internal/ceres/reorder_program.cc

// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 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.
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//   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
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// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
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//
// Author: sameeragarwal@google.com (Sameer Agarwal)

#include "ceres/reorder_program.h"

#include <algorithm>
#include <memory>
#include <numeric>
#include <vector>

#include "ceres/cxsparse.h"
#include "ceres/internal/port.h"
#include "ceres/ordered_groups.h"
#include "ceres/parameter_block.h"
#include "ceres/parameter_block_ordering.h"
#include "ceres/problem_impl.h"
#include "ceres/program.h"
#include "ceres/residual_block.h"
#include "ceres/solver.h"
#include "ceres/suitesparse.h"
#include "ceres/triplet_sparse_matrix.h"
#include "ceres/types.h"
#include "Eigen/SparseCore"

#ifdef CERES_USE_EIGEN_SPARSE
#include "Eigen/OrderingMethods"
#endif

#include "glog/logging.h"

namespace ceres {
namespace internal {

using std::map;
using std::set;
using std::string;
using std::vector;

namespace {

// Find the minimum index of any parameter block to the given
// residual.  Parameter blocks that have indices greater than
// size_of_first_elimination_group are considered to have an index
// equal to size_of_first_elimination_group.
static int MinParameterBlock(const ResidualBlock* residual_block,
                             int size_of_first_elimination_group) {
  int min_parameter_block_position = size_of_first_elimination_group;
  for (int i = 0; i < residual_block->NumParameterBlocks(); ++i) {
    ParameterBlock* parameter_block = residual_block->parameter_blocks()[i];
    if (!parameter_block->IsConstant()) {
      CHECK_NE(parameter_block->index(), -1)
          << "Did you forget to call Program::SetParameterOffsetsAndIndex()? "
          << "This is a Ceres bug; please contact the developers!";
      min_parameter_block_position = std::min(parameter_block->index(),
                                              min_parameter_block_position);
    }
  }
  return min_parameter_block_position;
}

#if EIGEN_VERSION_AT_LEAST(3, 2, 2) && defined(CERES_USE_EIGEN_SPARSE)
Eigen::SparseMatrix<int> CreateBlockJacobian(
    const TripletSparseMatrix& block_jacobian_transpose) {
  typedef Eigen::SparseMatrix<int> SparseMatrix;
  typedef Eigen::Triplet<int> Triplet;

  const int* rows = block_jacobian_transpose.rows();
  const int* cols = block_jacobian_transpose.cols();
  int num_nonzeros = block_jacobian_transpose.num_nonzeros();
  vector<Triplet> triplets;
  triplets.reserve(num_nonzeros);
  for (int i = 0; i < num_nonzeros; ++i) {
    triplets.push_back(Triplet(cols[i], rows[i], 1));
  }

  SparseMatrix block_jacobian(block_jacobian_transpose.num_cols(),
                              block_jacobian_transpose.num_rows());
  block_jacobian.setFromTriplets(triplets.begin(), triplets.end());
  return block_jacobian;
}
#endif

void OrderingForSparseNormalCholeskyUsingSuiteSparse(
    const TripletSparseMatrix& tsm_block_jacobian_transpose,
    const vector<ParameterBlock*>& parameter_blocks,
    const ParameterBlockOrdering& parameter_block_ordering,
    int* ordering) {
#ifdef CERES_NO_SUITESPARSE
  LOG(FATAL) << "Congratulations, you found a Ceres bug! "
             << "Please report this error to the developers.";
#else
  SuiteSparse ss;
  cholmod_sparse* block_jacobian_transpose =
      ss.CreateSparseMatrix(
          const_cast<TripletSparseMatrix*>(&tsm_block_jacobian_transpose));

