// matrix/jama-svd.h
// Copyright 2009-2011 Microsoft Corporation
// See ../../COPYING for clarification regarding multiple authors
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// http://www.apache.org/licenses/LICENSE-2.0
//
// THIS CODE IS PROVIDED *AS IS* BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
// KIND, EITHER EXPRESS OR IMPLIED, INCLUDING WITHOUT LIMITATION ANY IMPLIED
// WARRANTIES OR CONDITIONS OF TITLE, FITNESS FOR A PARTICULAR PURPOSE,
// MERCHANTABLITY OR NON-INFRINGEMENT.
// See the Apache 2 License for the specific language governing permissions and
// limitations under the License.
// This file consists of a port and modification of materials from
// JAMA: A Java Matrix Package
// under the following notice: This software is a cooperative product of
// The MathWorks and the National Institute of Standards and Technology (NIST)
// which has been released to the public. This notice and the original code are
// available at http://math.nist.gov/javanumerics/jama/domain.notice
#ifndef KALDI_MATRIX_JAMA_SVD_H_
#define KALDI_MATRIX_JAMA_SVD_H_ 1
#include "matrix/kaldi-matrix.h"
#include "matrix/sp-matrix.h"
#include "matrix/cblas-wrappers.h"
namespace kaldi {
#if defined(HAVE_ATLAS) || defined(USE_KALDI_SVD)
// using ATLAS as our math library, which doesn't have SVD -> need
// to implement it.
// This routine is a modified form of jama_svd.h which is part of the TNT distribution.
// (originally comes from JAMA).
/** Singular Value Decomposition.
* <P>
* For an m-by-n matrix A with m >= n, the singular value decomposition is
* an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
* an n-by-n orthogonal matrix V so that A = U*S*V'.
* <P>
* The singular values, sigma[k] = S(k, k), are ordered so that
* sigma[0] >= sigma[1] >= ... >= sigma[n-1].
* <P>
* The singular value decompostion always exists, so the constructor will
* never fail. The matrix condition number and the effective numerical
* rank can be computed from this decomposition.
* <p>
* (Adapted from JAMA, a Java Matrix Library, developed by jointly
* by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).
*/
template<typename Real>
bool MatrixBase<Real>::JamaSvd(VectorBase<Real> *s_in,
MatrixBase<Real> *U_in,
MatrixBase<Real> *V_in) { // Destructive!
KALDI_ASSERT(s_in != NULL && U_in != this && V_in != this);
int wantu = (U_in != NULL), wantv = (V_in != NULL);
Matrix<Real> Utmp, Vtmp;
MatrixBase<Real> &U = (U_in ? *U_in : Utmp), &V = (V_in ? *V_in : Vtmp);
VectorBase<Real> &s = *s_in;
int m = num_rows_, n = num_cols_;
KALDI_ASSERT(m>=n && m != 0 && n != 0);
if (wantu) KALDI_ASSERT((int)U.num_rows_ == m && (int)U.num_cols_ == n);
if (wantv) KALDI_ASSERT((int)V.num_rows_ == n && (int)V.num_cols_ == n);
KALDI_ASSERT((int)s.Dim() == n); // n<=m so n is min.
int nu = n;
U.SetZero(); // make sure all zero.
Vector<Real> e(n);
Vector<Real> work(m);
MatrixBase<Real> &A(*this);
Real *adata = A.Data(), *workdata = work.Data(), *edata = e.Data(),
*udata = U.Data(), *vdata = V.Data();
int astride = static_cast<int>(A.Stride()),
ustride = static_cast<int>(U.Stride()),
vstride = static_cast<int>(V.Stride());
int i = 0, j = 0, k = 0;
// Reduce A to bidiagonal form, storing the diagonal elements
// in s and the super-diagonal elements in e.
int nct = std::min(m-1, n);
int nrt = std::max(0, std::min(n-2, m));
for (k = 0; k < std::max(nct, nrt); k++) {
if (k < nct) {
// Compute the transformation for the k-th column and
// place the k-th diagonal in s(k).
