public void TestSVD()
{
IMatrix Sigma, U, Vstar;
SVD.Factorize(A, out U, out Sigma, out Vstar);
}
public override double[] Solve()
{
int m = A.RowCount;
int n = A.ColCount;
Debug.Assert(m >= n);
IVector t = new SparseVector(n);
for (int i = 0; i < m; ++i)
{
t[i] = 0;
}
IVector s = new SparseVector(n);
IVector sy = new SparseVector(n);
for (int i = 0; i < n; ++i)
{
s[i] = 0;
}
IVector s_old = s;
IMatrix U; // U is a m x m orthogonal matrix
IMatrix Vt; // V is a n x n orthogonal matrix
IMatrix Sigma; // Sigma is a m x n diagonal matrix with non-negative real numbers on its diagonal
SVD.Factorize(A, out U, out Sigma, out Vt); // A is a m x n matrix
IMatrix Ut = U.Transpose();
IMatrix V = Vt.Transpose();
//SigmaInv is obtained by replacing every non-zero diagonal entry by its reciprocal and transposing the resulting matrix
IMatrix SigmaInv = new SparseMatrix(m, n);
for (int i = 0; i < n; ++i) // assuming m >= n
{
double sigma_i = Sigma[i, i];
if (sigma_i < epsilon) // model matrix A is rank deficient
{
throw new Exception("Near rank-deficient model matrix");
}
SigmaInv[i, i] = 1.0 / sigma_i;
}
SigmaInv = SigmaInv.Transpose();
double[] W = new double[m];
for (int j = 0; j < mMaxIters; ++j)
{
Console.WriteLine("j: {0}", j);
IVector z = new SparseVector(m);
double[] g = new double[m];
double[] gprime = new double[m];
for (int k = 0; k < m; ++k)
{
g[k] = mLinkFunc.GetInvLink(t[k]);
gprime[k] = mLinkFunc.GetInvLinkDerivative(t[k]);
z[k] = t[k] + (b[k] - g[k]) / gprime[k];
}
int tiny_weight_count = 0;
for (int k = 0; k < m; ++k)
{
double w_kk = gprime[k] * gprime[k] / GetVariance(g[k]);
W[k] = w_kk;
if (w_kk < double.Epsilon * 2)
{
tiny_weight_count++;
}
}
if (tiny_weight_count > 0)
{
Console.WriteLine("Warning: tiny weights encountered, (diag(W)) is too small");
}
s_old = s;
IMatrix UtW = new SparseMatrix(m, m);
for (int k = 0; k < m; ++k)
{
for (int k2 = 0; k2 < m; ++k2)
{
UtW[k, k2] = Ut[k, k2] * W[k];
}
}
IMatrix UtWU = UtW.Multiply(U); // m x m positive definite matrix
IMatrix L; // m x m lower triangular matrix
Cholesky.Factorize(UtWU, out L);
IMatrix Lt = L.Transpose(); // m x m upper triangular matrix
IVector UtWz = UtW.Multiply(z); // m x 1 vector
// (Ut * W * U) * s = Ut * W * z
// L * Lt * s = Ut * W * z (Cholesky factorization on Ut * W * U)
// L * sy = Ut * W * z, Lt * s = sy
s = new SparseVector(n);
for (int i = 0; i < n; ++i)
{
s[i] = 0;
sy[i] = 0;
}
// forward solve sy for L * sy = Ut * W * z
for (int i = 0; i < n; ++i) // since m >= n
{
double cross_prod = 0;
for (int k = 0; k < i; ++k)
{
cross_prod += L[i, k] * sy[k];
}
sy[i] = (UtWz[i] - cross_prod) / L[i, i];
}
// backward solve s for Lt * s = sy
for (int i = n - 1; i >= 0; --i)
{
double cross_prod = 0;
for (int k = i + 1; k < n; ++k)
{
cross_prod += Lt[i, k] * s[k];
}
s[i] = (sy[i] - cross_prod) / Lt[i, i];
}
t = U.Multiply(s);
if ((s_old.Minus(s)).Norm(2) < mTol)
{
break;
}
}
IVector x = V.Multiply(SigmaInv).Multiply(Ut).Multiply(t);
mX = new double[n];
for (int i = 0; i < n; ++i)
{
mX[i] = x[i];
}
UpdateStatistics(W);
return X;
}