  // No CAMD or the user did not supply a useful ordering, then just
  // use regular AMD.
  if (parameter_block_ordering.NumGroups() <= 1 ||
      !SuiteSparse::IsConstrainedApproximateMinimumDegreeOrderingAvailable()) {
    ss.ApproximateMinimumDegreeOrdering(block_jacobian_transpose, &ordering[0]);
  } else {
    vector<int> constraints;
    for (int i = 0; i < parameter_blocks.size(); ++i) {
      constraints.push_back(
          parameter_block_ordering.GroupId(
              parameter_blocks[i]->mutable_user_state()));
    }

    // Renumber the entries of constraints to be contiguous integers
    // as CAMD requires that the group ids be in the range [0,
    // parameter_blocks.size() - 1].
    MapValuesToContiguousRange(constraints.size(), &constraints[0]);
    ss.ConstrainedApproximateMinimumDegreeOrdering(block_jacobian_transpose,
                                                   &constraints[0],
                                                   ordering);
  }

  VLOG(2) << "Block ordering stats: "
          << " flops: " << ss.mutable_cc()->fl
          << " lnz  : " << ss.mutable_cc()->lnz
          << " anz  : " << ss.mutable_cc()->anz;

  ss.Free(block_jacobian_transpose);
#endif  // CERES_NO_SUITESPARSE
}

void OrderingForSparseNormalCholeskyUsingCXSparse(
    const TripletSparseMatrix& tsm_block_jacobian_transpose,
    int* ordering) {
#ifdef CERES_NO_CXSPARSE
  LOG(FATAL) << "Congratulations, you found a Ceres bug! "
             << "Please report this error to the developers.";
#else  // CERES_NO_CXSPARSE
  // CXSparse works with J'J instead of J'. So compute the block
  // sparsity for J'J and compute an approximate minimum degree
  // ordering.
  CXSparse cxsparse;
  cs_di* block_jacobian_transpose;
  block_jacobian_transpose =
      cxsparse.CreateSparseMatrix(
            const_cast<TripletSparseMatrix*>(&tsm_block_jacobian_transpose));
  cs_di* block_jacobian = cxsparse.TransposeMatrix(block_jacobian_transpose);
  cs_di* block_hessian =
      cxsparse.MatrixMatrixMultiply(block_jacobian_transpose, block_jacobian);
  cxsparse.Free(block_jacobian);
  cxsparse.Free(block_jacobian_transpose);

  cxsparse.ApproximateMinimumDegreeOrdering(block_hessian, ordering);
  cxsparse.Free(block_hessian);
#endif  // CERES_NO_CXSPARSE
}


#if EIGEN_VERSION_AT_LEAST(3, 2, 2)
void OrderingForSparseNormalCholeskyUsingEigenSparse(
    const TripletSparseMatrix& tsm_block_jacobian_transpose,
    int* ordering) {
#ifndef CERES_USE_EIGEN_SPARSE
  LOG(FATAL) <<
      "SPARSE_NORMAL_CHOLESKY cannot be used with EIGEN_SPARSE "
      "because Ceres was not built with support for "
      "Eigen's SimplicialLDLT decomposition. "
      "This requires enabling building with -DEIGENSPARSE=ON.";
#else

  // This conversion from a TripletSparseMatrix to a Eigen::Triplet
  // matrix is unfortunate, but unavoidable for now. It is not a
  // significant performance penalty in the grand scheme of
  // things. The right thing to do here would be to get a compressed
  // row sparse matrix representation of the jacobian and go from
  // there. But that is a project for another day.
  typedef Eigen::SparseMatrix<int> SparseMatrix;

  const SparseMatrix block_jacobian =
      CreateBlockJacobian(tsm_block_jacobian_transpose);
  const SparseMatrix block_hessian =
      block_jacobian.transpose() * block_jacobian;

  Eigen::AMDOrdering<int> amd_ordering;
  Eigen::PermutationMatrix<Eigen::Dynamic, Eigen::Dynamic, int> perm;
  amd_ordering(block_hessian, perm);
  for (int i = 0; i < block_hessian.rows(); ++i) {
    ordering[i] = perm.indices()[i];
  }
#endif  // CERES_USE_EIGEN_SPARSE
}
#endif

