// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2013 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)
#ifndef CERES_PUBLIC_COVARIANCE_H_
#define CERES_PUBLIC_COVARIANCE_H_
#include <utility>
#include <vector>
#include "ceres/internal/port.h"
#include "ceres/internal/scoped_ptr.h"
#include "ceres/types.h"
#include "ceres/internal/disable_warnings.h"
namespace ceres {
class Problem;
namespace internal {
class CovarianceImpl;
} // namespace internal
// WARNING
// =======
// It is very easy to use this class incorrectly without understanding
// the underlying mathematics. Please read and understand the
// documentation completely before attempting to use this class.
//
//
// This class allows the user to evaluate the covariance for a
// non-linear least squares problem and provides random access to its
// blocks
//
// Background
// ==========
// One way to assess the quality of the solution returned by a
// non-linear least squares solve is to analyze the covariance of the
// solution.
//
// Let us consider the non-linear regression problem
//
// y = f(x) + N(0, I)
//
// i.e., the observation y is a random non-linear function of the
// independent variable x with mean f(x) and identity covariance. Then
// the maximum likelihood estimate of x given observations y is the
// solution to the non-linear least squares problem:
//
// x* = arg min_x |f(x)|^2
//
// And the covariance of x* is given by
//
// C(x*) = inverse[J'(x*)J(x*)]
//
// Here J(x*) is the Jacobian of f at x*. The above formula assumes
// that J(x*) has full column rank.
//
// If J(x*) is rank deficient, then the covariance matrix C(x*) is
// also rank deficient and is given by
//
// C(x*) = pseudoinverse[J'(x*)J(x*)]
//
// Note that in the above, we assumed that the covariance
// matrix for y was identity. This is an important assumption. If this
// is not the case and we have
//
// y = f(x) + N(0, S)
//
// Where S is a positive semi-definite matrix denoting the covariance
// of y, then the maximum likelihood problem to be solved is
//
// x* = arg min_x f'(x) inverse[S] f(x)
//
// and the corresponding covariance estimate of x* is given by
//
// C(x*) = inverse[J'(x*) inverse[S] J(x*)]
//
// So, if it is the case that the observations being fitted to have a
// covariance matrix not equal to identity, then it is the user's
// responsibility that the corresponding cost functions are correctly
// scaled, e.g. in the above case the cost function for this problem
// should evaluate S^{-1/2} f(x) instead of just f(x), where S^{-1/2}
// is the inverse square root of the covariance matrix S.
//
// This class allows the user to evaluate the covariance for a
// non-linear least squares problem and provides random access to its
// blocks. The computation assumes that the CostFunctions compute
// residuals such that their covariance is identity.
//
// Since the computation of the covariance matrix requires computing
// the inverse of a potentially large matrix, this can involve a
// rather large amount of time and memory. However, it is usually the
// case that the user is only interested in a small part of the
// covariance matrix. Quite often just the block diagonal. This class
// allows the user to specify the parts of the covariance matrix that
// she is interested in and then uses this information to only compute
// and store those parts of the covariance matrix.
//
// Rank of the Jacobian
// --------------------
// As we noted above, if the jacobian is rank deficient, then the
// inverse of J'J is not defined and instead a pseudo inverse needs to
// be computed.
//
// The rank deficiency in J can be structural -- columns which are
// always known to be zero or numerical -- depending on the exact
// values in the Jacobian.
//
// Structural rank deficiency occurs when the problem contains
// parameter blocks that are constant. This class correctly handles
// structural rank deficiency like that.
//
// Numerical rank deficiency, where the rank of the matrix cannot be
// predicted by its sparsity structure and requires looking at its
// numerical values is more complicated. Here again there are two
// cases.
//
// a. The rank deficiency arises from overparameterization. e.g., a
// four dimensional quaternion used to parameterize SO(3), which is
// a three dimensional manifold. In cases like this, the user should
// use an appropriate LocalParameterization. Not only will this lead
// to better numerical behaviour of the Solver, it will also expose
// the rank deficiency to the Covariance object so that it can
// handle it correctly.
