lt" maxIter)
{
double[][] Q;
double[][] R;
MatDecompQR(AA, out Q, out R);
double[][] Anew = MatProduct(R, Q);
V = MatProduct(V, Q);
// # check convergence (off-diagonal norm)
double[] tmp1 = MatDiag(Anew);
double[][] tmp2 = MatCreateDiag(tmp1);
double[][] tmp3 = MatSubtract(Anew, tmp2);
double norm = MatNorm(tmp3);
if (norm "lt" tol)
{
AA = Anew; break;
}
AA = Anew;
++iter;
} // iter
if (iter == maxIter)
Console.WriteLine("Warn: exceeded maxIter ");
evals = MatDiag(AA);
Evecs = V;
} // MatDecompEigen()
// ------------------------------------------------------
// 32 secondary helpers
// ------------------------------------------------------
private static int[] ArgSort(double[] vec)
{
int n = vec.Length;
int[] idxs = new int[n];
for (int i = 0; i "lt" n; ++i)
idxs[i] = i;
double[] dup = new double[n];
for (int i = 0; i "lt" n; ++i)
dup[i] = vec[i]; // don't modify vec
// special C# overload
Array.Sort(dup, idxs); // sort idxs based on dup vals
return idxs;
}
// ------------------------------------------------------
private static double[][] MatMake(int nr, int nc)
{
double[][] result = new double[nr][];
for (int i = 0; i "lt" nr; ++i)
result[i] = new double[nc];
return result;
}
// ------------------------------------------------------
private static double[][] MatIdentity(int n)
{
double[][] result = MatMake(n, n);
for (int i = 0; i "lt" n; ++i)
result[i][i] = 1.0;
return result;
}
// ------------------------------------------------------
private static double[][] MatCopy(double[][] A)
{
int m = A.Length; int n = A[0].Length;
double[][] result = MatMake(m, n);
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
result[i][j] = A[i][j];
return result;
}
// ------------------------------------------------------
private static double[] MatDiag(double[][] A)
{
// return diag elements of A as a vector
int m = A.Length; int n = A[0].Length; // m == n
double[] result = new double[m];
for (int i = 0; i "lt" m; ++i)
result[i] = A[i][i];
return result;
}
// ------------------------------------------------------
private static double[][] MatCreateDiag(double[] v)
{
// create a matrix using v as diagonal elements
int n = v.Length;
double[][] result = MatMake(n, n);
for (int i = 0;i "lt" n; ++i)
result[i][i] = v[i];
return result;
}
// ------------------------------------------------------
private static double[][] MatTranspose(double[][] A)
{
int m = A.Length; int n = A[0].Length;
double[][] result = MatMake(n, m); // note
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
result[j][i] = A[i][j];
return result;
}
// ------------------------------------------------------
private static double[][] MatSubtract(double[][] A,
double[][] B)
{
int m = A.Length; int n = A[0].Length;
double[][] result = MatMake(m, n);
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
result[i][j] = A[i][j] - B[i][j];
return result;
}
// ------------------------------------------------------
private static double[] MatColToEnd(double[][] A,
int k)
{
// last part of col k, from [k,k] to end
int m = A.Length; int n = A[0].Length;
double[] result = new double[m - k];
for (int i = 0; i "lt" m - k; ++i)
result[i] = A[i + k][k];
return result;
}
// ------------------------------------------------------
private static double[][] MatRowColToEnd(double[][] A,
int k)
{
// block of A, row k to end, col k to end
// A[k:, k:]
int m = A.Length; int n = A[0].Length;
double[][] result = MatMake(m-k, n-k);
for (int i = 0; i "lt" m - k; ++i)
for (int j = 0; j "lt" n - k; ++j)
result[i][j] = A[i + k][j + k];
return result;
}
// ------------------------------------------------------
private static double[] MatVecProduct(double[][] A,
double[] v)
{
// A * v
int m = A.Length; int n = A[0].Length;
double[] result = new double[m];
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
result[i] += A[i][j] * v[j];
return result;
}
private static double[] VecMatProduct(double[] v,
double[][] A)
{
// v * A
int m = A.Length; int n = A[0].Length;
double[] result = new double[n];
for (int j = 0; j "lt" n; ++j)
for (int i = 0; i "lt" m; ++i)
result[j] += v[i] * A[i][j];
return result;
}
// ------------------------------------------------------
private static double[][] MatScalarMult(double x,
double[][] A)
{
// x * A element-wise
int m = A.Length; int n = A[0].Length;
double[][] result = MatMake(m, n);
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
result[i][j] = x * A[i][j];
return result;
}
// ------------------------------------------------------
private static void MatSubtractCorner(double[][] A,
int k, double[][] C)
{
// subtract elements in C from lower right A
// A[k:, k:] -= C
int m = A.Length; int n = A[0].Length;
for (int i = 0; i "lt" m - k; ++i)
for (int j = 0; j "lt" n - k; ++j)
A[i + k][j + k] -= C[i][j];
return; // modify A in-place
}
// ------------------------------------------------------
private static void MatSubtractRows(double[][] A,
int k, double[][] C)
{
// A[:, k:] -= C
int m = A.Length; int n = A[0].Length;
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n-k; ++j)
A[i][j+k] -= C[i][j];
return; // modify A in-place
}
// ------------------------------------------------------
private static double[][]
MatExtractAllRowsColToEnd(double[][] A, int k)
{
// A[:, k:]
int m = A.Length; int n = A[0].Length;
double[][] result = MatMake(m, n - k);
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n - k; ++j)
result[i][j] = A[i][j + k];
return result;
}