}  // namespace

bool ApplyOrdering(const ProblemImpl::ParameterMap& parameter_map,
                   const ParameterBlockOrdering& ordering,
                   Program* program,
                   string* error) {
  const int num_parameter_blocks =  program->NumParameterBlocks();
  if (ordering.NumElements() != num_parameter_blocks) {
    *error = StringPrintf("User specified ordering does not have the same "
                          "number of parameters as the problem. The problem"
                          "has %d blocks while the ordering has %d blocks.",
                          num_parameter_blocks,
                          ordering.NumElements());
    return false;
  }

  vector<ParameterBlock*>* parameter_blocks =
      program->mutable_parameter_blocks();
  parameter_blocks->clear();

  const map<int, set<double*>>& groups = ordering.group_to_elements();
  for (const auto& p : groups) {
    const set<double*>& group = p.second;
    for (double* parameter_block_ptr : group) {
      auto it = parameter_map.find(parameter_block_ptr);
      if (it == parameter_map.end()) {
        *error = StringPrintf("User specified ordering contains a pointer "
                              "to a double that is not a parameter block in "
                              "the problem. The invalid double is in group: %d",
                              p.first);
        return false;
      }
      parameter_blocks->push_back(it->second);
    }
  }
  return true;
}

bool LexicographicallyOrderResidualBlocks(
    const int size_of_first_elimination_group,
    Program* program,
    string* error) {
  CHECK_GE(size_of_first_elimination_group, 1)
      << "Congratulations, you found a Ceres bug! Please report this error "
      << "to the developers.";

  // Create a histogram of the number of residuals for each E block. There is an
  // extra bucket at the end to catch all non-eliminated F blocks.
  vector<int> residual_blocks_per_e_block(size_of_first_elimination_group + 1);
  vector<ResidualBlock*>* residual_blocks = program->mutable_residual_blocks();
  vector<int> min_position_per_residual(residual_blocks->size());
  for (int i = 0; i < residual_blocks->size(); ++i) {
    ResidualBlock* residual_block = (*residual_blocks)[i];
    int position = MinParameterBlock(residual_block,
                                     size_of_first_elimination_group);
    min_position_per_residual[i] = position;
    DCHECK_LE(position, size_of_first_elimination_group);
    residual_blocks_per_e_block[position]++;
  }

  // Run a cumulative sum on the histogram, to obtain offsets to the start of
  // each histogram bucket (where each bucket is for the residuals for that
  // E-block).
  vector<int> offsets(size_of_first_elimination_group + 1);
  std::partial_sum(residual_blocks_per_e_block.begin(),
                   residual_blocks_per_e_block.end(),
                   offsets.begin());
  CHECK_EQ(offsets.back(), residual_blocks->size())
      << "Congratulations, you found a Ceres bug! Please report this error "
      << "to the developers.";

  CHECK(find(residual_blocks_per_e_block.begin(),
             residual_blocks_per_e_block.end() - 1, 0) !=
        residual_blocks_per_e_block.end())
      << "Congratulations, you found a Ceres bug! Please report this error "
      << "to the developers.";

  // Fill in each bucket with the residual blocks for its corresponding E block.
  // Each bucket is individually filled from the back of the bucket to the front
  // of the bucket. The filling order among the buckets is dictated by the
  // residual blocks. This loop uses the offsets as counters; subtracting one
  // from each offset as a residual block is placed in the bucket. When the
  // filling is finished, the offset pointerts should have shifted down one
  // entry (this is verified below).
  vector<ResidualBlock*> reordered_residual_blocks(
      (*residual_blocks).size(), static_cast<ResidualBlock*>(NULL));
  for (int i = 0; i < residual_blocks->size(); ++i) {
    int bucket = min_position_per_residual[i];

