Add a CG-based solver for rectangular least-square problems (bug #975).

Eigen/IterativeLinearSolvers

@@ -12,24 +12,26 @@
   * This module currently provides iterative methods to solve problems of the form \c A \c x = \c b, where \c A is a squared matrix, usually very large and sparse.
   * Those solvers are accessible via the following classes:
   *  - ConjugateGradient for selfadjoint (hermitian) matrices,
+  *  - LSCG for rectangular least-square problems,
   *  - BiCGSTAB for general square matrices.
   *
   * These iterative solvers are associated with some preconditioners:
   *  - IdentityPreconditioner - not really useful
   *  - DiagonalPreconditioner - also called JAcobi preconditioner, work very well on diagonal dominant matrices.
-  *  - IncompleteILUT - incomplete LU factorization with dual thresholding
+  *  - IncompleteLUT - incomplete LU factorization with dual thresholding
   *
   * Such problems can also be solved using the direct sparse decomposition modules: SparseCholesky, CholmodSupport, UmfPackSupport, SuperLUSupport.
   *
-  * \code
+    \code
-  * #include <Eigen/IterativeLinearSolvers>
+    #include <Eigen/IterativeLinearSolvers>
-  * \endcode
+    \endcode
   */
 
 #include "src/IterativeLinearSolvers/SolveWithGuess.h"
 #include "src/IterativeLinearSolvers/IterativeSolverBase.h"
 #include "src/IterativeLinearSolvers/BasicPreconditioners.h"
 #include "src/IterativeLinearSolvers/ConjugateGradient.h"
+#include "src/IterativeLinearSolvers/LeastSquareConjugateGradient.h"
 #include "src/IterativeLinearSolvers/BiCGSTAB.h"
 #include "src/IterativeLinearSolvers/IncompleteLUT.h"
 

Eigen/src/IterativeLinearSolvers/BasicPreconditioners.h

@@ -17,9 +17,9 @@ namespace Eigen {
   *
   * This class allows to approximately solve for A.x = b problems assuming A is a diagonal matrix.
   * In other words, this preconditioner neglects all off diagonal entries and, in Eigen's language, solves for:
-  * \code
+    \code
-  * A.diagonal().asDiagonal() . x = b
+    A.diagonal().asDiagonal() . x = b
-  * \endcode
+    \endcode
   *
   * \tparam _Scalar the type of the scalar.
   *
@@ -28,6 +28,7 @@ namespace Eigen {
   *
   * \note A variant that has yet to be implemented would attempt to preserve the norm of each column.
   *
+  * \sa class LeastSquareDiagonalPreconditioner, class ConjugateGradient
   */
 template <typename _Scalar>
 class DiagonalPreconditioner
@@ -100,6 +101,69 @@ class DiagonalPreconditioner
     bool m_isInitialized;
 };
 
+/** \ingroup IterativeLinearSolvers_Module
+  * \brief Jacobi preconditioner for LSCG
+  *
+  * This class allows to approximately solve for A' A x  = A' b problems assuming A' A is a diagonal matrix.
+  * In other words, this preconditioner neglects all off diagonal entries and, in Eigen's language, solves for:
+    \code
+    (A.adjoint() * A).diagonal().asDiagonal() * x = b
+    \endcode
+  *
+  * \tparam _Scalar the type of the scalar.
+  *
+  * The diagonal entries are pre-inverted and stored into a dense vector.
+  * 
+  * \sa class LSCG, class DiagonalPreconditioner
+  */
+template <typename _Scalar>
+class LeastSquareDiagonalPreconditioner : public DiagonalPreconditioner<_Scalar>
+{
+    typedef _Scalar Scalar;
+    typedef typename NumTraits<Scalar>::Real RealScalar;
+    typedef DiagonalPreconditioner<_Scalar> Base;
+    using Base::m_invdiag;
+  public:
+
+    LeastSquareDiagonalPreconditioner() : Base() {}
+
+    template<typename MatType>
+    explicit LeastSquareDiagonalPreconditioner(const MatType& mat) : Base()
+    {
+      compute(mat);
+    }
+
+    template<typename MatType>
+    LeastSquareDiagonalPreconditioner& analyzePattern(const MatType& )
+    {
+      return *this;
+    }
+    
+    template<typename MatType>
+    LeastSquareDiagonalPreconditioner& factorize(const MatType& mat)
+    {
+      // Compute the inverse squared-norm of each column of mat
+      m_invdiag.resize(mat.cols());
+      for(Index j=0; j<mat.outerSize(); ++j)
+      {
+        RealScalar sum = mat.innerVector(j).squaredNorm();
+        if(sum>0)
+          m_invdiag(j) = RealScalar(1)/sum;
+        else
+          m_invdiag(j) = RealScalar(1);
+      }
+      Base::m_isInitialized = true;
+      return *this;
+    }
+    
+    template<typename MatType>
+    LeastSquareDiagonalPreconditioner& compute(const MatType& mat)
+    {
+      return factorize(mat);
+    }
+
+  protected:
+};
 
