Block oriented fill reducing orderings.
By virtue of the modeling layer in Ceres being block oriented,
all the matrices used by Ceres are also block oriented.
When doing sparse direct factorization of these matrices, the
fill-reducing ordering algorithms can either be run on the
block or the scalar form of these matrices. Running it on the
block form exposes more of the super-nodal structure of the
matrix to the Cholesky factorization routines. This leads to
substantial gains in factorization performance.
This changelist adds support for approximate minimium degree
orderings to be computed on the block structure of the
Schur complement matrix. This affects, SchurComplementSolver
and VisibilityBasedPreconditioner and SparseNormalCholesky
when using SuiteSparse.
A bool, use_block_amd has been added to Solver::Options and
bundle_adjuster.cc has been updated to allow testing with it.
When combined with a multithreaded Schur elimination, speed ups
can be seen quite uniformly across the board. For some problems
this can be dramatic, reducing the factorization time from 70
seconds down to 17 seconds.
Change-Id: I15ebb0afcbc85ada032ec8d179ee3a2f7c8d3e46
diff --git a/examples/bundle_adjuster.cc b/examples/bundle_adjuster.cc
index 718fde9..1216a82 100644
--- a/examples/bundle_adjuster.cc
+++ b/examples/bundle_adjuster.cc
@@ -70,17 +70,20 @@
DEFINE_string(preconditioner_type, "jacobi", "Options are: "
"identity, jacobi, schur_jacobi, cluster_jacobi, "
"cluster_tridiagonal");
+DEFINE_string(sparse_linear_algebra_library, "suitesparse",
+ "Options are: suitesparse and cxsparse");
DEFINE_int32(num_iterations, 5, "Number of iterations");
DEFINE_int32(num_threads, 1, "Number of threads");
DEFINE_double(eta, 1e-2, "Default value for eta. Eta determines the "
"accuracy of each linear solve of the truncated newton step. "
"Changing this parameter can affect solve performance ");
-DEFINE_bool(use_schur_ordering, false, "Use automatic Schur ordering.");
+DEFINE_string(ordering_type, "schur", "Options are: schur, user, natural");
DEFINE_bool(use_quaternions, false, "If true, uses quaternions to represent "
"rotations. If false, angle axis is used");
DEFINE_bool(use_local_parameterization, false, "For quaternions, use a local "
"parameterization.");
DEFINE_bool(robustify, false, "Use a robust loss function");
+DEFINE_bool(use_block_amd, true, "Use a block oriented fill reducing ordering.");
namespace ceres {
namespace examples {
@@ -138,10 +141,31 @@
}
}
+ if (FLAGS_sparse_linear_algebra_library == "suitesparse") {
+ options->sparse_linear_algebra_library = SUITE_SPARSE;
+ } else if (FLAGS_sparse_linear_algebra_library == "cxsparse") {
+ options->sparse_linear_algebra_library = CX_SPARSE;
+ } else {
+ LOG(FATAL) << "Unknown sparse linear algebra library type.";
+ }
+
options->num_linear_solver_threads = FLAGS_num_threads;
}
void SetOrdering(BALProblem* bal_problem, Solver::Options* options) {
+ options->use_block_amd = FLAGS_use_block_amd;
+
+ // Only non-Schur solvers support the natural ordering for this
+ // problem.
+ if (FLAGS_ordering_type == "natural") {
+ if (options->linear_solver_type == SPARSE_SCHUR ||
+ options->linear_solver_type == DENSE_SCHUR ||
+ options->linear_solver_type == ITERATIVE_SCHUR) {
+ LOG(FATAL) << "Natural ordering with Schur type solver does not work.";
+ }
+ return;
+ }
+
// Bundle adjustment problems have a sparsity structure that makes
// them amenable to more specialized and much more efficient
// solution strategies. The SPARSE_SCHUR, DENSE_SCHUR and
@@ -156,7 +180,7 @@
// the right ParameterBlock ordering, or by manually specifying a
// suitable ordering vector and defining
// Options::num_eliminate_blocks.
- if (FLAGS_use_schur_ordering) {
+ if (FLAGS_ordering_type == "schur") {
options->ordering_type = ceres::SCHUR;
return;
}
@@ -272,7 +296,6 @@
BuildProblem(&bal_problem, &problem);
Solver::Options options;
SetSolverOptionsFromFlags(&bal_problem, &options);
-
Solver::Summary summary;
Solve(options, &problem, &summary);
std::cout << summary.FullReport() << "\n";
diff --git a/include/ceres/solver.h b/include/ceres/solver.h
index ed4c9b8..12351ab 100644
--- a/include/ceres/solver.h
+++ b/include/ceres/solver.h
@@ -83,6 +83,12 @@
sparse_linear_algebra_library = CX_SPARSE;
#endif
+#if defined(CERES_NO_SUITESPARSE)
+ use_block_amd = false;
+#else
+ use_block_amd = true;
+#endif
+
preconditioner_type = JACOBI;
num_linear_solver_threads = 1;
num_eliminate_blocks = 0;
@@ -207,6 +213,19 @@
// non-empty.
vector<double*> ordering;
+ // By virtue of the modeling layer in Ceres being block oriented,
+ // all the matrices used by Ceres are also block oriented. When
+ // doing sparse direct factorization of these matrices (for
+ // SPARSE_NORMAL_CHOLESKY, SPARSE_SCHUR and ITERATIVE in
+ // conjunction with CLUSTER_TRIDIAGONAL AND CLUSTER_JACOBI
+ // preconditioners), the fill-reducing ordering algorithms can
+ // either be run on the block or the scalar form of these matrices.