    // Decrement the cursor, which should now point at the next empty position.
    offsets[bucket]--;

    // Sanity.
    CHECK(reordered_residual_blocks[offsets[bucket]] == NULL)
        << "Congratulations, you found a Ceres bug! Please report this error "
        << "to the developers.";

    reordered_residual_blocks[offsets[bucket]] = (*residual_blocks)[i];
  }

  // Sanity check #1: The difference in bucket offsets should match the
  // histogram sizes.
  for (int i = 0; i < size_of_first_elimination_group; ++i) {
    CHECK_EQ(residual_blocks_per_e_block[i], offsets[i + 1] - offsets[i])
        << "Congratulations, you found a Ceres bug! Please report this error "
        << "to the developers.";
  }
  // Sanity check #2: No NULL's left behind.
  for (int i = 0; i < reordered_residual_blocks.size(); ++i) {
    CHECK(reordered_residual_blocks[i] != NULL)
        << "Congratulations, you found a Ceres bug! Please report this error "
        << "to the developers.";
  }

  // Now that the residuals are collected by E block, swap them in place.
  swap(*program->mutable_residual_blocks(), reordered_residual_blocks);
  return true;
}

// Pre-order the columns corresponding to the schur complement if
// possible.
void MaybeReorderSchurComplementColumnsUsingSuiteSparse(
    const ParameterBlockOrdering& parameter_block_ordering,
    Program* program) {
#ifndef CERES_NO_SUITESPARSE
  SuiteSparse ss;
  if (!SuiteSparse::IsConstrainedApproximateMinimumDegreeOrderingAvailable()) {
    return;
  }

  vector<int> constraints;
  vector<ParameterBlock*>& parameter_blocks =
      *(program->mutable_parameter_blocks());

  for (int i = 0; i < parameter_blocks.size(); ++i) {
    constraints.push_back(
        parameter_block_ordering.GroupId(
            parameter_blocks[i]->mutable_user_state()));
  }

  // Renumber the entries of constraints to be contiguous integers as
  // CAMD requires that the group ids be in the range [0,
  // parameter_blocks.size() - 1].
  MapValuesToContiguousRange(constraints.size(), &constraints[0]);

  // Compute a block sparse presentation of J'.
  std::unique_ptr<TripletSparseMatrix> tsm_block_jacobian_transpose(
      program->CreateJacobianBlockSparsityTranspose());

  cholmod_sparse* block_jacobian_transpose =
      ss.CreateSparseMatrix(tsm_block_jacobian_transpose.get());

  vector<int> ordering(parameter_blocks.size(), 0);
  ss.ConstrainedApproximateMinimumDegreeOrdering(block_jacobian_transpose,
                                                 &constraints[0],
                                                 &ordering[0]);
  ss.Free(block_jacobian_transpose);

  const vector<ParameterBlock*> parameter_blocks_copy(parameter_blocks);
  for (int i = 0; i < program->NumParameterBlocks(); ++i) {
    parameter_blocks[i] = parameter_blocks_copy[ordering[i]];
  }

  program->SetParameterOffsetsAndIndex();
#endif
}

void MaybeReorderSchurComplementColumnsUsingEigen(
    const int size_of_first_elimination_group,
    const ProblemImpl::ParameterMap& parameter_map,
    Program* program) {
#if !EIGEN_VERSION_AT_LEAST(3, 2, 2) || !defined(CERES_USE_EIGEN_SPARSE)
  return;
#else

  std::unique_ptr<TripletSparseMatrix> tsm_block_jacobian_transpose(
      program->CreateJacobianBlockSparsityTranspose());

  typedef Eigen::SparseMatrix<int> SparseMatrix;
  const SparseMatrix block_jacobian =
      CreateBlockJacobian(*tsm_block_jacobian_transpose);
  const int num_rows = block_jacobian.rows();
  const int num_cols = block_jacobian.cols();

