1 // Ceres Solver - A fast non-linear least squares minimizer 2 // Copyright 2010, 2011, 2012 Google Inc. All rights reserved. 3 // http://code.google.com/p/ceres-solver/ 4 // 5 // Redistribution and use in source and binary forms, with or without 6 // modification, are permitted provided that the following conditions are met: 7 // 8 // * Redistributions of source code must retain the above copyright notice, 9 // this list of conditions and the following disclaimer. 10 // * Redistributions in binary form must reproduce the above copyright notice, 11 // this list of conditions and the following disclaimer in the documentation 12 // and/or other materials provided with the distribution. 13 // * Neither the name of Google Inc. nor the names of its contributors may be 14 // used to endorse or promote products derived from this software without 15 // specific prior written permission. 16 // 17 // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" 18 // AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 19 // IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 20 // ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE 21 // LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR 22 // CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF 23 // SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS 24 // INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN 25 // CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 26 // ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE 27 // POSSIBILITY OF SUCH DAMAGE. 28 // 29 // Author: sameeragarwal (at) google.com (Sameer Agarwal) 30 31 #ifndef CERES_INTERNAL_SCHUR_ELIMINATOR_H_ 32 #define CERES_INTERNAL_SCHUR_ELIMINATOR_H_ 33 34 #include <map> 35 #include <vector> 36 #include "ceres/mutex.h" 37 #include "ceres/block_random_access_matrix.h" 38 #include "ceres/block_sparse_matrix.h" 39 #include "ceres/block_structure.h" 40 #include "ceres/linear_solver.h" 41 #include "ceres/internal/eigen.h" 42 #include "ceres/internal/scoped_ptr.h" 43 44 namespace ceres { 45 namespace internal { 46 47 // Classes implementing the SchurEliminatorBase interface implement 48 // variable elimination for linear least squares problems. Assuming 49 // that the input linear system Ax = b can be partitioned into 50 // 51 // E y + F z = b 52 // 53 // Where x = [y;z] is a partition of the variables. The paritioning 54 // of the variables is such that, E'E is a block diagonal matrix. Or 55 // in other words, the parameter blocks in E form an independent set 56 // of the of the graph implied by the block matrix A'A. Then, this 57 // class provides the functionality to compute the Schur complement 58 // system 59 // 60 // S z = r 61 // 62 // where 63 // 64 // S = F'F - F'E (E'E)^{-1} E'F and r = F'b - F'E(E'E)^(-1) E'b 65 // 66 // This is the Eliminate operation, i.e., construct the linear system 67 // obtained by eliminating the variables in E. 68 // 69 // The eliminator also provides the reverse functionality, i.e. given 70 // values for z it can back substitute for the values of y, by solving the 71 // linear system 72 // 73 // Ey = b - F z 74 // 75 // which is done by observing that 76 // 77 // y = (E'E)^(-1) [E'b - E'F z] 78 // 79 // The eliminator has a number of requirements. 80 // 81 // The rows of A are ordered so that for every variable block in y, 82 // all the rows containing that variable block occur as a vertically 83 // contiguous block. i.e the matrix A looks like 84 // 85 // E F chunk 86 // A = [ y1 0 0 0 | z1 0 0 0 z5] 1 87 // [ y1 0 0 0 | z1 z2 0 0 0] 1 88 // [ 0 y2 0 0 | 0 0 z3 0 0] 2 89 // [ 0 0 y3 0 | z1 z2 z3 z4 z5] 3 90 // [ 0 0 y3 0 | z1 0 0 0 z5] 3 91 // [ 0 0 0 y4 | 0 0 0 0 z5] 4 92 // [ 0 0 0 y4 | 0 z2 0 0 0] 4 93 // [ 0 0 0 y4 | 0 0 0 0 0] 4 94 // [ 0 0 0 0 | z1 0 0 0 0] non chunk blocks 95 // [ 0 0 0 0 | 0 0 z3 z4 z5] non chunk blocks 96 // 97 // This structure should be reflected in the corresponding 98 // CompressedRowBlockStructure object associated with A. The linear 99 // system Ax = b should either be well posed or the array D below 100 // should be non-null and the diagonal matrix corresponding to it 101 // should be non-singular. For simplicity of exposition only the case 102 // with a null D is described. 