//
// b. More general numerical rank deficiency in the Jacobian
// requires the computation of the so called Singular Value
// Decomposition (SVD) of J'J. We do not know how to do this for
// large sparse matrices efficiently. For small and moderate sized
// problems this is done using dense linear algebra.
//
// Gauge Invariance
// ----------------
// In structure from motion (3D reconstruction) problems, the
// reconstruction is ambiguous upto a similarity transform. This is
// known as a Gauge Ambiguity. Handling Gauges correctly requires the
// use of SVD or custom inversion algorithms. For small problems the
// user can use the dense algorithm. For more details see
//
// Ken-ichi Kanatani, Daniel D. Morris: Gauges and gauge
// transformations for uncertainty description of geometric structure
// with indeterminacy. IEEE Transactions on Information Theory 47(5):
// 2017-2028 (2001)
//
// Example Usage
// =============
//
// double x[3];
// double y[2];
//
// Problem problem;
// problem.AddParameterBlock(x, 3);
// problem.AddParameterBlock(y, 2);
// <Build Problem>
// <Solve Problem>
//
// Covariance::Options options;
// Covariance covariance(options);
//
// vector<pair<const double*, const double*> > covariance_blocks;
// covariance_blocks.push_back(make_pair(x, x));
// covariance_blocks.push_back(make_pair(y, y));
// covariance_blocks.push_back(make_pair(x, y));
//
// CHECK(covariance.Compute(covariance_blocks, &problem));
//
// double covariance_xx[3 * 3];
// double covariance_yy[2 * 2];
// double covariance_xy[3 * 2];
// covariance.GetCovarianceBlock(x, x, covariance_xx)
// covariance.GetCovarianceBlock(y, y, covariance_yy)
// covariance.GetCovarianceBlock(x, y, covariance_xy)
//
class CERES_EXPORT Covariance {
public:
struct CERES_EXPORT Options {
Options()
#ifndef CERES_NO_SUITESPARSE
: algorithm_type(SUITE_SPARSE_QR),
#else
: algorithm_type(EIGEN_SPARSE_QR),
#endif
min_reciprocal_condition_number(1e-14),
null_space_rank(0),
num_threads(1),
apply_loss_function(true) {
}
// Ceres supports three different algorithms for covariance
// estimation, which represent different tradeoffs in speed,
// accuracy and reliability.
//
// 1. DENSE_SVD uses Eigen's JacobiSVD to perform the
// computations. It computes the singular value decomposition
//
// U * S * V' = J
//
// and then uses it to compute the pseudo inverse of J'J as
//
// pseudoinverse[J'J]^ = V * pseudoinverse[S] * V'
//
// It is an accurate but slow method and should only be used
// for small to moderate sized problems. It can handle
// full-rank as well as rank deficient Jacobians.
//
// 2. EIGEN_SPARSE_QR uses the sparse QR factorization algorithm
// in Eigen to compute the decomposition
//
// Q * R = J
//
// [J'J]^-1 = [R*R']^-1
//
// It is a moderately fast algorithm for sparse matrices.
//
// 3. SUITE_SPARSE_QR uses the SuiteSparseQR sparse QR
// factorization algorithm. It uses dense linear algebra and is
// multi threaded, so for large sparse sparse matrices it is
// significantly faster than EIGEN_SPARSE_QR.
//
// Neither EIGEN_SPARSE_QR not SUITE_SPARSE_QR are capable of
// computing the covariance if the Jacobian is rank deficient.
CovarianceAlgorithmType algorithm_type;
// If the Jacobian matrix is near singular, then inverting J'J
// will result in unreliable results, e.g, if
//
// J = [1.0 1.0 ]
// [1.0 1.0000001 ]
//
// which is essentially a rank deficient matrix, we have
//
// inv(J'J) = [ 2.0471e+14 -2.0471e+14]
// [-2.0471e+14 2.0471e+14]
//
// This is not a useful result. Therefore, by default
// Covariance::Compute will return false if a rank deficient
// Jacobian is encountered. How rank deficiency is detected
// depends on the algorithm being used.
