internal/ceres/implicit_schur_complement_test.cc

// Ceres Solver - A fast non-linear least squares minimizer
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// Author: sameeragarwal@google.com (Sameer Agarwal)

#include "ceres/implicit_schur_complement.h"

#include <cstddef>
#include "Eigen/Dense"
#include "ceres/block_random_access_dense_matrix.h"
#include "ceres/block_sparse_matrix.h"
#include "ceres/casts.h"
#include "ceres/internal/eigen.h"
#include "ceres/internal/scoped_ptr.h"
#include "ceres/linear_least_squares_problems.h"
#include "ceres/linear_solver.h"
#include "ceres/schur_eliminator.h"
#include "ceres/triplet_sparse_matrix.h"
#include "ceres/types.h"
#include "glog/logging.h"
#include "gtest/gtest.h"

namespace ceres {
namespace internal {

using testing::AssertionResult;

const double kEpsilon = 1e-14;

class ImplicitSchurComplementTest : public ::testing::Test {
 protected :
  virtual void SetUp() {
    scoped_ptr<LinearLeastSquaresProblem> problem(
        CreateLinearLeastSquaresProblemFromId(2));

    CHECK_NOTNULL(problem.get());
    A_.reset(down_cast<BlockSparseMatrix*>(problem->A.release()));
    b_.reset(problem->b.release());
    D_.reset(problem->D.release());

    num_cols_ = A_->num_cols();
    num_rows_ = A_->num_rows();
    num_eliminate_blocks_ = problem->num_eliminate_blocks;
  }

  void ReducedLinearSystemAndSolution(double* D,
                                      Matrix* lhs,
                                      Vector* rhs,
                                      Vector* solution) {
    const CompressedRowBlockStructure* bs = A_->block_structure();
    const int num_col_blocks = bs->cols.size();
    std::vector<int> blocks(num_col_blocks - num_eliminate_blocks_, 0);
    for (int i = num_eliminate_blocks_; i < num_col_blocks; ++i) {
      blocks[i - num_eliminate_blocks_] = bs->cols[i].size;
    }

    BlockRandomAccessDenseMatrix blhs(blocks);
    const int num_schur_rows = blhs.num_rows();

    LinearSolver::Options options;
    options.elimination_groups.push_back(num_eliminate_blocks_);
    options.type = DENSE_SCHUR;

    scoped_ptr<SchurEliminatorBase> eliminator(
        SchurEliminatorBase::Create(options));
    CHECK_NOTNULL(eliminator.get());
    const bool kFullRankETE = true;
    eliminator->Init(num_eliminate_blocks_, kFullRankETE, bs);

    lhs->resize(num_schur_rows, num_schur_rows);
    rhs->resize(num_schur_rows);

    eliminator->Eliminate(A_.get(), b_.get(), D, &blhs, rhs->data());

    MatrixRef lhs_ref(blhs.mutable_values(), num_schur_rows, num_schur_rows);

    // lhs_ref is an upper triangular matrix. Construct a full version
    // of lhs_ref in lhs by transposing lhs_ref, choosing the strictly
    // lower triangular part of the matrix and adding it to lhs_ref.
    *lhs = lhs_ref;
    lhs->triangularView<Eigen::StrictlyLower>() =
        lhs_ref.triangularView<Eigen::StrictlyUpper>().transpose();

    solution->resize(num_cols_);
    solution->setZero();
    VectorRef schur_solution(solution->data() + num_cols_ - num_schur_rows,
                             num_schur_rows);
    schur_solution = lhs->selfadjointView<Eigen::Upper>().llt().solve(*rhs);
    eliminator->BackSubstitute(A_.get(), b_.get(), D,
                               schur_solution.data(), solution->data());
  }

  AssertionResult TestImplicitSchurComplement(double* D) {
    Matrix lhs;
    Vector rhs;
    Vector reference_solution;
    ReducedLinearSystemAndSolution(D, &lhs, &rhs, &reference_solution);

    LinearSolver::Options options;
    options.elimination_groups.push_back(num_eliminate_blocks_);
    options.preconditioner_type = JACOBI;
    ImplicitSchurComplement isc(options);
    isc.Init(*A_, D, b_.get());

    int num_sc_cols = lhs.cols();

    for (int i = 0; i < num_sc_cols; ++i) {
      Vector x(num_sc_cols);
      x.setZero();
      x(i) = 1.0;

      Vector y(num_sc_cols);
      y = lhs * x;

      Vector z(num_sc_cols);
      isc.RightMultiply(x.data(), z.data());

      // The i^th column of the implicit schur complement is the same as
      // the explicit schur complement.
      if ((y - z).norm() > kEpsilon) {
        return testing::AssertionFailure()
            << "Explicit and Implicit SchurComplements differ in "
            << "column " << i << ". explicit: " << y.transpose()
            << " implicit: " << z.transpose();
      }
    }

    // Compare the rhs of the reduced linear system
    if ((isc.rhs() - rhs).norm() > kEpsilon) {
      return testing::AssertionFailure()
            << "Explicit and Implicit SchurComplements differ in "
            << "rhs. explicit: " << rhs.transpose()
            << " implicit: " << isc.rhs().transpose();
    }

    // Reference solution to the f_block.
    const Vector reference_f_sol =
        lhs.selfadjointView<Eigen::Upper>().llt().solve(rhs);

    // Backsubstituted solution from the implicit schur solver using the
    // reference solution to the f_block.
    Vector sol(num_cols_);
    isc.BackSubstitute(reference_f_sol.data(), sol.data());
    if ((sol - reference_solution).norm() > kEpsilon) {
      return testing::AssertionFailure()
          << "Explicit and Implicit SchurComplements solutions differ. "
          << "explicit: " << reference_solution.transpose()
          << " implicit: " << sol.transpose();
    }

    return testing::AssertionSuccess();
  }

  int num_rows_;
  int num_cols_;
  int num_eliminate_blocks_;

  scoped_ptr<BlockSparseMatrix> A_;
  scoped_array<double> b_;
  scoped_array<double> D_;
};

// Verify that the Schur Complement matrix implied by the
// ImplicitSchurComplement class matches the one explicitly computed
// by the SchurComplement solver.
//
// We do this with and without regularization to check that the
// support for the LM diagonal is correct.
TEST_F(ImplicitSchurComplementTest, SchurMatrixValuesTest) {
  EXPECT_TRUE(TestImplicitSchurComplement(NULL));
  EXPECT_TRUE(TestImplicitSchurComplement(D_.get()));
}

}  // namespace internal
}  // namespace ceres