Add a dense Cholesky factorization based linear solver.

For problems with a small number of variables, but a large
number of residuals, it is sometimes beneficial to use the
Cholesky factorization on the normal equations, instead of
the dense QR factorization of the Jacobian, even though it
is numerically the better thing to do.

Change-Id: I3506b006195754018deec964e6e190b7e8c9ac8f

examples/bundle_adjuster.cc

@@ -77,8 +77,8 @@ 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_string(solver_type, "sparse_schur", "Options are: "
-              "sparse_schur, dense_schur, iterative_schur, cholesky, "
-              "dense_qr, and conjugate_gradients");
+              "sparse_schur, dense_schur, iterative_schur, sparse_cholesky, "
+              "dense_qr, dense_cholesky and conjugate_gradients");
 DEFINE_string(preconditioner_type, "jacobi", "Options are: "
               "identity, jacobi, schur_jacobi, cluster_jacobi, "
               "cluster_tridiagonal");
@@ -116,7 +116,7 @@ void SetLinearSolver(Solver::Options* options) {
     options->linear_solver_type = ceres::DENSE_SCHUR;
   } else if (FLAGS_solver_type == "iterative_schur") {
     options->linear_solver_type = ceres::ITERATIVE_SCHUR;
-  } else if (FLAGS_solver_type == "cholesky") {
+  } else if (FLAGS_solver_type == "sparse_cholesky") {
     options->linear_solver_type = ceres::SPARSE_NORMAL_CHOLESKY;
   } else if (FLAGS_solver_type == "cgnr") {
     options->linear_solver_type = ceres::CGNR;
@@ -126,6 +126,12 @@ void SetLinearSolver(Solver::Options* options) {
     // to store even the smallest of the bundle adjustment jacobian
     // arrays
     options->linear_solver_type = ceres::DENSE_QR;
+  } else if (FLAGS_solver_type == "dense_cholesky") {
+    // DENSE_NORMAL_CHOLESKY is included here for completeness, but
+    // actually using this option is a bad idea due to the amount of
+    // memory needed to store even the smallest of the bundle
+    // adjustment jacobian arrays
+    options->linear_solver_type = ceres::DENSE_NORMAL_CHOLESKY;
   } else {
     LOG(FATAL) << "Unknown ceres solver type: "
                << FLAGS_solver_type;

include/ceres/types.h

@@ -59,14 +59,18 @@ enum LinearSolverType {
   // normal equations A'A x = A'b. They are direct solvers and do not
   // assume any special problem structure.
 
-  // Solve the normal equations using a sparse cholesky solver; based
-  // on CHOLMOD.
-  SPARSE_NORMAL_CHOLESKY,
+  // Solve the normal equations using a dense Cholesky solver; based
+  // on Eigen.
+  DENSE_NORMAL_CHOLESKY,
 
   // Solve the normal equations using a dense QR solver; based on
   // Eigen.
   DENSE_QR,
 
+  // Solve the normal equations using a sparse cholesky solver; requires
+  // SuiteSparse or CXSparse.
+  SPARSE_NORMAL_CHOLESKY,
+
   // Specialized solvers, specific to problems with a generalized
   // bi-partitite structure.
 

internal/ceres/CMakeLists.txt

@@ -45,6 +45,7 @@ SET(CERES_INTERNAL_SRC
     conditioned_cost_function.cc
     conjugate_gradients_solver.cc
     corrector.cc
+    dense_normal_cholesky_solver.cc
     dense_qr_solver.cc
     dense_sparse_matrix.cc
     detect_structure.cc

internal/ceres/dense_normal_cholesky_solver.cc

@@ -0,0 +1,86 @@
+// 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 "ceres/dense_normal_cholesky_solver.h"
+
+#include <cstddef>
+
+#include "Eigen/Dense"
+#include "ceres/dense_sparse_matrix.h"
+#include "ceres/linear_solver.h"
+#include "ceres/internal/eigen.h"
+#include "ceres/internal/scoped_ptr.h"
+#include "ceres/types.h"
+
+namespace ceres {
+namespace internal {
+
+DenseNormalCholeskySolver::DenseNormalCholeskySolver(
+    const LinearSolver::Options& options)
+    : options_(options) {}
+
+LinearSolver::Summary DenseNormalCholeskySolver::SolveImpl(
+    DenseSparseMatrix* A,
+    const double* b,
+    const LinearSolver::PerSolveOptions& per_solve_options,
+    double* x) {
+  const int num_rows = A->num_rows();
+  const int num_cols = A->num_cols();
+
+  ConstAlignedMatrixRef Aref = A->matrix();
+  Matrix lhs(num_cols, num_cols);
+  lhs.setZero();
+
+  //   lhs += A'A
+  //
+  // Using rankUpdate instead of GEMM, exposes the fact that its the
+  // same matrix being multiplied with itself and that the product is
+  // symmetric.
+  lhs.selfadjointView<Eigen::Upper>().rankUpdate(Aref.transpose());
+
+  //   rhs = A'b
+  Vector rhs = Aref.transpose() * ConstVectorRef(b, num_rows);
+
+  if (per_solve_options.D != NULL) {
+    ConstVectorRef D(per_solve_options.D, num_cols);
+    lhs += D.array().square().matrix().asDiagonal();
+  }
+
+  VectorRef(x, num_cols) =
+      lhs.selfadjointView<Eigen::Upper>().ldlt().solve(rhs);
+
+  LinearSolver::Summary summary;
+  summary.num_iterations = 1;
+  summary.termination_type = TOLERANCE;
+  return summary;
+}
+
+}   // namespace internal
+}   // namespace ceres