+ // Running it on the block form exposes more of the super-nodal
+ // structure of the matrix to the factorization routines. Setting
+ // this parameter to true runs the ordering algorithms in block
+ // form. Currently this option only makes sense with
+ // sparse_linear_algebra_library = SUITE_SPARSE.
+ bool use_block_amd;
// Minimum number of iterations for which the linear solver should
// run, even if the convergence criterion is satisfied.
diff --git a/internal/ceres/CMakeLists.txt b/internal/ceres/CMakeLists.txt
index 2a0a6af..432346f 100644
--- a/internal/ceres/CMakeLists.txt
+++ b/internal/ceres/CMakeLists.txt
@@ -196,6 +196,7 @@
ENDIF (GFLAGS)
CERES_TEST(schur_ordering)
CERES_TEST(solver_impl)
+ CERES_TEST(suitesparse)
CERES_TEST(symmetric_linear_solver)
CERES_TEST(triplet_sparse_matrix)
CERES_TEST(trust_region_minimizer)
diff --git a/internal/ceres/compressed_row_jacobian_writer.cc b/internal/ceres/compressed_row_jacobian_writer.cc
index aa883b7..912c484 100644
--- a/internal/ceres/compressed_row_jacobian_writer.cc
+++ b/internal/ceres/compressed_row_jacobian_writer.cc
@@ -130,8 +130,24 @@
}
row_pos += num_residuals;
}
-
CHECK_EQ(num_jacobian_nonzeros, rows[total_num_residuals]);
+
+ // Populate the row and column block vectors for use by block
+ // oriented ordering algorithms. This is useful when
+ // Solver::Options::use_block_amd = true.
+ const vector<ParameterBlock*>& parameter_blocks = program_->parameter_blocks();
+ vector<int>& col_blocks = *(jacobian->mutable_col_blocks());
+ col_blocks.resize(parameter_blocks.size());
+ for (int i = 0; i < parameter_blocks.size(); ++i) {
+ col_blocks[i] = parameter_blocks[i]->LocalSize();
+ }
+
+ vector<int>& row_blocks = *(jacobian->mutable_row_blocks());
+ row_blocks.resize(residual_blocks.size());
+ for (int i = 0; i < residual_blocks.size(); ++i) {
+ row_blocks[i] = residual_blocks[i]->NumResiduals();
+ }
+
return jacobian;
}
diff --git a/internal/ceres/compressed_row_sparse_matrix.h b/internal/ceres/compressed_row_sparse_matrix.h
index c44f1db..7fb460a 100644
--- a/internal/ceres/compressed_row_sparse_matrix.h
+++ b/internal/ceres/compressed_row_sparse_matrix.h
@@ -31,11 +31,13 @@
#ifndef CERES_INTERNAL_COMPRESSED_ROW_SPARSE_MATRIX_H_
#define CERES_INTERNAL_COMPRESSED_ROW_SPARSE_MATRIX_H_
+#include <vector>
#include <glog/logging.h>
#include "ceres/sparse_matrix.h"
#include "ceres/triplet_sparse_matrix.h"
#include "ceres/internal/eigen.h"
#include "ceres/internal/macros.h"
+#include "ceres/internal/port.h"
#include "ceres/types.h"
namespace ceres {
@@ -110,6 +112,12 @@
const int* rows() const { return rows_.get(); }
int* mutable_rows() { return rows_.get(); }
+ const vector<int>& row_blocks() const { return row_blocks_; };
+ vector<int>* mutable_row_blocks() { return &row_blocks_; };
+
+ const vector<int>& col_blocks() const { return col_blocks_; };
+ vector<int>* mutable_col_blocks() { return &col_blocks_; };
+
private:
scoped_array<int> cols_;
scoped_array<int> rows_;
@@ -117,9 +125,16 @@
int num_rows_;
int num_cols_;
-
int max_num_nonzeros_;
+ // If the matrix has an underlying block structure, then it can also
+ // carry with it row and column block sizes. This is auxilliary and
+ // optional information for use by algorithms operating on the
+ // matrix. The class itself does not make use of this information in
+ // any way.
+ vector<int> row_blocks_;
+ vector<int> col_blocks_;
+
CERES_DISALLOW_COPY_AND_ASSIGN(CompressedRowSparseMatrix);
};
diff --git a/internal/ceres/linear_solver.h b/internal/ceres/linear_solver.h
index 853abcb..31f8874 100644
--- a/internal/ceres/linear_solver.h
+++ b/internal/ceres/linear_solver.h
@@ -72,6 +72,7 @@
: type(SPARSE_NORMAL_CHOLESKY),
preconditioner_type(JACOBI),
sparse_linear_algebra_library(SUITE_SPARSE),
+ use_block_amd(true),
min_num_iterations(1),
max_num_iterations(1),
num_threads(1),
@@ -88,6 +89,9 @@
SparseLinearAlgebraLibraryType sparse_linear_algebra_library;
+ // See solver.h for explanation of this option.
