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* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You 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
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
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package org.apache.ignite.ml.math.decompositions;
import org.apache.ignite.ml.math.Algebra;
import org.apache.ignite.ml.math.Destroyable;
import org.apache.ignite.ml.math.Matrix;
import static org.apache.ignite.ml.math.util.MatrixUtil.like;
/**
* Compute a singular value decomposition (SVD) of {@code (l x k)} matrix {@code m}.
* <p>This decomposition can be thought
* as an extension of {@link EigenDecomposition} to rectangular matrices. The factorization we get is following:</p>
* <p>{@code m = u * s * v^{*}}, where</p>
* <ul><li>{@code u} is a real or complex unitary matrix.</li>
* <li>{@code s} is a rectangular diagonal matrix with non-negative real numbers on diagonal
* (these numbers are singular values of {@code m}).</li>
* <li>{@code v} is a real or complex unitary matrix.</li></ul>
* <p>If {@code m} is real then {@code u} and {@code v} are also real.</p>
* <p>See also: <a href="https://en.wikipedia.org/wiki/Singular_value_decomposition">Wikipedia article on SVD</a>.</p>
* <p>Note: complex case is currently not supported.</p>
*/
public class SingularValueDecomposition implements Destroyable {
// U and V.
/** */
private final double[][] u;
/** */
private final double[][] v;
/** Singular values. */
private final double[] s;
/** Row dimension. */
private final int m;
/** Column dimension. */
private final int n;
/** */
private Matrix arg;
/** */
private boolean transpositionNeeded;
/**
* Singular value decomposition object.
*
* @param arg A rectangular matrix.
*/
public SingularValueDecomposition(Matrix arg) {
assert arg != null;
this.arg = arg;
if (arg.rowSize() < arg.columnSize())
transpositionNeeded = true;
double[][] a;
if (transpositionNeeded) {
// Use the transpose matrix.
m = arg.columnSize();
n = arg.rowSize();
a = new double[m][n];
for (int i = 0; i < m; i++)
for (int j = 0; j < n; j++)
a[i][j] = arg.get(j, i);
}
else {
m = arg.rowSize();
n = arg.columnSize();
a = new double[m][n];
for (int i = 0; i < m; i++)
for (int j = 0; j < n; j++)
a[i][j] = arg.get(i, j);
}
int nu = Math.min(m, n);
s = new double[Math.min(m + 1, n)];
u = new double[m][nu];
v = new double[n][n];
double[] e = new double[n];
double[] work = new double[m];
int nct = Math.min(m - 1, n);
int nrt = Math.max(0, Math.min(n - 2, m));
for (int k = 0; k < Math.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 (int i = k; i < m; i++)
s[k] = Algebra.hypot(s[k], a[i][k]);
if (s[k] != 0.0) {
if (a[k][k] < 0.0)
s[k] = -s[k];
for (int i = k; i < m; i++)
a[i][k] /= s[k];
a[k][k] += 1.0;
}
s[k] = -s[k];
}
for (int j = k + 1; j < n; j++) {
if (k < nct && s[k] != 0.0) {
// Apply the transformation.
double t = 0;
for (int i = k; i < m; i++)
t += a[i][k] * a[i][j];
t = -t / a[k][k];
for (int i = k; i < m; i++)
a[i][j] += t * a[i][k];
}
// Place the k-th row of A into e for the
// subsequent calculation of the row transformation.
e[j] = a[k][j];
}
if (k < nct)
// Place the transformation in U for subsequent back
// multiplication.
for (int 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 (int i = k + 1; i < n; i++)
e[k] = Algebra.hypot(e[k], e[i]);
if (e[k] != 0.0) {
if (e[k + 1] < 0.0)
e[k] = -e[k];
for (int 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 (int i = k + 1; i < m; i++)
work[i] = 0.0;
for (int j = k + 1; j < n; j++)
for (int i = k + 1; i < m; i++)
work[i] += e[j] * a[i][j];
for (int j = k + 1; j < n; j++) {
double t = -e[j] / e[k + 1];
for (int i = k + 1; i < m; i++)
a[i][j] += t * work[i];
}
}
// Place the transformation in V for subsequent
// back multiplication.
for (int i = k + 1; i < n; i++)
v[i][k] = e[i];
}
}
// Set up the final bi-diagonal matrix or order p.
int p = Math.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;
// Generate U.
for (int j = nct; j < nu; j++) {
for (int i = 0; i < m; i++)
u[i][j] = 0.0;
u[j][j] = 1.0;
}
for (int k = nct - 1; k >= 0; k--) {
if (s[k] != 0.0) {
for (int j = k + 1; j < nu; j++) {
double t = 0;
for (int i = k; i < m; i++)
t += u[i][k] * u[i][j];
t = -t / u[k][k];
for (int i = k; i < m; i++)
u[i][j] += t * u[i][k];
}
for (int i = k; i < m; i++)
u[i][k] = -u[i][k];
u[k][k] = 1.0 + u[k][k];
for (int i = 0; i < k - 1; i++)
u[i][k] = 0.0;
}
else {
for (int i = 0; i < m; i++)
u[i][k] = 0.0;
u[k][k] = 1.0;
}
}
// Generate V.