  // Vertically partition the jacobian in parameter blocks of type E
  // and F.
  const SparseMatrix E =
      block_jacobian.block(0,
                           0,
                           num_rows,
                           size_of_first_elimination_group);
  const SparseMatrix F =
      block_jacobian.block(0,
                           size_of_first_elimination_group,
                           num_rows,
                           num_cols - size_of_first_elimination_group);

  // Block sparsity pattern of the schur complement.
  const SparseMatrix block_schur_complement =
      F.transpose() * F - F.transpose() * E * E.transpose() * F;

  Eigen::AMDOrdering<int> amd_ordering;
  Eigen::PermutationMatrix<Eigen::Dynamic, Eigen::Dynamic, int> perm;
  amd_ordering(block_schur_complement, perm);

  const vector<ParameterBlock*>& parameter_blocks = program->parameter_blocks();
  vector<ParameterBlock*> ordering(num_cols);

  // The ordering of the first size_of_first_elimination_group does
  // not matter, so we preserve the existing ordering.
  for (int i = 0; i < size_of_first_elimination_group; ++i) {
    ordering[i] = parameter_blocks[i];
  }

  // For the rest of the blocks, use the ordering computed using AMD.
  for (int i = 0; i < block_schur_complement.cols(); ++i) {
    ordering[size_of_first_elimination_group + i] =
        parameter_blocks[size_of_first_elimination_group + perm.indices()[i]];
  }

  swap(*program->mutable_parameter_blocks(), ordering);
  program->SetParameterOffsetsAndIndex();
#endif
}

bool ReorderProgramForSchurTypeLinearSolver(
    const LinearSolverType linear_solver_type,
    const SparseLinearAlgebraLibraryType sparse_linear_algebra_library_type,
    const ProblemImpl::ParameterMap& parameter_map,
    ParameterBlockOrdering* parameter_block_ordering,
    Program* program,
    string* error) {
  if (parameter_block_ordering->NumElements() !=
      program->NumParameterBlocks()) {
    *error = StringPrintf(
        "The program has %d parameter blocks, but the parameter block "
        "ordering has %d parameter blocks.",
        program->NumParameterBlocks(),
        parameter_block_ordering->NumElements());
    return false;
  }

  if (parameter_block_ordering->NumGroups() == 1) {
    // If the user supplied an parameter_block_ordering with just one
    // group, it is equivalent to the user supplying NULL as an
    // parameter_block_ordering. Ceres is completely free to choose the
    // parameter block ordering as it sees fit. For Schur type solvers,
    // this means that the user wishes for Ceres to identify the
    // e_blocks, which we do by computing a maximal independent set.
    vector<ParameterBlock*> schur_ordering;
    const int size_of_first_elimination_group =
        ComputeStableSchurOrdering(*program, &schur_ordering);

    CHECK_EQ(schur_ordering.size(), program->NumParameterBlocks())
        << "Congratulations, you found a Ceres bug! Please report this error "
        << "to the developers.";

    // Update the parameter_block_ordering object.
    for (int i = 0; i < schur_ordering.size(); ++i) {
      double* parameter_block = schur_ordering[i]->mutable_user_state();
      const int group_id = (i < size_of_first_elimination_group) ? 0 : 1;
      parameter_block_ordering->AddElementToGroup(parameter_block, group_id);
    }

    // We could call ApplyOrdering but this is cheaper and
    // simpler.
    swap(*program->mutable_parameter_blocks(), schur_ordering);
  } else {
    // The user provided an ordering with more than one elimination
    // group.