+ bool use_block_amd;
+
// Number of internal iterations that the solver uses. This
// parameter only makes sense for iterative solvers like CG.
int min_num_iterations;
diff --git a/internal/ceres/schur_complement_solver.cc b/internal/ceres/schur_complement_solver.cc
index b30d6ed..5679a09 100644
--- a/internal/ceres/schur_complement_solver.cc
+++ b/internal/ceres/schur_complement_solver.cc
@@ -94,11 +94,11 @@
const time_t backsubstitute_time = time(NULL);
summary.termination_type = TOLERANCE;
- VLOG(2) << "time (sec) total: " << backsubstitute_time - start_time
- << " init: " << init_time - start_time
- << " eliminate: " << eliminate_time - init_time
- << " solve: " << solve_time - eliminate_time
- << " backsubstitute: " << backsubstitute_time - solve_time;
+ VLOG(2) << "time (sec) total: " << (backsubstitute_time - start_time)
+ << " init: " << (init_time - start_time)
+ << " eliminate: " << (eliminate_time - init_time)
+ << " solve: " << (solve_time - eliminate_time)
+ << " backsubstitute: " << (backsubstitute_time - solve_time);
return summary;
}
@@ -150,15 +150,15 @@
const LinearSolver::Options& options)
: SchurComplementSolver(options) {
#ifndef CERES_NO_SUITESPARSE
- symbolic_factor_ = NULL;
+ factor_ = NULL;
#endif // CERES_NO_SUITESPARSE
}
SparseSchurComplementSolver::~SparseSchurComplementSolver() {
#ifndef CERES_NO_SUITESPARSE
- if (symbolic_factor_ != NULL) {
- ss_.Free(symbolic_factor_);
- symbolic_factor_ = NULL;
+ if (factor_ != NULL) {
+ ss_.Free(factor_);
+ factor_ = NULL;
}
#endif // CERES_NO_SUITESPARSE
}
@@ -171,13 +171,13 @@
const int num_col_blocks = bs->cols.size();
const int num_row_blocks = bs->rows.size();
- vector<int> blocks(num_col_blocks - num_eliminate_blocks, 0);
+ blocks_.resize(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;
+ blocks_[i - num_eliminate_blocks] = bs->cols[i].size;
}
set<pair<int, int> > block_pairs;
- for (int i = 0; i < blocks.size(); ++i) {
+ for (int i = 0; i < blocks_.size(); ++i) {
block_pairs.insert(make_pair(i, i));
}
@@ -230,7 +230,7 @@
}
}
- set_lhs(new BlockRandomAccessSparseMatrix(blocks, block_pairs));
+ set_lhs(new BlockRandomAccessSparseMatrix(blocks_, block_pairs));
set_rhs(new double[lhs()->num_rows()]);
}
@@ -256,7 +256,8 @@
// CHOLMOD's sparse cholesky factorization routines.
bool SparseSchurComplementSolver::SolveReducedLinearSystemUsingSuiteSparse(
double* solution) {
- // Extract the TripletSparseMatrix that is used for actually storing S.
+ const time_t start_time = time(NULL);
+
TripletSparseMatrix* tsm =
const_cast<TripletSparseMatrix*>(
down_cast<const BlockRandomAccessSparseMatrix*>(lhs())->matrix());
@@ -273,17 +274,32 @@
// The matrix is symmetric, and the upper triangular part of the
// matrix contains the values.
cholmod_lhs->stype = 1;
+ const time_t lhs_time = time(NULL);
cholmod_dense* cholmod_rhs =
ss_.CreateDenseVector(const_cast<double*>(rhs()), num_rows, num_rows);
+ const time_t rhs_time = time(NULL);
// Symbolic factorization is computed if we don't already have one handy.
- if (symbolic_factor_ == NULL) {
- symbolic_factor_ = ss_.AnalyzeCholesky(cholmod_lhs);
+ if (factor_ == NULL) {
+ if (options().use_block_amd) {
+ factor_ = ss_.BlockAnalyzeCholesky(cholmod_lhs, blocks_, blocks_);
+ } else {
+ factor_ = ss_.AnalyzeCholesky(cholmod_lhs);
+ }
}
+ if (VLOG_IS_ON(2)) {
+ cholmod_print_common("Symbolic Analysis", ss_.mutable_cc());
+ }
+
+ CHECK_NOTNULL(factor_);
+
+ const time_t symbolic_time = time(NULL);
cholmod_dense* cholmod_solution =
- ss_.SolveCholesky(cholmod_lhs, symbolic_factor_, cholmod_rhs);
+ ss_.SolveCholesky(cholmod_lhs, factor_, cholmod_rhs);
+
+ const time_t solve_time = time(NULL);
ss_.Free(cholmod_lhs);
cholmod_lhs = NULL;
@@ -298,6 +314,13 @@
VectorRef(solution, num_rows)
= VectorRef(static_cast<double*>(cholmod_solution->x), num_rows);
ss_.Free(cholmod_solution);
+ const time_t final_time = time(NULL);
+ VLOG(2) << "time: " << (final_time - start_time)
+ << " lhs : " << (lhs_time - start_time)
+ << " rhs: " << (rhs_time - lhs_time)
+ << " analyze: " << (symbolic_time - rhs_time)
+ << " factor_and_solve: " << (solve_time - symbolic_time)
+ << " cleanup: " << (final_time - solve_time);
return true;
}
#else
diff --git a/internal/ceres/schur_complement_solver.h b/internal/ceres/schur_complement_solver.h
index 0a25ce1..a8f0852 100644
--- a/internal/ceres/schur_complement_solver.h
+++ b/internal/ceres/schur_complement_solver.h
@@ -31,6 +31,8 @@
#ifndef CERES_INTERNAL_SCHUR_COMPLEMENT_SOLVER_H_
#define CERES_INTERNAL_SCHUR_COMPLEMENT_SOLVER_H_
+#include <set>
+#include <utility>
#include "ceres/block_random_access_matrix.h"
#include "ceres/block_sparse_matrix.h"
#include "ceres/block_structure.h"
@@ -158,11 +160,14 @@
bool SolveReducedLinearSystemUsingSuiteSparse(double* solution);
bool SolveReducedLinearSystemUsingCXSparse(double* solution);
+ // Size of the blocks in the Schur complement.