//
// 1. DENSE_SVD
//
// min_sigma / max_sigma < sqrt(min_reciprocal_condition_number)
//
// where min_sigma and max_sigma are the minimum and maxiumum
// singular values of J respectively.
//
// 2. SUITE_SPARSE_QR and EIGEN_SPARSE_QR
//
// rank(J) < num_col(J)
//
// Here rank(J) is the estimate of the rank of J returned by the
// sparse QR factorization algorithm. It is a fairly reliable
// indication of rank deficiency.
//
double min_reciprocal_condition_number;
// When using DENSE_SVD, the user has more control in dealing with
// singular and near singular covariance matrices.
//
// As mentioned above, when the covariance matrix is near
// singular, instead of computing the inverse of J'J, the
// Moore-Penrose pseudoinverse of J'J should be computed.
//
// If J'J has the eigen decomposition (lambda_i, e_i), where
// lambda_i is the i^th eigenvalue and e_i is the corresponding
// eigenvector, then the inverse of J'J is
//
// inverse[J'J] = sum_i e_i e_i' / lambda_i
//
// and computing the pseudo inverse involves dropping terms from
// this sum that correspond to small eigenvalues.
//
// How terms are dropped is controlled by
// min_reciprocal_condition_number and null_space_rank.
//
// If null_space_rank is non-negative, then the smallest
// null_space_rank eigenvalue/eigenvectors are dropped
// irrespective of the magnitude of lambda_i. If the ratio of the
// smallest non-zero eigenvalue to the largest eigenvalue in the
// truncated matrix is still below
// min_reciprocal_condition_number, then the Covariance::Compute()
// will fail and return false.
//
// Setting null_space_rank = -1 drops all terms for which
//
// lambda_i / lambda_max < min_reciprocal_condition_number.
//
// This option has no effect on the SUITE_SPARSE_QR and
// EIGEN_SPARSE_QR algorithms.
int null_space_rank;
int num_threads;
// Even though the residual blocks in the problem may contain loss
// functions, setting apply_loss_function to false will turn off
// the application of the loss function to the output of the cost
// function and in turn its effect on the covariance.
//
// TODO(sameergaarwal): Expand this based on Jim's experiments.
bool apply_loss_function;
};
explicit Covariance(const Options& options);
~Covariance();
// Compute a part of the covariance matrix.
//
// The vector covariance_blocks, indexes into the covariance matrix
// block-wise using pairs of parameter blocks. This allows the
// covariance estimation algorithm to only compute and store these
// blocks.
//
// Since the covariance matrix is symmetric, if the user passes
// (block1, block2), then GetCovarianceBlock can be called with
// block1, block2 as well as block2, block1.
//
// covariance_blocks cannot contain duplicates. Bad things will
// happen if they do.
//
// Note that the list of covariance_blocks is only used to determine
// what parts of the covariance matrix are computed. The full
// Jacobian is used to do the computation, i.e. they do not have an
// impact on what part of the Jacobian is used for computation.
//
// The return value indicates the success or failure of the
// covariance computation. Please see the documentation for
// Covariance::Options for more on the conditions under which this
// function returns false.
bool Compute(
const vector<pair<const double*, const double*> >& covariance_blocks,
Problem* problem);
// Return the block of the covariance matrix corresponding to
// parameter_block1 and parameter_block2.
//
// Compute must be called before the first call to
// GetCovarianceBlock and the pair <parameter_block1,
// parameter_block2> OR the pair <parameter_block2,
// parameter_block1> must have been present in the vector
// covariance_blocks when Compute was called. Otherwise
// GetCovarianceBlock will return false.
//
// covariance_block must point to a memory location that can store a
// parameter_block1_size x parameter_block2_size matrix. The
// returned covariance will be a row-major matrix.
bool GetCovarianceBlock(const double* parameter_block1,
const double* parameter_block2,
double* covariance_block) const;
private:
internal::scoped_ptr<internal::CovarianceImpl> impl_;
};
} // namespace ceres
#include "ceres/internal/reenable_warnings.h"
#endif // CERES_PUBLIC_COVARIANCE_H_