// Compute 2-norm of k-th column without under/overflow.
s(k) = 0;
for (i = k; i < m; i++) {
s(k) = hypot(s(k), A(i, k));
}
if (s(k) != 0.0) {
if (A(k, k) < 0.0) {
s(k) = -s(k);
}
for (i = k; i < m; i++) {
A(i, k) /= s(k);
}
A(k, k) += 1.0;
}
s(k) = -s(k);
}
for (j = k+1; j < n; j++) {
if ((k < nct) && (s(k) != 0.0)) {
// Apply the transformation.
Real t = cblas_Xdot(m - k, adata + astride*k + k, astride,
adata + astride*k + j, astride);
/*for (i = k; i < m; i++) {
t += adata[i*astride + k]*adata[i*astride + j]; // A(i, k)*A(i, j); // 3
}*/
t = -t/A(k, k);
cblas_Xaxpy(m - k, t, adata + k*astride + k, astride,
adata + k*astride + j, astride);
/*for (i = k; i < m; i++) {
adata[i*astride + j] += t*adata[i*astride + k]; // A(i, j) += t*A(i, k); // 5
}*/
}
// Place the k-th row of A into e for the
// subsequent calculation of the row transformation.
e(j) = A(k, j);
}
if (wantu & (k < nct)) {
// Place the transformation in U for subsequent back
// multiplication.
for (i = k; i < m; i++) {
U(i, k) = A(i, k);
}
}
if (k < nrt) {
// Compute the k-th row transformation and place the
// k-th super-diagonal in e(k).
// Compute 2-norm without under/overflow.
e(k) = 0;
for (i = k+1; i < n; i++) {
e(k) = hypot(e(k), e(i));
}
if (e(k) != 0.0) {
if (e(k+1) < 0.0) {
e(k) = -e(k);
}
for (i = k+1; i < n; i++) {
e(i) /= e(k);
}
e(k+1) += 1.0;
}
e(k) = -e(k);
if ((k+1 < m) & (e(k) != 0.0)) {
// Apply the transformation.
for (i = k+1; i < m; i++) {
work(i) = 0.0;
}
for (j = k+1; j < n; j++) {
for (i = k+1; i < m; i++) {
workdata[i] += edata[j] * adata[i*astride + j]; // work(i) += e(j)*A(i, j); // 5
}
}
for (j = k+1; j < n; j++) {
Real t(-e(j)/e(k+1));
cblas_Xaxpy(m - (k+1), t, workdata + (k+1), 1,
adata + (k+1)*astride + j, astride);
/*
for (i = k+1; i < m; i++) {
adata[i*astride + j] += t*workdata[i]; // A(i, j) += t*work(i); // 5
}*/
}
}
if (wantv) {
// Place the transformation in V for subsequent
// back multiplication.
for (i = k+1; i < n; i++) {
V(i, k) = e(i);
}
}
}
}
// Set up the final bidiagonal matrix or order p.
int p = std::min(n, m+1);
if (nct < n) {
s(nct) = A(nct, nct);
}
if (m < p) {
s(p-1) = 0.0;
}
if (nrt+1 < p) {
e(nrt) = A(nrt, p-1);
}
e(p-1) = 0.0;
// If required, generate U.
if (wantu) {
for (j = nct; j < nu; j++) {
for (i = 0; i < m; i++) {
U(i, j) = 0.0;
}
U(j, j) = 1.0;
}
for (k = nct-1; k >= 0; k--) {
if (s(k) != 0.0) {
for (j = k+1; j < nu; j++) {
Real t = cblas_Xdot(m - k, udata + k*ustride + k, ustride, udata + k*ustride + j, ustride);
//for (i = k; i < m; i++) {
// t += udata[i*ustride + k]*udata[i*ustride + j]; // t += U(i, k)*U(i, j); // 8
// }
t = -t/U(k, k);
cblas_Xaxpy(m - k, t, udata + ustride*k + k, ustride,
udata + k*ustride + j, ustride);
/*for (i = k; i < m; i++) {
udata[i*ustride + j] += t*udata[i*ustride + k]; // U(i, j) += t*U(i, k); // 4
}*/
}
for (i = k; i < m; i++ ) {
U(i, k) = -U(i, k);
}
U(k, k) = 1.0 + U(k, k);
for (i = 0; i < k-1; i++) {
U(i, k) = 0.0;
}
} else {
for (i = 0; i < m; i++) {
U(i, k) = 0.0;
}
U(k, k) = 1.0;
}
}
}
// If required, generate V.