+ vector<int> blocks_;
+
#ifndef CERES_NO_SUITESPARSE
SuiteSparse ss_;
// Symbolic factorization of the reduced linear system. Precomputed
// once and reused in subsequent calls.
- cholmod_factor* symbolic_factor_;
+ cholmod_factor* factor_;
#endif // CERES_NO_SUITESPARSE
CERES_DISALLOW_COPY_AND_ASSIGN(SparseSchurComplementSolver);
};
diff --git a/internal/ceres/solver_impl.cc b/internal/ceres/solver_impl.cc
index 4eb3d1f..fde5811 100644
--- a/internal/ceres/solver_impl.cc
+++ b/internal/ceres/solver_impl.cc
@@ -481,6 +481,7 @@
linear_solver_options.preconditioner_type = options->preconditioner_type;
linear_solver_options.sparse_linear_algebra_library =
options->sparse_linear_algebra_library;
+ linear_solver_options.use_block_amd = options->use_block_amd;
#ifdef CERES_NO_SUITESPARSE
if (linear_solver_options.preconditioner_type == SCHUR_JACOBI) {
diff --git a/internal/ceres/sparse_normal_cholesky_solver.cc b/internal/ceres/sparse_normal_cholesky_solver.cc
index 5a082de..4191f3c 100644
--- a/internal/ceres/sparse_normal_cholesky_solver.cc
+++ b/internal/ceres/sparse_normal_cholesky_solver.cc
@@ -46,8 +46,6 @@
#include "ceres/internal/scoped_ptr.h"
#include "ceres/types.h"
-
-
namespace ceres {
namespace internal {
@@ -55,15 +53,15 @@
const LinearSolver::Options& options)
: options_(options) {
#ifndef CERES_NO_SUITESPARSE
- symbolic_factor_ = NULL;
+ factor_ = NULL;
#endif
}
SparseNormalCholeskySolver::~SparseNormalCholeskySolver() {
#ifndef CERES_NO_SUITESPARSE
- if (symbolic_factor_ != NULL) {
- ss_.Free(symbolic_factor_);
- symbolic_factor_ = NULL;
+ if (factor_ != NULL) {
+ ss_.Free(factor_);
+ factor_ = NULL;
}
#endif
}
@@ -183,13 +181,25 @@
cholmod_dense* rhs = ss_.CreateDenseVector(Atb.data(), num_cols, num_cols);
const time_t init_time = time(NULL);
- if (symbolic_factor_ == NULL) {
- symbolic_factor_ = CHECK_NOTNULL(ss_.AnalyzeCholesky(lhs.get()));
+ if (factor_ == NULL) {
+ if (options_.use_block_amd) {
+ factor_ = ss_.BlockAnalyzeCholesky(lhs.get(),
+ A->col_blocks(),
+ A->row_blocks());
+ } else {
+ factor_ = ss_.AnalyzeCholesky(lhs.get());
+ }
}
+ if (VLOG_IS_ON(2)) {
+ cholmod_print_common("Symbolic Analysis", ss_.mutable_cc());
+ }
+
+ CHECK_NOTNULL(factor_);
+
const time_t symbolic_time = time(NULL);
- cholmod_dense* sol = ss_.SolveCholesky(lhs.get(), symbolic_factor_, rhs);
+ cholmod_dense* sol = ss_.SolveCholesky(lhs.get(), factor_, rhs);
const time_t solve_time = time(NULL);
ss_.Free(rhs);
@@ -209,11 +219,11 @@
}
const time_t cleanup_time = time(NULL);
- VLOG(2) << "time (sec) total: " << cleanup_time - start_time
- << " init: " << init_time - start_time
- << " symbolic: " << symbolic_time - init_time
- << " solve: " << solve_time - symbolic_time
- << " cleanup: " << cleanup_time - solve_time;
+ VLOG(2) << "time (sec) total: " << (cleanup_time - start_time)
+ << " init: " << (init_time - start_time)
+ << " symbolic: " << (symbolic_time - init_time)
+ << " solve: " << (solve_time - symbolic_time)
+ << " cleanup: " << (cleanup_time - solve_time);
return summary;
}
#else
diff --git a/internal/ceres/sparse_normal_cholesky_solver.h b/internal/ceres/sparse_normal_cholesky_solver.h
index e38f39b..6366f86 100644
--- a/internal/ceres/sparse_normal_cholesky_solver.h
+++ b/internal/ceres/sparse_normal_cholesky_solver.h
@@ -77,7 +77,7 @@
#ifndef CERES_NO_SUITESPARSE
SuiteSparse ss_;
// Cached factorization
- cholmod_factor* symbolic_factor_;
+ cholmod_factor* factor_;
#endif // CERES_NO_SUITESPARSE
diff --git a/internal/ceres/suitesparse.cc b/internal/ceres/suitesparse.cc
index 1cf6a74..c02d305 100644
--- a/internal/ceres/suitesparse.cc
+++ b/internal/ceres/suitesparse.cc
@@ -29,15 +29,14 @@
// Author: sameeragarwal@google.com (Sameer Agarwal)
#ifndef CERES_NO_SUITESPARSE
-
#include "ceres/suitesparse.h"
+#include <vector>
#include "cholmod.h"
#include "ceres/compressed_row_sparse_matrix.h"
#include "ceres/triplet_sparse_matrix.h"
namespace ceres {
namespace internal {
-
cholmod_sparse* SuiteSparse::CreateSparseMatrix(TripletSparseMatrix* A) {
cholmod_triplet triplet;
@@ -111,6 +110,13 @@
}
cholmod_factor* SuiteSparse::AnalyzeCholesky(cholmod_sparse* A) {
+ // Cholmod can try multiple re-ordering strategies to find a fill
+ // reducing ordering. Here we just tell it use AMD with automatic
+ // matrix dependence choice of supernodal versus simplicial
+ // factorization.