 /** \ingroup IterativeLinearSolvers_Module
   * \brief A naive preconditioner which approximates any matrix as the identity matrix

Eigen/src/IterativeLinearSolvers/ConjugateGradient.h

@@ -60,29 +60,29 @@ void conjugate_gradient(const MatrixType& mat, const Rhs& rhs, Dest& x,
   }
   
   VectorType p(n);
-  p = precond.solve(residual);      //initial search direction
+  p = precond.solve(residual);      // initial search direction
 
   VectorType z(n), tmp(n);
   RealScalar absNew = numext::real(residual.dot(p));  // the square of the absolute value of r scaled by invM
   Index i = 0;
   while(i < maxIters)
   {
-    tmp.noalias() = mat * p;              // the bottleneck of the algorithm
+    tmp.noalias() = mat * p;                    // the bottleneck of the algorithm
 
-    Scalar alpha = absNew / p.dot(tmp);   // the amount we travel on dir
+    Scalar alpha = absNew / p.dot(tmp);         // the amount we travel on dir
-    x += alpha * p;                       // update solution
+    x += alpha * p;                             // update solution
-    residual -= alpha * tmp;              // update residue
+    residual -= alpha * tmp;                    // update residual
     
     residualNorm2 = residual.squaredNorm();
     if(residualNorm2 < threshold)
       break;
     
-    z = precond.solve(residual);          // approximately solve for "A z = residual"
+    z = precond.solve(residual);                // approximately solve for "A z = residual"
 
     RealScalar absOld = absNew;
     absNew = numext::real(residual.dot(z));     // update the absolute value of r
-    RealScalar beta = absNew / absOld;            // calculate the Gram-Schmidt value used to create the new search direction
+    RealScalar beta = absNew / absOld;          // calculate the Gram-Schmidt value used to create the new search direction
-    p = z + beta * p;                             // update search direction
+    p = z + beta * p;                           // update search direction
     i++;
   }
   tol_error = sqrt(residualNorm2 / rhsNorm2);
@@ -122,24 +122,24 @@ struct traits<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> >
   * and NumTraits<Scalar>::epsilon() for the tolerance.
   * 
   * This class can be used as the direct solver classes. Here is a typical usage example:
-  * \code
+    \code
-  * int n = 10000;
+    int n = 10000;
-  * VectorXd x(n), b(n);
+    VectorXd x(n), b(n);
-  * SparseMatrix<double> A(n,n);
+    SparseMatrix<double> A(n,n);
-  * // fill A and b
+    // fill A and b
-  * ConjugateGradient<SparseMatrix<double> > cg;
+    ConjugateGradient<SparseMatrix<double> > cg;
-  * cg.compute(A);
+    cg.compute(A);
-  * x = cg.solve(b);
+    x = cg.solve(b);
-  * std::cout << "#iterations:     " << cg.iterations() << std::endl;
+    std::cout << "#iterations:     " << cg.iterations() << std::endl;
-  * std::cout << "estimated error: " << cg.error()      << std::endl;
+    std::cout << "estimated error: " << cg.error()      << std::endl;
-  * // update b, and solve again
+    // update b, and solve again
-  * x = cg.solve(b);
+    x = cg.solve(b);
-  * \endcode
+    \endcode
   * 
   * By default the iterations start with x=0 as an initial guess of the solution.
   * One can control the start using the solveWithGuess() method.
   * 
-  * \sa class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
+  * \sa class LSCG, class SimplicialCholesky, DiagonalPreconditioner, IdentityPreconditioner
   */
 template< typename _MatrixType, int _UpLo, typename _Preconditioner>
 class ConjugateGradient : public IterativeSolverBase<ConjugateGradient<_MatrixType,_UpLo,_Preconditioner> >