if (wantv) {
for (k = n-1; k >= 0; k--) {
if ((k < nrt) & (e(k) != 0.0)) {
for (j = k+1; j < nu; j++) {
Real t = cblas_Xdot(n - (k+1), vdata + (k+1)*vstride + k, vstride,
vdata + (k+1)*vstride + j, vstride);
/*Real t (0.0);
for (i = k+1; i < n; i++) {
t += vdata[i*vstride + k]*vdata[i*vstride + j]; // t += V(i, k)*V(i, j); // 7
}*/
t = -t/V(k+1, k);
cblas_Xaxpy(n - (k+1), t, vdata + (k+1)*vstride + k, vstride,
vdata + (k+1)*vstride + j, vstride);
/*for (i = k+1; i < n; i++) {
vdata[i*vstride + j] += t*vdata[i*vstride + k]; // V(i, j) += t*V(i, k); // 7
}*/
}
}
for (i = 0; i < n; i++) {
V(i, k) = 0.0;
}
V(k, k) = 1.0;
}
}
// Main iteration loop for the singular values.
int pp = p-1;
int iter = 0;
// note: -52.0 is from Jama code; the -23 is the extension
// to float, because mantissa length in (double, float)
// is (52, 23) bits respectively.
Real eps(pow(2.0, sizeof(Real) == 4 ? -23.0 : -52.0));
// Note: the -966 was taken from Jama code, but the -120 is a guess
// of how to extend this to float... the exponent in double goes
// from -1022 .. 1023, and in float from -126..127. I'm not sure
// what the significance of 966 is, so -120 just represents a number
// that's a bit less negative than -126. If we get convergence
// failure in float only, this may mean that we have to make the
// -120 value less negative.
Real tiny(pow(2.0, sizeof(Real) == 4 ? -120.0: -966.0 ));
while (p > 0) {
int k = 0;
int kase = 0;
if (iter == 500 || iter == 750) {
KALDI_WARN << "Svd taking a long time: making convergence criterion less exact.";
eps = pow(static_cast<Real>(0.8), eps);
tiny = pow(static_cast<Real>(0.8), tiny);
}
if (iter > 1000) {
KALDI_WARN << "Svd not converging on matrix of size " << m << " by " <<n;
return false;
}
// This section of the program inspects for
// negligible elements in the s and e arrays. On
// completion the variables kase and k are set as follows.
// kase = 1 if s(p) and e(k-1) are negligible and k < p
// kase = 2 if s(k) is negligible and k < p
// kase = 3 if e(k-1) is negligible, k < p, and
// s(k), ..., s(p) are not negligible (qr step).
// kase = 4 if e(p-1) is negligible (convergence).
for (k = p-2; k >= -1; k--) {
if (k == -1) {
break;
}
if (std::abs(e(k)) <=
tiny + eps*(std::abs(s(k)) + std::abs(s(k+1)))) {
e(k) = 0.0;
break;
}
}
if (k == p-2) {
kase = 4;
} else {
int ks;
for (ks = p-1; ks >= k; ks--) {
if (ks == k) {
break;
}
Real t( (ks != p ? std::abs(e(ks)) : 0.) +
(ks != k+1 ? std::abs(e(ks-1)) : 0.));
if (std::abs(s(ks)) <= tiny + eps*t) {
s(ks) = 0.0;
break;
}
}
if (ks == k) {
kase = 3;
} else if (ks == p-1) {
kase = 1;
} else {
kase = 2;
k = ks;
}
}
k++;
// Perform the task indicated by kase.
switch (kase) {
// Deflate negligible s(p).
case 1: {
Real f(e(p-2));
e(p-2) = 0.0;
for (j = p-2; j >= k; j--) {
Real t( hypot(s(j), f));
Real cs(s(j)/t);
Real sn(f/t);
s(j) = t;
if (j != k) {
f = -sn*e(j-1);
e(j-1) = cs*e(j-1);
}
if (wantv) {
for (i = 0; i < n; i++) {
t = cs*V(i, j) + sn*V(i, p-1);
V(i, p-1) = -sn*V(i, j) + cs*V(i, p-1);
V(i, j) = t;
}
}
}
}
break;
// Split at negligible s(k).
case 2: {
Real f(e(k-1));
e(k-1) = 0.0;
for (j = k; j < p; j++) {
Real t(hypot(s(j), f));
Real cs( s(j)/t);
Real sn(f/t);
s(j) = t;
f = -sn*e(j);
e(j) = cs*e(j);
if (wantu) {
for (i = 0; i < m; i++) {
t = cs*U(i, j) + sn*U(i, k-1);
U(i, k-1) = -sn*U(i, j) + cs*U(i, k-1);
U(i, j) = t;
}
}
}
}
break;
// Perform one qr step.
case 3: {
// Calculate the shift.