+ cc_.nmethods = 1;
+ cc_.method[0].ordering = CHOLMOD_AMD;
+ cc_.supernodal = CHOLMOD_AUTO;
cholmod_factor* factor = cholmod_analyze(A, &cc_);
CHECK_EQ(cc_.status, CHOLMOD_OK)
<< "Cholmod symbolic analysis failed " << cc_.status;
@@ -118,6 +124,146 @@
return factor;
}
+cholmod_factor* SuiteSparse::BlockAnalyzeCholesky(
+ cholmod_sparse* A,
+ const vector<int>& row_blocks,
+ const vector<int>& col_blocks) {
+ vector<int> ordering;
+ if (!BlockAMDOrdering(A, row_blocks, col_blocks, &ordering)) {
+ return NULL;
+ }
+ return AnalyzeCholeskyWithUserOrdering(A, ordering);
+}
+
+cholmod_factor* SuiteSparse::AnalyzeCholeskyWithUserOrdering(cholmod_sparse* A,
+ const vector<int>& ordering) {
+ CHECK_EQ(ordering.size(), A->nrow);
+ cc_.nmethods = 1 ;
+ cc_.method[0].ordering = CHOLMOD_GIVEN;
+ cholmod_factor* factor =
+ cholmod_analyze_p(A, const_cast<int*>(&ordering[0]), NULL, 0, &cc_);
+ CHECK_EQ(cc_.status, CHOLMOD_OK)
+ << "Cholmod symbolic analysis failed " << cc_.status;
+ CHECK_NOTNULL(factor);
+ return factor;
+}
+
+bool SuiteSparse::BlockAMDOrdering(const cholmod_sparse* A,
+ const vector<int>& row_blocks,
+ const vector<int>& col_blocks,
+ vector<int>* ordering) {
+ const int num_row_blocks = row_blocks.size();
+ const int num_col_blocks = col_blocks.size();
+
+ // Arrays storing the compressed column structure of the matrix
+ // incoding the block sparsity of A.
+ vector<int> block_cols;
+ vector<int> block_rows;
+
+ ScalarMatrixToBlockMatrix(A,
+ row_blocks,
+ col_blocks,
+ &block_rows,
+ &block_cols);
+
+ cholmod_sparse_struct block_matrix;
+ block_matrix.nrow = num_row_blocks;
+ block_matrix.ncol = num_col_blocks;
+ block_matrix.nzmax = block_rows.size();
+ block_matrix.p = reinterpret_cast<void*>(&block_cols[0]);
+ block_matrix.i = reinterpret_cast<void*>(&block_rows[0]);
+ block_matrix.x = NULL;
+ block_matrix.stype = A->stype;
+ block_matrix.itype = CHOLMOD_INT;
+ block_matrix.xtype = CHOLMOD_PATTERN;
+ block_matrix.dtype = CHOLMOD_DOUBLE;
+ block_matrix.sorted = 1;
+ block_matrix.packed = 1;
+
+ vector<int> block_ordering(num_row_blocks);
+ if (!cholmod_amd(&block_matrix, NULL, 0, &block_ordering[0], &cc_)) {
+ return false;
+ }
+
+ BlockOrderingToScalarOrdering(row_blocks, block_ordering, ordering);
+ return true;
+}
+
+void SuiteSparse::ScalarMatrixToBlockMatrix(const cholmod_sparse* A,
+ const vector<int>& row_blocks,
+ const vector<int>& col_blocks,
+ vector<int>* block_rows,
+ vector<int>* block_cols) {
+ CHECK_NOTNULL(block_rows)->clear();
+ CHECK_NOTNULL(block_cols)->clear();
+ const int num_row_blocks = row_blocks.size();
+ const int num_col_blocks = col_blocks.size();
+
+ vector<int> row_block_starts(num_row_blocks);
+ for (int i = 0, cursor = 0; i < num_row_blocks; ++i) {
+ row_block_starts[i] = cursor;
+ cursor += row_blocks[i];
+ }
+
+ // The reinterpret_cast is needed here because CHOLMOD stores arrays
+ // as void*.