// -----------------------------------------------------
private static double[][]
MatExtractFirstCols(double[][] A, int k)
{
// A[:, :n]
int m = A.Length; int n = A[0].Length;
double[][] result = MatMake(m, k);
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" k; ++j)
result[i][j] = A[i][j];
return result;
}
// ------------------------------------------------------
private static double[][]
MatExtractFirstRows(double[][] A, int k)
{
// A[:n, :]
int m = A.Length; int n = A[0].Length;
double[][] result = MatMake(k, n);
for (int i = 0; i "lt" k; ++i)
for (int j = 0; j "lt" n; ++j)
result[i][j] = A[i][j];
return result;
}
// ------------------------------------------------------
private static double[][] MatProduct(double[][] A,
double[][] B)
{
int aRows = A.Length;
int aCols = A[0].Length;
int bRows = B.Length;
int bCols = B[0].Length;
if (aCols != bRows)
throw new Exception("Non-conformable matrices");
double[][] result = MatMake(aRows, bCols);
for (int i = 0; i "lt" aRows; ++i) // each row of A
for (int j = 0; j "lt" bCols; ++j) // each col of B
for (int k = 0; k "lt" aCols; ++k)
result[i][j] += A[i][k] * B[k][j];
return result;
}
// ------------------------------------------------------
private static double[][] MatRemoveCols(double[][] A,
int[] idxs)
{
// remove the columns specified in idxs
int m = A.Length; int n = A[0].Length;
int countNonzeos = 0;
for (int i = 0; i "lt" idxs.Length; ++i)
if (idxs[i] == 1) ++countNonzeos;
double[][] result = MatMake(m, countNonzeos);
int k = 0; // destination col
for (int j = 0; j "lt" n; ++j)
{
if (idxs[j] == 0) continue; // skip this col
for (int i = 0; i "lt" m; ++i)
{
result[i][k] = A[i][j];
}
++k;
}
return result;
}
// ------------------------------------------------------
private static double[][] MatDivVector(double[][] A,
double[] v)
{
// v must be all non-zero
// divide each row of A by v, element-wise
int m = A.Length; int n = A[0].Length;
double[][] result = MatMake(m, n);
for (int i = 0; i "lt" m; ++i)
for (int j= 0; j "lt" n; ++j)
result[i][j] = A[i][j] / v[j];
return result;
}
// ------------------------------------------------------
private static double MatNorm(double[][] A)
{
// treat A as one big vector
int m = A.Length; int n = A[0].Length;
double sum = 0.0;
for (int i = 0; i "lt" m; ++i)
for (int j = 0; j "lt" n; ++j)
sum += A[i][j] * A[i][j];
return Math.Sqrt(sum);
}
private static double VecNorm(double[] v)
{
// standard vector norm
double sum = 0.0;
for (int i = 0; i "lt" v.Length; ++i)
sum += v[i] * v[i];
return Math.Sqrt(sum);
}
// ------------------------------------------------------
private static double[][]
MatSortedUsingIdxs(double[][] A, int[] idxs)
{
// arrange columns of A using order in idxs
int m = A.Length; int n = A[0].Length;
double[][] result = MatMake(m, n);
for (int i = 0; i "lt" m; ++i)
{
for (int j = 0; j "lt" n; ++j)
{
int srcCol = idxs[j];
result[i][j] = A[i][srcCol];
}
}
return result;
}
// ------------------------------------------------------
private static double[]
VecSortedUsingIdxs(double[] v, int[] idxs)
{
// arrange values in v using order specified in idxs
int n = v.Length;
double[] result = new double[n];
for (int i = 0; i "lt" n; ++i)
result[i] = v[idxs[i]];
return result;
}
// ------------------------------------------------------
private static double[] VecCopy(double[] v)
{
double[] result = new double[v.Length];
for (int i = 0; i "lt" v.Length; ++i)
result[i] = v[i];
return result;
}
// ------------------------------------------------------
private static double[][] VecOuter(double[] v1,
double[] v2)
{
// vector outer product
double[][] result = MatMake(v1.Length, v2.Length);
for (int i = 0; i "lt" v1.Length; ++i)
for (int j = 0; j "lt" v2.Length; ++j)
result[i][j] = v1[i] * v2[j];
return result;
}
// ------------------------------------------------------
private static int[] VecReversed(int[] v)
{
// return vector of v elements in reversed order
int n = v.Length;
int[] result = new int[n];
for (int i = 0; i "lt" n; ++i)
result[i] = v[n-1-i];
return result;
}
// ------------------------------------------------------
private static double[] VecClip(double[] v, double minVal)
{
// used to make sure no negative values
int n = v.Length;
double[] result = new double[n];
for (int i = 0; i "lt" n; ++i)
{
if (v[i] "lt" minVal)
result[i] = minVal;
else
result[i] = v[i];
}
return result;
}
// ------------------------------------------------------
private static double[] VecSqrt(double[] v)
{
// sqrt each element in v
// assumes v has no negative values
int n = v.Length;
double[] result = new double[n];
for (int i = 0; i "lt" n; ++i)
result[i] = Math.Sqrt(v[i]);
return result;
}
// ------------------------------------------------------
private static int[]
VecIndicesWhereNonNegative(double[] v, double tol)
{
// indices of non-negative values (within tolerance)
int n = v.Length;
int[] result = new int[n];
for (int i = 0; i "lt" n; ++i)
{
if (Math.Abs(v[i]) "gt" tol)
result[i] = 1;
}
return result;
}
// ------------------------------------------------------
private static double[] VecRemoveZeros(double[] v,
int[] idxs)
{
// if idx = 1, val is non-zero, if 0 val is near zero
int n = v.Length;
List"lt"double"gt" lst = new List"lt"double"gt"();
for (int i = 0; i "lt" n; ++i)
{
if (idxs[i] == 1)
lst.Add(v[i]);
}
return lst.ToArray();
}
// ------------------------------------------------------
} // class SVD
} // ns
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