Real scale = std::max(std::max(std::max(std::max(
std::abs(s(p-1)), std::abs(s(p-2))), std::abs(e(p-2))),
std::abs(s(k))), std::abs(e(k)));
Real sp = s(p-1)/scale;
Real spm1 = s(p-2)/scale;
Real epm1 = e(p-2)/scale;
Real sk = s(k)/scale;
Real ek = e(k)/scale;
Real b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;
Real c = (sp*epm1)*(sp*epm1);
Real shift = 0.0;
if ((b != 0.0) || (c != 0.0)) {
shift = std::sqrt(b*b + c);
if (b < 0.0) {
shift = -shift;
}
shift = c/(b + shift);
}
Real f = (sk + sp)*(sk - sp) + shift;
Real g = sk*ek;
// Chase zeros.
for (j = k; j < p-1; j++) {
Real t = hypot(f, g);
Real cs = f/t;
Real sn = g/t;
if (j != k) {
e(j-1) = t;
}
f = cs*s(j) + sn*e(j);
e(j) = cs*e(j) - sn*s(j);
g = sn*s(j+1);
s(j+1) = cs*s(j+1);
if (wantv) {
cblas_Xrot(n, vdata + j, vstride, vdata + j+1, vstride, cs, sn);
/*for (i = 0; i < n; i++) {
t = cs*vdata[i*vstride + j] + sn*vdata[i*vstride + j+1]; // t = cs*V(i, j) + sn*V(i, j+1); // 13
vdata[i*vstride + j+1] = -sn*vdata[i*vstride + j] + cs*vdata[i*vstride + j+1]; // V(i, j+1) = -sn*V(i, j) + cs*V(i, j+1); // 5
vdata[i*vstride + j] = t; // V(i, j) = t; // 4
}*/
}
t = hypot(f, g);
cs = f/t;
sn = g/t;
s(j) = t;
f = cs*e(j) + sn*s(j+1);
s(j+1) = -sn*e(j) + cs*s(j+1);
g = sn*e(j+1);
e(j+1) = cs*e(j+1);
if (wantu && (j < m-1)) {
cblas_Xrot(m, udata + j, ustride, udata + j+1, ustride, cs, sn);
/*for (i = 0; i < m; i++) {
t = cs*udata[i*ustride + j] + sn*udata[i*ustride + j+1]; // t = cs*U(i, j) + sn*U(i, j+1); // 7
udata[i*ustride + j+1] = -sn*udata[i*ustride + j] +cs*udata[i*ustride + j+1]; // U(i, j+1) = -sn*U(i, j) + cs*U(i, j+1); // 8
udata[i*ustride + j] = t; // U(i, j) = t; // 1
}*/
}
}
e(p-2) = f;
iter = iter + 1;
}
break;
// Convergence.
case 4: {
// Make the singular values positive.
if (s(k) <= 0.0) {
s(k) = (s(k) < 0.0 ? -s(k) : 0.0);
if (wantv) {
for (i = 0; i <= pp; i++) {
V(i, k) = -V(i, k);
}
}
}
// Order the singular values.
while (k < pp) {
if (s(k) >= s(k+1)) {
break;
}
Real t = s(k);
s(k) = s(k+1);
s(k+1) = t;
if (wantv && (k < n-1)) {
for (i = 0; i < n; i++) {
t = V(i, k+1); V(i, k+1) = V(i, k); V(i, k) = t;
}
}
if (wantu && (k < m-1)) {
for (i = 0; i < m; i++) {
t = U(i, k+1); U(i, k+1) = U(i, k); U(i, k) = t;
}
}
k++;
}
iter = 0;
p--;
}
break;
}
}
return true;
}
#endif // defined(HAVE_ATLAS) || defined(USE_KALDI_SVD)
} // namespace kaldi
#endif // KALDI_MATRIX_JAMA_SVD_H_