+ const int* scalar_cols = reinterpret_cast<const int*>(A->p);
+ const int* scalar_rows = reinterpret_cast<const int*>(A->i);
+
+ // This loop extracts the block sparsity of the scalar sparse matrix
+ // A. It does so by iterating over the columns, but only considering
+ // the columns corresponding to the first element of each column
+ // block. Within each column, the inner loop iterates over the rows,
+ // and detects the presence of a row block by checking for the
+ // presence of a non-zero entry corresponding to its first element.
+ block_cols->push_back(0);
+ int c = 0;
+ for (int col_block = 0; col_block < num_col_blocks; ++col_block) {
+ int column_size = 0;
+ for (int idx = scalar_cols[c]; idx < scalar_cols[c + 1]; ++idx) {
+ vector<int>::const_iterator it = lower_bound(row_block_starts.begin(),
+ row_block_starts.end(),
+ scalar_rows[idx]);
+ DCHECK(it != row_block_starts.end());
+ // Only consider the first row of each row block.
+ if (*it != scalar_rows[idx]) {
+ continue;
+ }
+
+ block_rows->push_back(it - row_block_starts.begin());
+ ++column_size;
+ }
+ block_cols->push_back(block_cols->back() + column_size);
+ c += col_blocks[col_block];
+ }
+}
+
+void SuiteSparse::BlockOrderingToScalarOrdering(
+ const vector<int>& blocks,
+ const vector<int>& block_ordering,
+ vector<int>* scalar_ordering) {
+ CHECK_EQ(blocks.size(), block_ordering.size());
+ const int num_blocks = blocks.size();
+
+ // block_starts = [0, block1, block1 + block2 ..]
+ vector<int> block_starts(num_blocks);
+ for (int i = 0, cursor = 0; i < num_blocks ; ++i) {
+ block_starts[i] = cursor;
+ cursor += blocks[i];
+ }
+
+ scalar_ordering->resize(block_starts.back() + blocks.back());
+ int cursor = 0;
+ for (int i = 0; i < num_blocks; ++i) {
+ const int block_id = block_ordering[i];
+ const int block_size = blocks[block_id];
+ int block_position = block_starts[block_id];
+ for (int j = 0; j < block_size; ++j) {
+ (*scalar_ordering)[cursor++] = block_position++;
+ }
+ }
+}
+
bool SuiteSparse::Cholesky(cholmod_sparse* A, cholmod_factor* L) {
CHECK_NOTNULL(A);
CHECK_NOTNULL(L);
diff --git a/internal/ceres/suitesparse.h b/internal/ceres/suitesparse.h
index 091e67a..eb691c0 100644
--- a/internal/ceres/suitesparse.h
+++ b/internal/ceres/suitesparse.h
@@ -37,6 +37,7 @@
#include <cstring>
#include <string>
+#include <vector>
#include <glog/logging.h>
#include "cholmod.h"
@@ -105,12 +106,35 @@
cholmod_sdmult(A, 0, alpha_, beta_, x, y, &cc_);
}
- // Analyze the sparsity structure of the matrix A compute the
- // symbolic factorization of A. A is not modified, only the pattern
- // of non-zeros of A is used, the actual numerical values in A are
- // of no consequence. Caller owns the result.
+ // Find an ordering of A or AA' (if A is unsymmetric) that minimizes
+ // the fill-in in the Cholesky factorization of the corresponding
+ // matrix. This is done by using the AMD algorithm.
+ //
+ // Using this ordering, the symbolic Cholesky factorization of A (or
+ // AA') is computed and returned.
+ //
+ // A is not modified, only the pattern of non-zeros of A is used,
+ // the actual numerical values in A are of no consequence.
+ //
+ // Caller owns the result.
cholmod_factor* AnalyzeCholesky(cholmod_sparse* A);
+ cholmod_factor* BlockAnalyzeCholesky(cholmod_sparse* A,
+ const vector<int>& row_blocks,
+ const vector<int>& col_blocks);
+
+ // If A is symmetric, then compute the symbolic Cholesky
+ // factorization of A(ordering, ordering). If A is unsymmetric, then
+ // compute the symbolic factorization of
+ // A(ordering,:) A(ordering,:)'.
+ //
+ // A is not modified, only the pattern of non-zeros of A is used,
+ // the actual numerical values in A are of no consequence.
+ //
+ // Caller owns the result.
+ cholmod_factor* AnalyzeCholeskyWithUserOrdering(cholmod_sparse* A,
+ const vector<int>& ordering);
+
// Use the symbolic factorization in L, to find the numerical
// factorization for the matrix A or AA^T. Return true if
// successful, false otherwise. L contains the numeric factorization
@@ -129,6 +153,56 @@
cholmod_factor* L,
cholmod_dense* b);
+ // By virtue of the modeling layer in Ceres being block oriented,
+ // all the matrices used by Ceres are also block oriented. When
+ // doing sparse direct factorization of these matrices the
+ // fill-reducing ordering algorithms (in particular AMD) can either
+ // be run on the block or the scalar form of these matrices. The two
+ // SuiteSparse::AnalyzeCholesky methods allows the the client to
+ // compute the symbolic factorization of a matrix by either using
+ // AMD on the matrix or a user provided ordering of the rows.
