/// <summary>
/// Moore–Penrose pseudoinverse
/// If A = U • Σ • VT is the singular value decomposition of A, then A† = V • Σ† • UT.
/// For a diagonal matrix such as Σ, we get the pseudoinverse by taking the reciprocal of each non-zero element
/// on the diagonal, leaving the zeros in place, and transposing the resulting matrix.
/// In numerical computation, only elements larger than some small tolerance are taken to be nonzero,
/// and the others are replaced by zeros. For example, in the MATLAB or NumPy function pinv,
/// the tolerance is taken to be t = ε • max(m,n) • max(Σ), where ε is the machine epsilon. (Wikipedia)
/// </summary>
/// <param name="M">The matrix to pseudoinverse</param>
/// <returns>The pseudoinverse of this Matrix</returns>
public static Matrix PseudoInverse(this Matrix M)
{
Svd D = M.Svd(true);
Matrix W = (Matrix)D.W();
Vector s = (Vector)D.S();
// The first element of W has the maximum value.
double tolerance = Precision.EpsilonOf(2) * Math.Max(M.RowCount, M.ColumnCount) * W[0, 0];
for (int i = 0; i < s.Count; i++)
{
if (s[i] < tolerance)
{
s[i] = 0;
}
else
{
s[i] = 1 / s[i];
}
}
W.SetDiagonal(s);
// (U * W * VT)T is equivalent with V * WT * UT
return((Matrix)(D.U() * W * D.VT()).Transpose());
}
private static Matrix <double> PseudoInverse(Svd <double> svd)
{
Matrix <double> W = svd.W();
Vector <double> s = svd.S();
// The first element of W has the maximum value.
double tolerance = Precision.EpsilonOf(2) * Math.Max(svd.U().RowCount, svd.VT().ColumnCount) * W[0, 0];
for (int i = 0; i < s.Count; i++)
{
if (s[i] < tolerance)
{
s[i] = 0;
}
else
{
s[i] = 1 / s[i];
}
}
W.SetDiagonal(s);
// (U * W * VT)T is equivalent with V * WT * UT
return((svd.U() * W * svd.VT()).Transpose());
}
public void TestMakeLowRankMatrix()
{
Matrix <double> x = SampleGenerator.MakeLowRankMatrix(
numSamples: 50,
numFeatures: 25,
effectiveRank: 5,
tailStrength: 0.01,
randomState: new Random(0));
Assert.AreEqual(50, x.RowCount, "X shape mismatch");
Assert.AreEqual(25, x.ColumnCount, "X shape mismatch");
Svd svd = x.Svd(true);
double sum = svd.S().Sum();
Assert.IsTrue(Math.Abs(sum - 5) < 0.1, "X rank is not approximately 5");
}
public override void Estimate(List <Point3D> datas)
{
double sum_x = 0;
double sum_y = 0;
double sum_z = 0;
foreach (Point3D temp in datas)
{
sum_x += temp.x;
sum_y += temp.y;
sum_z += temp.z;
}
sum_x /= datas.Count;
sum_y /= datas.Count;
sum_z /= datas.Count;
DenseMatrix jacobian = new DenseMatrix(datas.Count, 3);
foreach (Point3D temp in datas)
{
Vector <double> gradient = new DenseVector(3);
gradient[0] = temp.x - sum_x;
gradient[1] = temp.y - sum_y;
gradient[2] = temp.z - sum_z;
jacobian.SetRow(datas.IndexOf(temp), gradient);
}
Svd svd = jacobian.Svd(true);
// get matrix of left singular vectors with first n columns of U
Matrix <double> U1 = svd.U().SubMatrix(0, datas.Count, 0, 3);
// get matrix of singular values
Matrix <double> S = new DiagonalMatrix(3, 3, svd.S().ToArray());
// get matrix of right singular vectors
Matrix <double> V = svd.VT().Transpose();
Vector <double> parameters = new DenseVector(3);
parameters = V.Column(0);
x = sum_x;
y = sum_y;
z = sum_z;
i = parameters[0];
j = parameters[1];
k = parameters[2];
}
