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-// 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_
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