+ //
+ // But since the underlying matrices are block oriented, it is worth
+ // running AMD on just the block structre of these matrices and then
+ // lifting these block orderings to a full scalar ordering. This
+ // preserves the block structure of the permuted matrix, and exposes
+ // more of the super-nodal structure of the matrix to the numerical
+ // factorization routines.
+ //
+ // Find the block oriented AMD ordering of a matrix A, whose row and
+ // column blocks are given by row_blocks, and col_blocks
+ // respectively. The matrix may or may not be symmetric. The entries
+ // of col_blocks do not need to sum to the number of columns in
+ // A. If this is the case, only the first sum(col_blocks) are used
+ // to compute the ordering.
+ bool BlockAMDOrdering(const cholmod_sparse* A,
+ const vector<int>& row_blocks,
+ const vector<int>& col_blocks,
+ vector<int>* ordering);
+
+ // Given a set of blocks and a permutation of these blocks, compute
+ // the corresponding "scalar" ordering, where the scalar ordering of
+ // size sum(blocks).
+ static void BlockOrderingToScalarOrdering(const vector<int>& blocks,
+ const vector<int>& block_ordering,
+ vector<int>* scalar_ordering);
+
+ // Extract the block sparsity pattern of the scalar sparse matrix
+ // A and return it in compressed column form. The compressed column
+ // form is stored in two vectors block_rows, and block_cols, which
+ // correspond to the row and column arrays in a compressed column sparse
+ // matrix.
+ //
+ // If c_ij is the block in the matrix A corresponding to row block i
+ // and column block j, then it is expected that A contains at least
+ // one non-zero entry corresponding to the top left entry of c_ij,
+ // as that entry is used to detect the presence of a non-zero c_ij.
+ static void ScalarMatrixToBlockMatrix(const cholmod_sparse* A,
+ const vector<int>& row_blocks,
+ const vector<int>& col_blocks,
+ vector<int>* block_rows,
+ vector<int>* block_cols);
+
void Free(cholmod_sparse* m) { cholmod_free_sparse(&m, &cc_); }
void Free(cholmod_dense* m) { cholmod_free_dense(&m, &cc_); }
void Free(cholmod_factor* m) { cholmod_free_factor(&m, &cc_); }
diff --git a/internal/ceres/suitesparse_test.cc b/internal/ceres/suitesparse_test.cc
new file mode 100644
index 0000000..9211788
--- /dev/null
+++ b/internal/ceres/suitesparse_test.cc
@@ -0,0 +1,194 @@
+// Ceres Solver - A fast non-linear least squares minimizer
+// Copyright 2012 Google Inc. All rights reserved.
+// http://code.google.com/p/ceres-solver/
+//
+// 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 <algorithm>
+#include <glog/logging.h>
+#include "gtest/gtest.h"
+#include "ceres/suitesparse.h"
+#include "ceres/triplet_sparse_matrix.h"
+#include "ceres/internal/port.h"
+
+namespace ceres {
+namespace internal {
+
+TEST(SuiteSparse, BlockPermutationToScalarPermutation) {
+ vector<int> blocks;
+ // Block structure
+ // 0 --1- ---2--- ---3--- 4
+ // [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
+ blocks.push_back(1);
+ blocks.push_back(2);
+ blocks.push_back(3);
+ blocks.push_back(3);
+ blocks.push_back(1);
+
+ // Block ordering
+ // [1, 0, 2, 4, 5]
+ vector<int> block_ordering;
+ block_ordering.push_back(1);
+ block_ordering.push_back(0);
+ block_ordering.push_back(2);
+ block_ordering.push_back(4);
+ block_ordering.push_back(3);
+
+ // Expected ordering
+ // [1, 2, 0, 3, 4, 5, 9, 6, 7, 8]
+ vector<int> expected_scalar_ordering;
+ expected_scalar_ordering.push_back(1);
+ expected_scalar_ordering.push_back(2);
+ expected_scalar_ordering.push_back(0);
+ expected_scalar_ordering.push_back(3);
+ expected_scalar_ordering.push_back(4);
+ expected_scalar_ordering.push_back(5);
+ expected_scalar_ordering.push_back(9);
+ expected_scalar_ordering.push_back(6);
+ expected_scalar_ordering.push_back(7);
+ expected_scalar_ordering.push_back(8);
+
+ vector<int> scalar_ordering;
+ SuiteSparse::BlockOrderingToScalarOrdering(blocks,
+ block_ordering,
+ &scalar_ordering);
+ EXPECT_EQ(scalar_ordering.size(), expected_scalar_ordering.size());
+ for (int i = 0; i < expected_scalar_ordering.size(); ++i) {
+ EXPECT_EQ(scalar_ordering[i], expected_scalar_ordering[i]);
+ }
+}
+
+// Helper function to fill the sparsity pattern of a TripletSparseMatrix.
+int FillBlock(const vector<int>& row_blocks,
+ const vector<int>& col_blocks,
+ const int row_block_id,
+ const int col_block_id,
+ int* rows,
+ int* cols) {
+ int row_pos = 0;
+ for (int i = 0; i < row_block_id; ++i) {
+ row_pos += row_blocks[i];
+ }
+
+ int col_pos = 0;
+ for (int i = 0; i < col_block_id; ++i) {
+ col_pos += col_blocks[i];
+ }
+
+ int offset = 0;
+ for (int r = 0; r < row_blocks[row_block_id]; ++r) {
+ for (int c = 0; c < col_blocks[col_block_id]; ++c, ++offset) {
+ rows[offset] = row_pos + r;
+ cols[offset] = col_pos + c;
+ }
+ }
+ return offset;
+}
+
+TEST(SuiteSparse, ScalarMatrixToBlockMatrix) {
+ // Block sparsity.