for (int k = n - 1; k >= 0; k--) {
if (k < nrt && e[k] != 0.0) {
for (int j = k + 1; j < nu; j++) {
double t = 0;
for (int i = k + 1; i < n; i++)
t += v[i][k] * v[i][j];
t = -t / v[k + 1][k];
for (int i = k + 1; i < n; i++)
v[i][j] += t * v[i][k];
}
}
for (int 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;
double eps = Math.pow(2.0, -52.0);
double tiny = Math.pow(2.0, -966.0);
while (p > 0) {
int k;
for (k = p - 2; k >= -1; k--) {
if (k == -1)
break;
if (Math.abs(e[k]) <= tiny + eps * (Math.abs(s[k]) + Math.abs(s[k + 1]))) {
e[k] = 0.0;
break;
}
}
int kase;
if (k == p - 2)
kase = 4;
else {
int ks;
for (ks = p - 1; ks >= k; ks--) {
if (ks == k)
break;
double t =
(ks != p ? Math.abs(e[ks]) : 0.) +
(ks != k + 1 ? Math.abs(e[ks - 1]) : 0.);
if (Math.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: {
double f = e[p - 2];
e[p - 2] = 0.0;
for (int j = p - 2; j >= k; j--) {
double t = Algebra.hypot(s[j], f);
double cs = s[j] / t;
double sn = f / t;
s[j] = t;
if (j != k) {
f = -sn * e[j - 1];
e[j - 1] = cs * e[j - 1];
}
for (int 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: {
double f = e[k - 1];
e[k - 1] = 0.0;
for (int j = k; j < p; j++) {
double t = Algebra.hypot(s[j], f);
double cs = s[j] / t;
double sn = f / t;
s[j] = t;
f = -sn * e[j];
e[j] = cs * e[j];
for (int 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.
double scale = Math.max(Math.max(Math.max(Math.max(
Math.abs(s[p - 1]), Math.abs(s[p - 2])), Math.abs(e[p - 2])),
Math.abs(s[k])), Math.abs(e[k]));
double sp = s[p - 1] / scale;
double spm1 = s[p - 2] / scale;
double epm1 = e[p - 2] / scale;
double sk = s[k] / scale;
double ek = e[k] / scale;
double b = ((spm1 + sp) * (spm1 - sp) + epm1 * epm1) / 2.0;
double c = sp * epm1 * sp * epm1;
double shift = 0.0;
if (b != 0.0 || c != 0.0) {
shift = Math.sqrt(b * b + c);
if (b < 0.0)
shift = -shift;
shift = c / (b + shift);
}
double f = (sk + sp) * (sk - sp) + shift;
double g = sk * ek;
// Chase zeros.
for (int j = k; j < p - 1; j++) {
double t = Algebra.hypot(f, g);
double cs = f / t;
double 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];
for (int i = 0; i < n; i++) {
t = cs * v[i][j] + sn * v[i][j + 1];
v[i][j + 1] = -sn * v[i][j] + cs * v[i][j + 1];
v[i][j] = t;
}
t = Algebra.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 (j < m - 1)
for (int i = 0; i < m; i++) {
t = cs * u[i][j] + sn * u[i][j + 1];
u[i][j + 1] = -sn * u[i][j] + cs * u[i][j + 1];
u[i][j] = t;
}
}
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;
for (int 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;
double t = s[k];
s[k] = s[k + 1];
s[k + 1] = t;
if (k < n - 1)
for (int i = 0; i < n; i++) {
t = v[i][k + 1];
v[i][k + 1] = v[i][k];
v[i][k] = t;
}
if (k < m - 1)
for (int 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;
default:
throw new IllegalStateException();
}
}
}
/**
* Gets the two norm condition number, which is {@code max(S) / min(S)} .
*/
public double cond() {
return s[0] / s[Math.min(m, n) - 1];
}
/**
* @return the diagonal matrix of singular values.
*/
public Matrix getS() {
double[][] s = new double[n][n];
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++)
s[i][j] = 0.0;
s[i][i] = this.s[i];
}
return like(arg, n, n).assign(s);
}
/**
* Gets the diagonal of {@code S}, which is a one-dimensional array of
* singular values.
*
* @return diagonal of {@code S}.
*/
public double[] getSingularValues() {
return s;
}
/**
* Gets the left singular vectors {@code U}.
*
* @return {@code U}
*/
public Matrix getU() {
if (transpositionNeeded)
return like(arg, v.length, v.length).assign(v);
else {
int numCols = Math.min(m + 1, n);
Matrix r = like(arg, m, numCols);
for (int i = 0; i < m; i++)
for (int j = 0; j < numCols; j++)
r.set(i, j, u[i][j]);
return r;
}
}
/**
* Gets the right singular vectors {@code V}.
*
* @return {@code V}
*/
public Matrix getV() {
if (transpositionNeeded) {
int numCols = Math.min(m + 1, n);
Matrix r = like(arg, m, numCols);
for (int i = 0; i < m; i++)
for (int j = 0; j < numCols; j++)
r.set(i, j, u[i][j]);
return r;
}
else
return like(arg, v.length, v.length).assign(v);
}
/**
* Gets the two norm, which is {@code max(S)}.
*/
public double norm2() {
return s[0];
}
/**
* Gets effective numerical matrix rank.
*/
public int rank() {
double eps = Math.pow(2.0, -52.0);
double tol = Math.max(m, n) * s[0] * eps;
int r = 0;
for (double value : s)
if (value > tol)
r++;
return r;
}
/**
* Gets [n × n] covariance matrix.
*
* @param minSingularVal Value below which singular values are ignored.
*/
Matrix getCovariance(double minSingularVal) {
Matrix j = like(arg, s.length, s.length);
Matrix vMat = like(arg, v.length, v.length).assign(v);
for (int i = 0; i < s.length; i++)
j.set(i, i, s[i] >= minSingularVal ? 1 / (s[i] * s[i]) : 0.0);
return vMat.times(j).times(vMat.transpose());
}
}