internal/ceres/dense_normal_cholesky_solver.h

@@ -0,0 +1,95 @@
+// 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)
+//
+// Solve dense rectangular systems Ax = b by forming the normal
+// equations and solving them using the Cholesky factorization.
+
+#ifndef CERES_INTERNAL_DENSE_NORMAL_CHOLESKY_SOLVER_H_
+#define CERES_INTERNAL_DENSE_NORMAL_CHOLESKY_SOLVER_H_
+
+#include "ceres/linear_solver.h"
+#include "ceres/internal/macros.h"
+
+namespace ceres {
+namespace internal {
+
+class DenseSparseMatrix;
+
+// This class implements the LinearSolver interface for solving
+// rectangular/unsymmetric (well constrained) linear systems of the
+// form
+//
+//   Ax = b
+//
+// Since there does not usually exist a solution that satisfies these
+// equations, the solver instead solves the linear least squares
+// problem
+//
+//   min_x |Ax - b|^2
+//
+// Setting the gradient of the above optimization problem to zero
+// gives us the normal equations
+//
+//   A'Ax = A'b
+//
+// A'A is a positive definite matrix (hopefully), and the resulting
+// linear system can be solved using Cholesky factorization.
+//
+// If the PerSolveOptions struct has a non-null array D, then the
+// augmented/regularized linear system
+//
+//   [    A    ]x = [b]
+//   [ diag(D) ]    [0]
+//
+// is solved.
+//
+// This class uses the LDLT factorization routines from the Eigen
+// library. This solver always returns a solution, it is the user's
+// responsibility to judge if the solution is good enough for their
+// purposes.
+class DenseNormalCholeskySolver: public DenseSparseMatrixSolver {
+ public:
+  explicit DenseNormalCholeskySolver(const LinearSolver::Options& options);
+
+ private:
+  virtual LinearSolver::Summary SolveImpl(
+      DenseSparseMatrix* A,
+      const double* b,
+      const LinearSolver::PerSolveOptions& per_solve_options,
+      double* x);
+
+  const LinearSolver::Options options_;
+  CERES_DISALLOW_COPY_AND_ASSIGN(DenseNormalCholeskySolver);
+};
+
+}  // namespace internal
+}  // namespace ceres
+
+#endif  // CERES_INTERNAL_DENSE_NORMAL_CHOLESKY_SOLVER_H_

internal/ceres/dense_qr_solver.cc

@@ -33,8 +33,8 @@
 #include <cstddef>
 
 #include "Eigen/Dense"
+#include "ceres/dense_sparse_matrix.h"
 #include "ceres/linear_solver.h"
-#include "ceres/triplet_sparse_matrix.h"
 #include "ceres/internal/eigen.h"
 #include "ceres/internal/scoped_ptr.h"
 #include "ceres/types.h"

internal/ceres/dense_qr_solver.h

@@ -28,7 +28,7 @@
 //
 // Author: sameeragarwal@google.com (Sameer Agarwal)
 //
-// Solve dense rectangular systems Ax = b using the QR factoriztion.
+// Solve dense rectangular systems Ax = b using the QR factorization.
 #ifndef CERES_INTERNAL_DENSE_QR_SOLVER_H_
 #define CERES_INTERNAL_DENSE_QR_SOLVER_H_
 

internal/ceres/evaluator.cc

@@ -51,6 +51,7 @@ Evaluator* Evaluator::Create(const Evaluator::Options& options,
                              string* error) {
   switch (options.linear_solver_type) {
     case DENSE_QR:
+    case DENSE_NORMAL_CHOLESKY:
       return new ProgramEvaluator<ScratchEvaluatePreparer,
                                   DenseJacobianWriter>(options,
                                                        program);

internal/ceres/linear_solver.cc

@@ -31,6 +31,7 @@
 #include "ceres/linear_solver.h"
 
