internal/ceres/implicit_schur_complement_test.cc - ceres-solver - Git at Google

 // Ceres Solver - A fast non-linear least squares minimizer
 // Copyright 2023 Google Inc. All rights reserved.
 // http://ceres-solver.org/
 //
 // Redistribution and use in source and binary forms, with or without
 // modification, are permitted provided that the following conditions are met:
 //
 // * Redistributions of source code must retain the above copyright notice,
 //   this list of conditions and the following disclaimer.
 // * Redistributions in binary form must reproduce the above copyright notice,
 //   this list of conditions and the following disclaimer in the documentation
 //   and/or other materials provided with the distribution.
 // * Neither the name of Google Inc. nor the names of its contributors may be
 //   used to endorse or promote products derived from this software without
 //   specific prior written permission.
 //
 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
 // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
 // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
 // CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
 // SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
 // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
 // CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
 // POSSIBILITY OF SUCH DAMAGE.
 //
 // 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 "glog/logging.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);

     CHECK(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);
     CHECK(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
	// Ceres Solver - A fast non-linear least squares minimizer
	// Copyright 2023 Google Inc. All rights reserved.
	// http://ceres-solver.org/
	//
	// Redistribution and use in source and binary forms, with or without
	// modification, are permitted provided that the following conditions are met:
	//
	// * Redistributions of source code must retain the above copyright notice,
	// this list of conditions and the following disclaimer.
	// * Redistributions in binary form must reproduce the above copyright notice,
	// this list of conditions and the following disclaimer in the documentation
	// and/or other materials provided with the distribution.
	// * Neither the name of Google Inc. nor the names of its contributors may be
	// used to endorse or promote products derived from this software without
	// specific prior written permission.
	//
	// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
	// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
	// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
	// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
	// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
	// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
	// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
	// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
	// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
	// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
	// POSSIBILITY OF SUCH DAMAGE.
	//
	// 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 "glog/logging.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);

	CHECK(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);
	CHECK(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