+ //
+ // [1 2 3 2]
+ // [1] x x
+ // [2] x x
+ // [1] x x
+ // num_nonzeros = 1 + 3 + 4 + 4 + 1 + 2 = 15
+
+ vector<int> col_blocks;
+ col_blocks.push_back(1);
+ col_blocks.push_back(2);
+ col_blocks.push_back(3);
+ col_blocks.push_back(2);
+
+ vector<int> row_blocks;
+ row_blocks.push_back(1);
+ row_blocks.push_back(2);
+ row_blocks.push_back(1);
+
+ TripletSparseMatrix tsm(4, 8, 15);
+ int* rows = tsm.mutable_rows();
+ int* cols = tsm.mutable_cols();
+ fill(tsm.mutable_values(), tsm.mutable_values() + 15, 1.0);
+ int offset = 0;
+
+#define CERES_TEST_FILL_BLOCK(r, c) \
+ offset += FillBlock(row_blocks, col_blocks, \
+ row_block_id, col_block_id, \
+ rows + offset, cols + offset);
+
+ CERES_TEST_FILL_BLOCK(0, 0);
+ CERES_TEST_FILL_BLOCK(2, 0);
+ CERES_TEST_FILL_BLOCK(1, 1);
+ CERES_TEST_FILL_BLOCK(2, 1);
+ CERES_TEST_FILL_BLOCK(0, 2);
+ CERES_TEST_FILL_BLOCK(1, 3);
+#undef CERES_TEST_FILL_BLOCK
+
+ tsm.set_num_nonzeros(offset);
+
+ SuiteSparse ss;
+ scoped_ptr<cholmod_sparse> ccsm(ss.CreateSparseMatrix(&tsm));
+
+ vector<int> expected_block_rows;
+ expected_block_rows.push_back(0);
+ expected_block_rows.push_back(2);
+ expected_block_rows.push_back(1);
+ expected_block_rows.push_back(2);
+ expected_block_rows.push_back(0);
+ expected_block_rows.push_back(1);
+
+ vector<int> expected_block_cols;
+ expected_block_cols.push_back(0);
+ expected_block_cols.push_back(2);
+ expected_block_cols.push_back(4);
+ expected_block_cols.push_back(5);
+ expected_block_cols.push_back(6);
+
+ vector<int> block_rows;
+ vector<int> block_cols;
+ SuiteSparse::ScalarMatrixToBlockMatrix(ccsm.get(),
+ row_blocks,
+ col_blocks,
+ &block_rows,
+ &block_cols);
+
+ EXPECT_EQ(block_cols.size(), expected_block_cols.size());
+ EXPECT_EQ(block_rows.size(), expected_block_rows.size());
+
+ for (int i = 0; i < expected_block_cols.size(); ++i) {
+ EXPECT_EQ(block_cols[i], expected_block_cols[i]);
+ }
+
+ for (int i = 0; i < expected_block_rows.size(); ++i) {
+ EXPECT_EQ(block_rows[i], expected_block_rows[i]);
+ }
+
+ ss.Free(ccsm.release());
+}
+
+} // namespace internal
+} // namespace ceres
diff --git a/internal/ceres/unsymmetric_linear_solver_test.cc b/internal/ceres/unsymmetric_linear_solver_test.cc
index a7ee23e..35cec80 100644
--- a/internal/ceres/unsymmetric_linear_solver_test.cc
+++ b/internal/ceres/unsymmetric_linear_solver_test.cc
@@ -62,6 +62,7 @@
LinearSolver::Options options;
options.type = linear_solver_type;
options.sparse_linear_algebra_library = sparse_linear_algebra_library;
+ options.use_block_amd = false;
scoped_ptr<LinearSolver> solver(LinearSolver::Create(options));
LinearSolver::PerSolveOptions per_solve_options;
diff --git a/internal/ceres/visibility_based_preconditioner.cc b/internal/ceres/visibility_based_preconditioner.cc
index a930595..5a01006 100644
--- a/internal/ceres/visibility_based_preconditioner.cc
+++ b/internal/ceres/visibility_based_preconditioner.cc
@@ -33,7 +33,6 @@
#include <algorithm>
#include <functional>
#include <iterator>
-#include <numeric>
#include <set>
#include <utility>
#include <vector>
@@ -447,12 +446,21 @@
// matrix contains the values.
lhs->stype = 1;
- // Symbolic factorization is computed if we don't already have one
- // handy.
+ // Symbolic factorization is computed if we don't already have one handy.
if (factor_ == NULL) {
- factor_ = ss_.AnalyzeCholesky(lhs);
+ if (options_.use_block_amd) {
+ factor_ = ss_.BlockAnalyzeCholesky(lhs, block_size_, block_size_);
+ } else {
+ factor_ = ss_.AnalyzeCholesky(lhs);
+ }
}
+ if (VLOG_IS_ON(2)) {
+ cholmod_print_common("Symbolic Analysis", ss_.mutable_cc());
+ }
+
+ CHECK_NOTNULL(factor_);
+
bool status = ss_.Cholesky(lhs, factor_);
ss_.Free(lhs);
return status;