 #include "ceres/cgnr_solver.h"
+#include "ceres/dense_normal_cholesky_solver.h"
 #include "ceres/dense_qr_solver.h"
 #include "ceres/iterative_schur_complement_solver.h"
 #include "ceres/schur_complement_solver.h"
@@ -78,6 +79,9 @@ LinearSolver* LinearSolver::Create(const LinearSolver::Options& options) {
     case DENSE_QR:
       return new DenseQRSolver(options);
 
+    case DENSE_NORMAL_CHOLESKY:
+      return new DenseNormalCholeskySolver(options);
+
     default:
       LOG(FATAL) << "Unknown linear solver type :"
                  << options.type;

internal/ceres/solver_impl_test.cc

@@ -577,6 +577,11 @@ TEST(SolverImpl, CreateLinearSolverNormalOperation) {
   EXPECT_EQ(options.linear_solver_type, DENSE_QR);
   EXPECT_TRUE(solver.get() != NULL);
 
+  options.linear_solver_type = DENSE_NORMAL_CHOLESKY;
+  solver.reset(SolverImpl::CreateLinearSolver(&options, &error));
+  EXPECT_EQ(options.linear_solver_type, DENSE_NORMAL_CHOLESKY);
+  EXPECT_TRUE(solver.get() != NULL);
+
 #ifndef CERES_NO_SUITESPARSE
   options.linear_solver_type = SPARSE_NORMAL_CHOLESKY;
   options.sparse_linear_algebra_library = SUITE_SPARSE;

internal/ceres/system_test.cc

@@ -279,7 +279,8 @@ TEST(SystemTest, PowellsFunction) {
                                  sparse_linear_algebra_library,           \
                                  ordering))
 
-  CONFIGURE(DENSE_QR,    SUITE_SPARSE, NATURAL);
+  CONFIGURE(DENSE_QR, SUITE_SPARSE, NATURAL);
+  CONFIGURE(DENSE_NORMAL_CHOLESKY, SUITE_SPARSE, NATURAL);
   CONFIGURE(DENSE_SCHUR, SUITE_SPARSE, SCHUR);
 
 #ifndef CERES_NO_SUITESPARSE

internal/ceres/types.cc

@@ -37,8 +37,9 @@ namespace ceres {
 
 const char* LinearSolverTypeToString(LinearSolverType solver_type) {
   switch (solver_type) {
-    CASESTR(SPARSE_NORMAL_CHOLESKY);
+    CASESTR(DENSE_NORMAL_CHOLESKY);
     CASESTR(DENSE_QR);
+    CASESTR(SPARSE_NORMAL_CHOLESKY);
     CASESTR(DENSE_SCHUR);
     CASESTR(SPARSE_SCHUR);
     CASESTR(ITERATIVE_SCHUR);

internal/ceres/unsymmetric_linear_solver_test.cc

@@ -73,7 +73,8 @@ class UnsymmetricLinearSolverTest : public ::testing::Test {
 
     scoped_ptr<SparseMatrix> transformed_A;
 
-    if (linear_solver_type == DENSE_QR) {
+    if (linear_solver_type == DENSE_QR ||
+        linear_solver_type == DENSE_NORMAL_CHOLESKY) {
       transformed_A.reset(new DenseSparseMatrix(*A_));
     } else if (linear_solver_type == SPARSE_NORMAL_CHOLESKY) {
       transformed_A.reset(new   CompressedRowSparseMatrix(*A_));
@@ -118,6 +119,10 @@ TEST_F(UnsymmetricLinearSolverTest, DenseQR) {
   TestSolver(DENSE_QR, SUITE_SPARSE);
 }
 
+TEST_F(UnsymmetricLinearSolverTest, DenseNormalCholesky) {
+  TestSolver(DENSE_NORMAL_CHOLESKY, SUITE_SPARSE);
+}
+
 #ifndef CERES_NO_SUITESPARSE
 TEST_F(UnsymmetricLinearSolverTest, SparseNormalCholeskyUsingSuiteSparse) {
   TestSolver(SPARSE_NORMAL_CHOLESKY, SUITE_SPARSE);

jni/Android.mk

@@ -105,6 +105,7 @@ LOCAL_SRC_FILES := $(CERES_SRC_PATH)/array_utils.cc \
                    $(CERES_SRC_PATH)/conditioned_cost_function.cc \
                    $(CERES_SRC_PATH)/conjugate_gradients_solver.cc \
                    $(CERES_SRC_PATH)/corrector.cc \
+                   $(CERES_SRC_PATH)/dense_normal_cholesky_solver.cc \
                    $(CERES_SRC_PATH)/dense_qr_solver.cc \
                    $(CERES_SRC_PATH)/dense_sparse_matrix.cc \
                    $(CERES_SRC_PATH)/detect_structure.cc \