// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://ceres-solver.org/
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// Author: sameeragarwal@google.com (Sameer Agarwal)
#ifndef CERES_INTERNAL_SCHUR_ELIMINATOR_H_
#define CERES_INTERNAL_SCHUR_ELIMINATOR_H_
#include <map>
#include <vector>
#include "ceres/mutex.h"
#include "ceres/block_random_access_matrix.h"
#include "ceres/block_sparse_matrix.h"
#include "ceres/block_structure.h"
#include "ceres/linear_solver.h"
#include "ceres/internal/eigen.h"
#include "ceres/internal/scoped_ptr.h"
namespace ceres {
namespace internal {
// Classes implementing the SchurEliminatorBase interface implement
// variable elimination for linear least squares problems. Assuming
// that the input linear system Ax = b can be partitioned into
//
// E y + F z = b
//
// Where x = [y;z] is a partition of the variables. The paritioning
// of the variables is such that, E'E is a block diagonal matrix. Or
// in other words, the parameter blocks in E form an independent set
// of the of the graph implied by the block matrix A'A. Then, this
// class provides the functionality to compute the Schur complement
// system
//
// S z = r
//
// where
//
// S = F'F - F'E (E'E)^{-1} E'F and r = F'b - F'E(E'E)^(-1) E'b
//
// This is the Eliminate operation, i.e., construct the linear system
// obtained by eliminating the variables in E.
//
// The eliminator also provides the reverse functionality, i.e. given
// values for z it can back substitute for the values of y, by solving the
// linear system
//
// Ey = b - F z
//
// which is done by observing that
//
// y = (E'E)^(-1) [E'b - E'F z]
//
// The eliminator has a number of requirements.
//
// The rows of A are ordered so that for every variable block in y,
// all the rows containing that variable block occur as a vertically
// contiguous block. i.e the matrix A looks like
//
// E F chunk
// A = [ y1 0 0 0 | z1 0 0 0 z5] 1
// [ y1 0 0 0 | z1 z2 0 0 0] 1
// [ 0 y2 0 0 | 0 0 z3 0 0] 2
// [ 0 0 y3 0 | z1 z2 z3 z4 z5] 3
// [ 0 0 y3 0 | z1 0 0 0 z5] 3
// [ 0 0 0 y4 | 0 0 0 0 z5] 4
// [ 0 0 0 y4 | 0 z2 0 0 0] 4
// [ 0 0 0 y4 | 0 0 0 0 0] 4
// [ 0 0 0 0 | z1 0 0 0 0] non chunk blocks
// [ 0 0 0 0 | 0 0 z3 z4 z5] non chunk blocks
//
// This structure should be reflected in the corresponding
// CompressedRowBlockStructure object associated with A. The linear
// system Ax = b should either be well posed or the array D below
// should be non-null and the diagonal matrix corresponding to it
// should be non-singular. For simplicity of exposition only the case
// with a null D is described.
//
// The usual way to do the elimination is as follows. Starting with
//
// E y + F z = b
//
// we can form the normal equations,
//
// E'E y + E'F z = E'b
// F'E y + F'F z = F'b
//
// multiplying both sides of the first equation by (E'E)^(-1) and then
// by F'E we get
//
// F'E y + F'E (E'E)^(-1) E'F z = F'E (E'E)^(-1) E'b
// F'E y + F'F z = F'b
//
// now subtracting the two equations we get
//
// [FF' - F'E (E'E)^(-1) E'F] z = F'b - F'E(E'E)^(-1) E'b
//
// Instead of forming the normal equations and operating on them as
// general sparse matrices, the algorithm here deals with one
// parameter block in y at a time. The rows corresponding to a single
// parameter block yi are known as a chunk, and the algorithm operates
// on one chunk at a time. The mathematics remains the same since the
// reduced linear system can be shown to be the sum of the reduced
// linear systems for each chunk. This can be seen by observing two
// things.
//
// 1. E'E is a block diagonal matrix.
//
// 2. When E'F is computed, only the terms within a single chunk
// interact, i.e for y1 column blocks when transposed and multiplied
// with F, the only non-zero contribution comes from the blocks in
// chunk1.