103 // 104 // The usual way to do the elimination is as follows. Starting with 105 // 106 // E y + F z = b 107 // 108 // we can form the normal equations, 109 // 110 // E'E y + E'F z = E'b 111 // F'E y + F'F z = F'b 112 // 113 // multiplying both sides of the first equation by (E'E)^(-1) and then 114 // by F'E we get 115 // 116 // F'E y + F'E (E'E)^(-1) E'F z = F'E (E'E)^(-1) E'b 117 // F'E y + F'F z = F'b 118 // 119 // now subtracting the two equations we get 120 // 121 // [FF' - F'E (E'E)^(-1) E'F] z = F'b - F'E(E'E)^(-1) E'b 122 // 123 // Instead of forming the normal equations and operating on them as 124 // general sparse matrices, the algorithm here deals with one 125 // parameter block in y at a time. The rows corresponding to a single 126 // parameter block yi are known as a chunk, and the algorithm operates 127 // on one chunk at a time. The mathematics remains the same since the 128 // reduced linear system can be shown to be the sum of the reduced 129 // linear systems for each chunk. This can be seen by observing two 130 // things. 131 // 132 // 1. E'E is a block diagonal matrix. 133 // 134 // 2. When E'F is computed, only the terms within a single chunk 135 // interact, i.e for y1 column blocks when transposed and multiplied 136 // with F, the only non-zero contribution comes from the blocks in 137 // chunk1. 138 // 139 // Thus, the reduced linear system 140 // 141 // FF' - F'E (E'E)^(-1) E'F 142 // 143 // can be re-written as 144 // 145 // sum_k F_k F_k' - F_k'E_k (E_k'E_k)^(-1) E_k' F_k 146 // 147 // Where the sum is over chunks and E_k'E_k is dense matrix of size y1 148 // x y1. 149 // 150 // Advanced usage. Uptil now it has been assumed that the user would 151 // be interested in all of the Schur Complement S. However, it is also 152 // possible to use this eliminator to obtain an arbitrary submatrix of 153 // the full Schur complement. When the eliminator is generating the 154 // blocks of S, it asks the RandomAccessBlockMatrix instance passed to 155 // it if it has storage for that block. If it does, the eliminator 156 // computes/updates it, if not it is skipped. This is useful when one 157 // is interested in constructing a preconditioner based on the Schur 158 // Complement, e.g., computing the block diagonal of S so that it can 159 // be used as a preconditioner for an Iterative Substructuring based 160 // solver [See Agarwal et al, Bundle Adjustment in the Large, ECCV 161 // 2008 for an example of such use]. 162 // 163 // Example usage: Please see schur_complement_solver.cc 164 class SchurEliminatorBase { 165 public: 166 virtual ~SchurEliminatorBase() {} 167 168 // Initialize the eliminator. It is the user's responsibilty to call 169 // this function before calling Eliminate or BackSubstitute. It is 170 // also the caller's responsibilty to ensure that the 171 // CompressedRowBlockStructure object passed to this method is the 172 // same one (or is equivalent to) the one associated with the 173 // BlockSparseMatrix objects below. 174 virtual void Init(int num_eliminate_blocks, 175 const CompressedRowBlockStructure* bs) = 0; 176 177 // Compute the Schur complement system from the augmented linear 178 // least squares problem [A;D] x = [b;0]. The left hand side and the 179 // right hand side of the reduced linear system are returned in lhs 180 // and rhs respectively. 181 // 182 // It is the caller's responsibility to construct and initialize 183 // lhs. Depending upon the structure of the lhs object passed here, 184 // the full or a submatrix of the Schur complement will be computed. 185 // 186 // Since the Schur complement is a symmetric matrix, only the upper 187 // triangular part of the Schur complement is computed. 188 virtual void Eliminate(const BlockSparseMatrix* A, 189 const double* b, 190 const double* D, 191 BlockRandomAccessMatrix* lhs, 192 double* rhs) = 0; 193 194 // Given values for the variables z in the F block of A, solve for 195 // the optimal values of the variables y corresponding to the E 196 // block in A. 197 virtual void BackSubstitute(const BlockSparseMatrix* A, 198 const double* b, 199 const double* D, 200 const double* z, 201 double* y) = 0; 202 // Factory 203 static SchurEliminatorBase* Create(const LinearSolver::Options& options); 204 }; 205 206 // Templated implementation of the SchurEliminatorBase interface. The 207 // templating is on the sizes of the row, e and f blocks sizes in the 208 // input matrix. In many problems, the sizes of one or more of these 209 // blocks are constant, in that case, its worth passing these 210 // parameters as template arguments so that they are visible to the 211 // compiler and can be used for compile time optimization of the low 212 // level linear algebra routines. 213 // 214 // This implementation is mulithreaded using OpenMP. The level of 215 // parallelism is controlled by LinearSolver::Options::num_threads. 