    // Verify that the first elimination group is an independent set.
    const set<double*>& first_elimination_group =
        parameter_block_ordering
        ->group_to_elements()
        .begin()
        ->second;
    if (!program->IsParameterBlockSetIndependent(first_elimination_group)) {
      *error =
          StringPrintf("The first elimination group in the parameter block "
                       "ordering of size %zd is not an independent set",
                       first_elimination_group.size());
      return false;
    }

    if (!ApplyOrdering(parameter_map,
                       *parameter_block_ordering,
                       program,
                       error)) {
      return false;
    }
  }

  program->SetParameterOffsetsAndIndex();

  const int size_of_first_elimination_group =
      parameter_block_ordering->group_to_elements().begin()->second.size();

  if (linear_solver_type == SPARSE_SCHUR) {
    if (sparse_linear_algebra_library_type == SUITE_SPARSE) {
      MaybeReorderSchurComplementColumnsUsingSuiteSparse(
          *parameter_block_ordering,
          program);
    } else if (sparse_linear_algebra_library_type == EIGEN_SPARSE) {
      MaybeReorderSchurComplementColumnsUsingEigen(
          size_of_first_elimination_group,
          parameter_map,
          program);
    }
  }

  // Schur type solvers also require that their residual blocks be
  // lexicographically ordered.
  if (!LexicographicallyOrderResidualBlocks(size_of_first_elimination_group,
                                            program,
                                            error)) {
    return false;
  }

  return true;
}

bool ReorderProgramForSparseNormalCholesky(
    const SparseLinearAlgebraLibraryType sparse_linear_algebra_library_type,
    const ParameterBlockOrdering& parameter_block_ordering,
    Program* program,
    string* error) {
  if (parameter_block_ordering.NumElements() != program->NumParameterBlocks()) {
    *error = StringPrintf(
        "The program has %d parameter blocks, but the parameter block "
        "ordering has %d parameter blocks.",
        program->NumParameterBlocks(),
        parameter_block_ordering.NumElements());
    return false;
  }

  // Compute a block sparse presentation of J'.
  std::unique_ptr<TripletSparseMatrix> tsm_block_jacobian_transpose(
      program->CreateJacobianBlockSparsityTranspose());

  vector<int> ordering(program->NumParameterBlocks(), 0);
  vector<ParameterBlock*>& parameter_blocks =
      *(program->mutable_parameter_blocks());

  if (sparse_linear_algebra_library_type == SUITE_SPARSE) {
    OrderingForSparseNormalCholeskyUsingSuiteSparse(
        *tsm_block_jacobian_transpose,
        parameter_blocks,
        parameter_block_ordering,
        &ordering[0]);
  } else if (sparse_linear_algebra_library_type == CX_SPARSE) {
    OrderingForSparseNormalCholeskyUsingCXSparse(
        *tsm_block_jacobian_transpose,
        &ordering[0]);
  } else if (sparse_linear_algebra_library_type == ACCELERATE_SPARSE) {
    // Accelerate does not provide a function to perform reordering without
    // performing a full symbolic factorisation.  As such, we have nothing
    // to gain from trying to reorder the problem here, as it will happen
    // in AppleAccelerateCholesky::Factorize() (once) and reordering here
    // would involve performing two symbolic factorisations instead of one
    // which would have a negative overall impact on performance.
    return true;

  } else if (sparse_linear_algebra_library_type == EIGEN_SPARSE) {
#if EIGEN_VERSION_AT_LEAST(3, 2, 2)
       OrderingForSparseNormalCholeskyUsingEigenSparse(
        *tsm_block_jacobian_transpose,
        &ordering[0]);
#else
    // For Eigen versions less than 3.2.2, there is nothing to do as
    // older versions of Eigen do not expose a method for doing
    // symbolic analysis on pre-ordered matrices, so a block
    // pre-ordering is a bit pointless.

    return true;
#endif
  }

  // Apply ordering.
  const vector<ParameterBlock*> parameter_blocks_copy(parameter_blocks);
  for (int i = 0; i < program->NumParameterBlocks(); ++i) {
    parameter_blocks[i] = parameter_blocks_copy[ordering[i]];
  }

  program->SetParameterOffsetsAndIndex();
  return true;
}

}  // namespace internal
}  // namespace ceres