//
// Thus, the reduced linear system
//
// FF' - F'E (E'E)^(-1) E'F
//
// can be re-written as
//
// sum_k F_k F_k' - F_k'E_k (E_k'E_k)^(-1) E_k' F_k
//
// Where the sum is over chunks and E_k'E_k is dense matrix of size y1
// x y1.
//
// Advanced usage. Uptil now it has been assumed that the user would
// be interested in all of the Schur Complement S. However, it is also
// possible to use this eliminator to obtain an arbitrary submatrix of
// the full Schur complement. When the eliminator is generating the
// blocks of S, it asks the RandomAccessBlockMatrix instance passed to
// it if it has storage for that block. If it does, the eliminator
// computes/updates it, if not it is skipped. This is useful when one
// is interested in constructing a preconditioner based on the Schur
// Complement, e.g., computing the block diagonal of S so that it can
// be used as a preconditioner for an Iterative Substructuring based
// solver [See Agarwal et al, Bundle Adjustment in the Large, ECCV
// 2008 for an example of such use].
//
// Example usage: Please see schur_complement_solver.cc
class SchurEliminatorBase {
public:
virtual ~SchurEliminatorBase() {}
// Initialize the eliminator. It is the user's responsibilty to call
// this function before calling Eliminate or BackSubstitute. It is
// also the caller's responsibilty to ensure that the
// CompressedRowBlockStructure object passed to this method is the
// same one (or is equivalent to) the one associated with the
// BlockSparseMatrix objects below.
//
// assume_full_rank_ete controls how the eliminator inverts with the
// diagonal blocks corresponding to e blocks in A'A. If
// assume_full_rank_ete is true, then a Cholesky factorization is
// used to compute the inverse, otherwise a singular value
// decomposition is used to compute the pseudo inverse.
virtual void Init(int num_eliminate_blocks,
bool assume_full_rank_ete,
const CompressedRowBlockStructure* bs) = 0;
// Compute the Schur complement system from the augmented linear
// least squares problem [A;D] x = [b;0]. The left hand side and the
// right hand side of the reduced linear system are returned in lhs
// and rhs respectively.
//
// It is the caller's responsibility to construct and initialize
// lhs. Depending upon the structure of the lhs object passed here,
// the full or a submatrix of the Schur complement will be computed.
//
// Since the Schur complement is a symmetric matrix, only the upper
// triangular part of the Schur complement is computed.
virtual void Eliminate(const BlockSparseMatrix* A,
const double* b,
const double* D,
BlockRandomAccessMatrix* lhs,
double* rhs) = 0;
// Given values for the variables z in the F block of A, solve for
// the optimal values of the variables y corresponding to the E
// block in A.
virtual void BackSubstitute(const BlockSparseMatrix* A,
const double* b,
const double* D,
const double* z,
double* y) = 0;
// Factory
static SchurEliminatorBase* Create(const LinearSolver::Options& options);
};
// Templated implementation of the SchurEliminatorBase interface. The
// templating is on the sizes of the row, e and f blocks sizes in the
// input matrix. In many problems, the sizes of one or more of these
// blocks are constant, in that case, its worth passing these
// parameters as template arguments so that they are visible to the
// compiler and can be used for compile time optimization of the low
// level linear algebra routines.
//
// This implementation is mulithreaded using OpenMP. The level of
// parallelism is controlled by LinearSolver::Options::num_threads.
template <int kRowBlockSize = Eigen::Dynamic,
int kEBlockSize = Eigen::Dynamic,
int kFBlockSize = Eigen::Dynamic >
class SchurEliminator : public SchurEliminatorBase {
public:
explicit SchurEliminator(const LinearSolver::Options& options)
: num_threads_(options.num_threads) {
}
// SchurEliminatorBase Interface
virtual ~SchurEliminator();
virtual void Init(int num_eliminate_blocks,
bool assume_full_rank_ete,
const CompressedRowBlockStructure* bs);
virtual void Eliminate(const BlockSparseMatrix* A,
const double* b,
const double* D,
BlockRandomAccessMatrix* lhs,
double* rhs);
virtual void BackSubstitute(const BlockSparseMatrix* A,
const double* b,
const double* D,
const double* z,
double* y);
private:
// Chunk objects store combinatorial information needed to
// efficiently eliminate a whole chunk out of the least squares
// problem. Consider the first chunk in the example matrix above.