Eigen/src/IterativeLinearSolvers/LeastSquareConjugateGradient.h

@@ -0,0 +1,213 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2015 Gael Guennebaud <gael.guennebaud@inria.fr>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#ifndef EIGEN_LEAST_SQUARE_CONJUGATE_GRADIENT_H
+#define EIGEN_LEAST_SQUARE_CONJUGATE_GRADIENT_H
+
+namespace Eigen { 
+
+namespace internal {
+
+/** \internal Low-level conjugate gradient algorithm for least-square problems
+  * \param mat The matrix A
+  * \param rhs The right hand side vector b
+  * \param x On input and initial solution, on output the computed solution.
+  * \param precond A preconditioner being able to efficiently solve for an
+  *                approximation of A'Ax=b (regardless of b)
+  * \param iters On input the max number of iteration, on output the number of performed iterations.
+  * \param tol_error On input the tolerance error, on output an estimation of the relative error.
+  */
+template<typename MatrixType, typename Rhs, typename Dest, typename Preconditioner>
+EIGEN_DONT_INLINE
+void least_square_conjugate_gradient(const MatrixType& mat, const Rhs& rhs, Dest& x,
+                                     const Preconditioner& precond, Index& iters,
+                                     typename Dest::RealScalar& tol_error)
+{
+  using std::sqrt;
+  using std::abs;
+  typedef typename Dest::RealScalar RealScalar;
+  typedef typename Dest::Scalar Scalar;
+  typedef Matrix<Scalar,Dynamic,1> VectorType;
+  
+  RealScalar tol = tol_error;
+  Index maxIters = iters;
+  
+  Index m = mat.rows(), n = mat.cols();
+
+  VectorType residual        = rhs - mat * x;
+  VectorType normal_residual = mat.adjoint() * residual;
+
+  RealScalar rhsNorm2 = (mat.adjoint()*rhs).squaredNorm();
+  if(rhsNorm2 == 0) 
+  {
+    x.setZero();
+    iters = 0;
+    tol_error = 0;
+    return;
+  }
+  RealScalar threshold = tol*tol*rhsNorm2;
+  RealScalar residualNorm2 = normal_residual.squaredNorm();
+  if (residualNorm2 < threshold)
+  {
+    iters = 0;
+    tol_error = sqrt(residualNorm2 / rhsNorm2);
+    return;
+  }
+  
+  VectorType p(n);
+  p = precond.solve(normal_residual);                         // initial search direction
+
+  VectorType z(n), tmp(m);
+  RealScalar absNew = numext::real(normal_residual.dot(p));  // the square of the absolute value of r scaled by invM
+  Index i = 0;
+  while(i < maxIters)
+  {
+    tmp.noalias() = mat * p;
+
+    Scalar alpha = absNew / tmp.squaredNorm();      // the amount we travel on dir
+    x += alpha * p;                                 // update solution
+    residual -= alpha * tmp;                        // update residual
+    normal_residual = mat.adjoint() * residual;     // update residual of the normal equation
+    
+    residualNorm2 = normal_residual.squaredNorm();
+    if(residualNorm2 < threshold)
+      break;
+    
+    z = precond.solve(normal_residual);             // approximately solve for "A'A z = normal_residual"
+
+    RealScalar absOld = absNew;
+    absNew = numext::real(normal_residual.dot(z));  // update the absolute value of r
+    RealScalar beta = absNew / absOld;              // calculate the Gram-Schmidt value used to create the new search direction
+    p = z + beta * p;                               // update search direction
+    i++;
+  }
+  tol_error = sqrt(residualNorm2 / rhsNorm2);
+  iters = i;
+}
+
+}
+
+template< typename _MatrixType,
+          typename _Preconditioner = LeastSquareDiagonalPreconditioner<typename _MatrixType::Scalar> >
+class LSCG;
+
+namespace internal {
+
+template< typename _MatrixType, typename _Preconditioner>
+struct traits<LSCG<_MatrixType,_Preconditioner> >
+{
+  typedef _MatrixType MatrixType;
+  typedef _Preconditioner Preconditioner;
+};
+
+}
+
