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// Ceres Solver - A fast non-linear least squares minimizer
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// http://ceres-solver.org/
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// Author: sameeragarwal@google.com (Sameer Agarwal)
#include "ceres/implicit_schur_complement.h"
#include <cstddef>
#include <memory>
#include "Eigen/Dense"
#include "ceres/block_random_access_dense_matrix.h"
#include "ceres/block_sparse_matrix.h"
#include "ceres/casts.h"
#include "ceres/context_impl.h"
#include "ceres/internal/eigen.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 "gtest/gtest.h"
namespace ceres::internal {
using testing::AssertionResult;
const double kEpsilon = 1e-14;
class ImplicitSchurComplementTest : public ::testing::Test {
protected:
void SetUp() final {
auto problem = CreateLinearLeastSquaresProblemFromId(2);
ASSERT_TRUE(problem != nullptr);
A_.reset(down_cast<BlockSparseMatrix*>(problem->A.release()));
b_ = std::move(problem->b);
D_ = std::move(problem->D);
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();
auto blocks = Tail(bs->cols, num_col_blocks - num_eliminate_blocks_);
BlockRandomAccessDenseMatrix blhs(blocks, &context_, 1);
const int num_schur_rows = blhs.num_rows();
LinearSolver::Options options;
options.elimination_groups.push_back(num_eliminate_blocks_);
options.type = DENSE_SCHUR;
ContextImpl context;
options.context = &context;
std::unique_ptr<SchurEliminatorBase> eliminator =
SchurEliminatorBase::Create(options);
ASSERT_TRUE(eliminator != nullptr);
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(
BlockSparseMatrixData(*A_), 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(BlockSparseMatrixData(*A_),
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;
ContextImpl context;
options.context = &context;
ImplicitSchurComplement isc(options);
isc.Init(*A_, D, b_.get());
const int num_f_cols = lhs.cols();
const int num_e_cols = num_cols_ - num_f_cols;
Matrix A_dense, E, F, DE, DF;
A_->ToDenseMatrix(&A_dense);
E = A_dense.leftCols(A_->num_cols() - num_f_cols);
F = A_dense.rightCols(num_f_cols);
if (D) {
DE = VectorRef(D, num_e_cols).asDiagonal();
DF = VectorRef(D + num_e_cols, num_f_cols).asDiagonal();
} else {
DE = Matrix::Zero(num_e_cols, num_e_cols);
DF = Matrix::Zero(num_f_cols, num_f_cols);
}
// Z = (block_diagonal(F'F))^-1 F'E (E'E)^-1 E'F
// Here, assuming that block_diagonal(F'F) == diagonal(F'F)
Matrix Z_reference =
(F.transpose() * F + DF).diagonal().asDiagonal().inverse() *
F.transpose() * E * (E.transpose() * E + DE).inverse() * E.transpose() *
F;
for (int i = 0; i < num_f_cols; ++i) {
Vector x(num_f_cols);
x.setZero();
x(i) = 1.0;
Vector y(num_f_cols);
y = lhs * x;
Vector z(num_f_cols);
isc.RightMultiplyAndAccumulate(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();
}
y.setZero();
y = Z_reference * x;
z.setZero();
isc.InversePowerSeriesOperatorRightMultiplyAccumulate(x.data(), z.data());
// The i^th column of operator Z stored implicitly is the same as its
// explicit version.
if ((y - z).norm() > kEpsilon) {
return testing::AssertionFailure()
<< "Explicit and Implicit operators used to approximate the "
"inversion of schur complement via power series expansion "
"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();
}
ContextImpl context_;
int num_rows_;
int num_cols_;
int num_eliminate_blocks_;
std::unique_ptr<BlockSparseMatrix> A_;
std::unique_ptr<double[]> b_;
std::unique_ptr<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(nullptr));
EXPECT_TRUE(TestImplicitSchurComplement(D_.get()));
}
} // namespace ceres::internal