//
// [ y1 0 0 0 | z1 0 0 0 z5]
// [ y1 0 0 0 | z1 z2 0 0 0]
//
// One of the intermediate quantities that needs to be calculated is
// for each row the product of the y block transposed with the
// non-zero z block, and the sum of these blocks across rows. A
// temporary array "buffer_" is used for computing and storing them
// and the buffer_layout maps the indices of the z-blocks to
// position in the buffer_ array. The size of the chunk is the
// number of row blocks/residual blocks for the particular y block
// being considered.
//
// For the example chunk shown above,
//
// size = 2
//
// The entries of buffer_layout will be filled in the following order.
//
// buffer_layout[z1] = 0
// buffer_layout[z5] = y1 * z1
// buffer_layout[z2] = y1 * z1 + y1 * z5
typedef std::map<int, int> BufferLayoutType;
struct Chunk {
Chunk() : size(0) {}
int size;
int start;
BufferLayoutType buffer_layout;
};
void ChunkDiagonalBlockAndGradient(
const Chunk& chunk,
const BlockSparseMatrix* A,
const double* b,
int row_block_counter,
typename EigenTypes<kEBlockSize, kEBlockSize>::Matrix* eet,
double* g,
double* buffer,
BlockRandomAccessMatrix* lhs);
void UpdateRhs(const Chunk& chunk,
const BlockSparseMatrix* A,
const double* b,
int row_block_counter,
const double* inverse_ete_g,
double* rhs);
void ChunkOuterProduct(const CompressedRowBlockStructure* bs,
const Matrix& inverse_eet,
const double* buffer,
const BufferLayoutType& buffer_layout,
BlockRandomAccessMatrix* lhs);
void EBlockRowOuterProduct(const BlockSparseMatrix* A,
int row_block_index,
BlockRandomAccessMatrix* lhs);
void NoEBlockRowsUpdate(const BlockSparseMatrix* A,
const double* b,
int row_block_counter,
BlockRandomAccessMatrix* lhs,
double* rhs);
void NoEBlockRowOuterProduct(const BlockSparseMatrix* A,
int row_block_index,
BlockRandomAccessMatrix* lhs);
int num_threads_;
int num_eliminate_blocks_;
bool assume_full_rank_ete_;
// Block layout of the columns of the reduced linear system. Since
// the f blocks can be of varying size, this vector stores the
// position of each f block in the row/col of the reduced linear
// system. Thus lhs_row_layout_[i] is the row/col position of the
// i^th f block.
std::vector<int> lhs_row_layout_;
// Combinatorial structure of the chunks in A. For more information
// see the documentation of the Chunk object above.
std::vector<Chunk> chunks_;
// TODO(sameeragarwal): The following two arrays contain per-thread
// storage. They should be refactored into a per thread struct.
// Buffer to store the products of the y and z blocks generated
// during the elimination phase. buffer_ is of size num_threads *
// buffer_size_. Each thread accesses the chunk
//
// [thread_id * buffer_size_ , (thread_id + 1) * buffer_size_]
//
scoped_array<double> buffer_;
// Buffer to store per thread matrix matrix products used by
// ChunkOuterProduct. Like buffer_ it is of size num_threads *
// buffer_size_. Each thread accesses the chunk
//
// [thread_id * buffer_size_ , (thread_id + 1) * buffer_size_ -1]
//
scoped_array<double> chunk_outer_product_buffer_;
int buffer_size_;
int uneliminated_row_begins_;
// Locks for the blocks in the right hand side of the reduced linear
// system.
std::vector<Mutex*> rhs_locks_;
};
} // namespace internal
} // namespace ceres
#endif // CERES_INTERNAL_SCHUR_ELIMINATOR_H_