+/** \ingroup IterativeLinearSolvers_Module
+  * \brief A conjugate gradient solver for sparse (or dense) least-square problems
+  *
+  * This class allows to solve for A x = b linear problems using an iterative conjugate gradient algorithm.
+  * The matrix A can be non symmetric and rectangular, but the matrix A' A should be positive-definite to guaranty stability.
+  * Otherwise, the SparseLU or SparseQR classes might be preferable.
+  * The matrix A and the vectors x and b can be either dense or sparse.
+  *
+  * \tparam _MatrixType the type of the matrix A, can be a dense or a sparse matrix.
+  * \tparam _Preconditioner the type of the preconditioner. Default is LeastSquareDiagonalPreconditioner
+  *
+  * The maximal number of iterations and tolerance value can be controlled via the setMaxIterations()
+  * and setTolerance() methods. The defaults are the size of the problem for the maximal number of iterations
+  * and NumTraits<Scalar>::epsilon() for the tolerance.
+  * 
+  * This class can be used as the direct solver classes. Here is a typical usage example:
+    \code
+    int m=1000000, n = 10000;
+    VectorXd x(n), b(m);
+    SparseMatrix<double> A(m,n);
+    // fill A and b
+    LSCG<SparseMatrix<double> > lscg;
+    lscg.compute(A);
+    x = lscg.solve(b);
+    std::cout << "#iterations:     " << lscg.iterations() << std::endl;
+    std::cout << "estimated error: " << lscg.error()      << std::endl;
+    // update b, and solve again
+    x = lscg.solve(b);
+    \endcode
+  * 
+  * By default the iterations start with x=0 as an initial guess of the solution.
+  * One can control the start using the solveWithGuess() method.
+  * 
+  * \sa class ConjugateGradient, SparseLU, SparseQR
+  */
+template< typename _MatrixType, typename _Preconditioner>
+class LSCG : public IterativeSolverBase<LSCG<_MatrixType,_Preconditioner> >
+{
+  typedef IterativeSolverBase<LSCG> Base;
+  using Base::mp_matrix;
+  using Base::m_error;
+  using Base::m_iterations;
+  using Base::m_info;
+  using Base::m_isInitialized;
+public:
+  typedef _MatrixType MatrixType;
+  typedef typename MatrixType::Scalar Scalar;
+  typedef typename MatrixType::RealScalar RealScalar;
+  typedef _Preconditioner Preconditioner;
+
+public:
+
+  /** Default constructor. */
+  LSCG() : Base() {}
+
+  /** Initialize the solver with matrix \a A for further \c Ax=b solving.
+    * 
+    * This constructor is a shortcut for the default constructor followed
+    * by a call to compute().
+    * 
+    * \warning this class stores a reference to the matrix A as well as some
+    * precomputed values that depend on it. Therefore, if \a A is changed
+    * this class becomes invalid. Call compute() to update it with the new
+    * matrix A, or modify a copy of A.
+    */
+  explicit LSCG(const MatrixType& A) : Base(A) {}
+
+  ~LSCG() {}
+
+  /** \internal */
+  template<typename Rhs,typename Dest>
+  void _solve_with_guess_impl(const Rhs& b, Dest& x) const
+  {
+    m_iterations = Base::maxIterations();
+    m_error = Base::m_tolerance;
+
+    for(Index j=0; j<b.cols(); ++j)
+    {
+      m_iterations = Base::maxIterations();
+      m_error = Base::m_tolerance;
+
+      typename Dest::ColXpr xj(x,j);
+      internal::least_square_conjugate_gradient(mp_matrix, b.col(j), xj, Base::m_preconditioner, m_iterations, m_error);
+    }
+
+    m_isInitialized = true;
+    m_info = m_error <= Base::m_tolerance ? Success : NoConvergence;
+  }
+  
+  /** \internal */
+  using Base::_solve_impl;
+  template<typename Rhs,typename Dest>
+  void _solve_impl(const MatrixBase<Rhs>& b, Dest& x) const
+  {
+    x.setZero();
+    _solve_with_guess_impl(b.derived(),x);
+  }
+
+};
+
+} // end namespace Eigen
+
+#endif // EIGEN_LEAST_SQUARE_CONJUGATE_GRADIENT_H