216 template <int kRowBlockSize = Eigen::Dynamic, 217 int kEBlockSize = Eigen::Dynamic, 218 int kFBlockSize = Eigen::Dynamic > 219 class SchurEliminator : public SchurEliminatorBase { 220 public: 221 explicit SchurEliminator(const LinearSolver::Options& options) 222 : num_threads_(options.num_threads) { 223 } 224 225 // SchurEliminatorBase Interface 226 virtual ~SchurEliminator(); 227 virtual void Init(int num_eliminate_blocks, 228 const CompressedRowBlockStructure* bs); 229 virtual void Eliminate(const BlockSparseMatrix* A, 230 const double* b, 231 const double* D, 232 BlockRandomAccessMatrix* lhs, 233 double* rhs); 234 virtual void BackSubstitute(const BlockSparseMatrix* A, 235 const double* b, 236 const double* D, 237 const double* z, 238 double* y); 239 240 private: 241 // Chunk objects store combinatorial information needed to 242 // efficiently eliminate a whole chunk out of the least squares 243 // problem. Consider the first chunk in the example matrix above. 244 // 245 // [ y1 0 0 0 | z1 0 0 0 z5] 246 // [ y1 0 0 0 | z1 z2 0 0 0] 247 // 248 // One of the intermediate quantities that needs to be calculated is 249 // for each row the product of the y block transposed with the 250 // non-zero z block, and the sum of these blocks across rows. A 251 // temporary array "buffer_" is used for computing and storing them 252 // and the buffer_layout maps the indices of the z-blocks to 253 // position in the buffer_ array. The size of the chunk is the 254 // number of row blocks/residual blocks for the particular y block 255 // being considered. 256 // 257 // For the example chunk shown above, 258 // 259 // size = 2 260 // 261 // The entries of buffer_layout will be filled in the following order. 262 // 263 // buffer_layout[z1] = 0 264 // buffer_layout[z5] = y1 * z1 265 // buffer_layout[z2] = y1 * z1 + y1 * z5 266 typedef map<int, int> BufferLayoutType; 267 struct Chunk { 268 Chunk() : size(0) {} 269 int size; 270 int start; 271 BufferLayoutType buffer_layout; 272 }; 273 274 void ChunkDiagonalBlockAndGradient( 275 const Chunk& chunk, 276 const BlockSparseMatrix* A, 277 const double* b, 278 int row_block_counter, 279 typename EigenTypes<kEBlockSize, kEBlockSize>::Matrix* eet, 280 double* g, 281 double* buffer, 282 BlockRandomAccessMatrix* lhs); 283 284 void UpdateRhs(const Chunk& chunk, 285 const BlockSparseMatrix* A, 286 const double* b, 287 int row_block_counter, 288 const double* inverse_ete_g, 289 double* rhs); 290 291 void ChunkOuterProduct(const CompressedRowBlockStructure* bs, 292 const Matrix& inverse_eet, 293 const double* buffer, 294 const BufferLayoutType& buffer_layout, 295 BlockRandomAccessMatrix* lhs); 296 void EBlockRowOuterProduct(const BlockSparseMatrix* A, 297 int row_block_index, 298 BlockRandomAccessMatrix* lhs); 299 300 301 void NoEBlockRowsUpdate(const BlockSparseMatrix* A, 302 const double* b, 303 int row_block_counter, 304 BlockRandomAccessMatrix* lhs, 305 double* rhs); 306 307 void NoEBlockRowOuterProduct(const BlockSparseMatrix* A, 308 int row_block_index, 309 BlockRandomAccessMatrix* lhs); 310 311 int num_eliminate_blocks_; 312 313 // Block layout of the columns of the reduced linear system. Since 314 // the f blocks can be of varying size, this vector stores the 315 // position of each f block in the row/col of the reduced linear 316 // system. Thus lhs_row_layout_[i] is the row/col position of the 317 // i^th f block. 318 vector<int> lhs_row_layout_; 319 320 // Combinatorial structure of the chunks in A. For more information 321 // see the documentation of the Chunk object above. 322 vector<Chunk> chunks_; 323 324 // TODO(sameeragarwal): The following two arrays contain per-thread 325 // storage. They should be refactored into a per thread struct. 326 327 // Buffer to store the products of the y and z blocks generated 328 // during the elimination phase. buffer_ is of size num_threads * 329 // buffer_size_. Each thread accesses the chunk 330 // 331 // [thread_id * buffer_size_ , (thread_id + 1) * buffer_size_] 332 // 333 scoped_array<double> buffer_; 334 335 // Buffer to store per thread matrix matrix products used by 336 // ChunkOuterProduct. Like buffer_ it is of size num_threads * 337 // buffer_size_. Each thread accesses the chunk 338 // 339 // [thread_id * buffer_size_ , (thread_id + 1) * buffer_size_ -1] 340 // 341 scoped_array<double> chunk_outer_product_buffer_; 342 343 int buffer_size_; 344 int num_threads_; 345 int uneliminated_row_begins_; 346 347 // Locks for the blocks in the right hand side of the reduced linear 348 // system. 349 vector<Mutex*> rhs_locks_; 350 }; 351 352 } // namespace internal 353 } // namespace ceres 354 355 #endif // CERES_INTERNAL_SCHUR_ELIMINATOR_H_ 356