test/CMakeLists.txt

@@ -234,6 +234,7 @@ ei_add_test(sparse_permutations)
 ei_add_test(simplicial_cholesky)
 ei_add_test(conjugate_gradient)
 ei_add_test(bicgstab)
+ei_add_test(lscg)
 ei_add_test(sparselu)
 ei_add_test(sparseqr)
 ei_add_test(umeyama)

test/lscg.cpp

@@ -0,0 +1,29 @@
+// This file is part of Eigen, a lightweight C++ template library
+// for linear algebra.
+//
+// Copyright (C) 2011 Gael Guennebaud <g.gael@free.fr>
+//
+// This Source Code Form is subject to the terms of the Mozilla
+// Public License v. 2.0. If a copy of the MPL was not distributed
+// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
+
+#include "sparse_solver.h"
+#include <Eigen/IterativeLinearSolvers>
+
+template<typename T> void test_lscg_T()
+{
+  LSCG<SparseMatrix<T> > lscg_colmajor_diag;
+  LSCG<SparseMatrix<T>, IdentityPreconditioner> lscg_colmajor_I;
+
+  CALL_SUBTEST( check_sparse_square_solving(lscg_colmajor_diag)  );
+  CALL_SUBTEST( check_sparse_square_solving(lscg_colmajor_I)     );
+  
+  CALL_SUBTEST( check_sparse_leastsquare_solving(lscg_colmajor_diag)  );
+  CALL_SUBTEST( check_sparse_leastsquare_solving(lscg_colmajor_I)     );
+}
+
+void test_lscg()
+{
+  CALL_SUBTEST_1(test_lscg_T<double>());
+  CALL_SUBTEST_2(test_lscg_T<std::complex<double> >());
+}

test/sparse_solver.h

@@ -17,9 +17,9 @@ void check_sparse_solving(Solver& solver, const typename Solver::MatrixType& A,
   typedef typename Mat::Scalar Scalar;
   typedef typename Mat::StorageIndex StorageIndex;
 
-  DenseRhs refX = dA.lu().solve(db);
+  DenseRhs refX = dA.householderQr().solve(db);
   {
-    Rhs x(b.rows(), b.cols());
+    Rhs x(A.cols(), b.cols());
     Rhs oldb = b;
 
     solver.compute(A);
@@ -94,7 +94,7 @@ void check_sparse_solving(Solver& solver, const typename Solver::MatrixType& A,
   
   // test dense Block as the result and rhs:
   {
-    DenseRhs x(db.rows(), db.cols());
+    DenseRhs x(refX.rows(), refX.cols());
     DenseRhs oldb(db);
     x.setZero();
     x.block(0,0,x.rows(),x.cols()) = solver.solve(db.block(0,0,db.rows(),db.cols()));
@@ -119,7 +119,7 @@ void check_sparse_solving_real_cases(Solver& solver, const typename Solver::Matr
   typedef typename Mat::Scalar Scalar;
   typedef typename Mat::RealScalar RealScalar;
   
-  Rhs x(b.rows(), b.cols());
+  Rhs x(A.cols(), b.cols());
   
   solver.compute(A);
   if (solver.info() != Success)
@@ -410,3 +410,53 @@ template<typename Solver> void check_sparse_square_abs_determinant(Solver& solve
   }
 }
 
+template<typename Solver, typename DenseMat>
+void generate_sparse_leastsquare_problem(Solver&, typename Solver::MatrixType& A, DenseMat& dA, int maxSize = 300, int options = ForceNonZeroDiag)
+{
+  typedef typename Solver::MatrixType Mat;
+  typedef typename Mat::Scalar Scalar;
+
+  int rows = internal::random<int>(1,maxSize);
+  int cols = internal::random<int>(1,rows);
+  double density = (std::max)(8./(rows*cols), 0.01);
+  
+  A.resize(rows,cols);
+  dA.resize(rows,cols);
+
+  initSparse<Scalar>(density, dA, A, options);
+}
+
+template<typename Solver> void check_sparse_leastsquare_solving(Solver& solver)
+{
+  typedef typename Solver::MatrixType Mat;
+  typedef typename Mat::Scalar Scalar;
+  typedef SparseMatrix<Scalar,ColMajor> SpMat;
+  typedef Matrix<Scalar,Dynamic,Dynamic> DenseMatrix;
+  typedef Matrix<Scalar,Dynamic,1> DenseVector;
+
+  int rhsCols = internal::random<int>(1,16);
+
+  Mat A;
+  DenseMatrix dA;
+  for (int i = 0; i < g_repeat; i++) {
+    generate_sparse_leastsquare_problem(solver, A, dA);
+
+    A.makeCompressed();
+    DenseVector b = DenseVector::Random(A.rows());
+    DenseMatrix dB(A.rows(),rhsCols);
+    SpMat B(A.rows(),rhsCols);
+    double density = (std::max)(8./(A.rows()*rhsCols), 0.1);
+    initSparse<Scalar>(density, dB, B, ForceNonZeroDiag);
+    B.makeCompressed();
+    check_sparse_solving(solver, A, b,  dA, b);
+    check_sparse_solving(solver, A, dB, dA, dB);
+    check_sparse_solving(solver, A, B,  dA, dB);
+    
+    // check only once
+    if(i==0)
+    {
+      b = DenseVector::Zero(A.rows());
+      check_sparse_solving(solver, A, b, dA, b);
+    }
+  }
+}