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// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2023 Google Inc. All rights reserved.
// http://ceres-solver.org/
//
// 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: mierle@gmail.com (Keir Mierle)
//
// WARNING WARNING WARNING
// WARNING WARNING WARNING Tiny solver is experimental and will change.
// WARNING WARNING WARNING
//
// A tiny least squares solver using Levenberg-Marquardt, intended for solving
// small dense problems with low latency and low overhead. The implementation
// takes care to do all allocation up front, so that no memory is allocated
// during solving. This is especially useful when solving many similar problems;
// for example, inverse pixel distortion for every pixel on a grid.
//
// Note: This code has no dependencies beyond Eigen, including on other parts of
// Ceres, so it is possible to take this file alone and put it in another
// project without the rest of Ceres.
//
// Algorithm based off of:
//
// [1] K. Madsen, H. Nielsen, O. Tingleoff.
// Methods for Non-linear Least Squares Problems.
// http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3215/pdf/imm3215.pdf
#ifndef CERES_PUBLIC_TINY_SOLVER_H_
#define CERES_PUBLIC_TINY_SOLVER_H_
#include <cassert>
#include <cmath>
#include "Eigen/Dense"
namespace ceres {
// To use tiny solver, create a class or struct that allows computing the cost
// function (described below). This is similar to a ceres::CostFunction, but is
// different to enable statically allocating all memory for the solver
// (specifically, enum sizes). Key parts are the Scalar typedef, the enums to
// describe problem sizes (needed to remove all heap allocations), and the
// operator() overload to evaluate the cost and (optionally) jacobians.
//
// struct TinySolverCostFunctionTraits {
// typedef double Scalar;
// enum {
// NUM_RESIDUALS = <int> OR Eigen::Dynamic,
// NUM_PARAMETERS = <int> OR Eigen::Dynamic,
// };
// bool operator()(const double* parameters,
// double* residuals,
// double* jacobian) const;
//
// int NumResiduals() const; -- Needed if NUM_RESIDUALS == Eigen::Dynamic.
// int NumParameters() const; -- Needed if NUM_PARAMETERS == Eigen::Dynamic.
// };
//
// For operator(), the size of the objects is:
//
// double* parameters -- NUM_PARAMETERS or NumParameters()
// double* residuals -- NUM_RESIDUALS or NumResiduals()
// double* jacobian -- NUM_RESIDUALS * NUM_PARAMETERS in column-major format
// (Eigen's default); or nullptr if no jacobian
// requested.
//
// An example (fully statically sized):
//
// struct MyCostFunctionExample {
// typedef double Scalar;
// enum {
// NUM_RESIDUALS = 2,
// NUM_PARAMETERS = 3,
// };
// bool operator()(const double* parameters,
// double* residuals,
// double* jacobian) const {
// residuals[0] = x + 2*y + 4*z;
// residuals[1] = y * z;
// if (jacobian) {
// jacobian[0 * 2 + 0] = 1; // First column (x).
// jacobian[0 * 2 + 1] = 0;
//
// jacobian[1 * 2 + 0] = 2; // Second column (y).
// jacobian[1 * 2 + 1] = z;
//
// jacobian[2 * 2 + 0] = 4; // Third column (z).
// jacobian[2 * 2 + 1] = y;
// }
// return true;
// }
// };
//
// The solver supports either statically or dynamically sized cost
// functions. If the number of residuals is dynamic then the Function
// must define:
//
// int NumResiduals() const;
//
// If the number of parameters is dynamic then the Function must
// define:
//
// int NumParameters() const;
//
template <typename Function,
typename LinearSolver =
Eigen::LDLT<Eigen::Matrix<typename Function::Scalar, //
Function::NUM_PARAMETERS, //
Function::NUM_PARAMETERS>>>
class TinySolver {
public:
// This class needs to have an Eigen aligned operator new as it contains
// fixed-size Eigen types.
EIGEN_MAKE_ALIGNED_OPERATOR_NEW
enum {
NUM_RESIDUALS = Function::NUM_RESIDUALS,
NUM_PARAMETERS = Function::NUM_PARAMETERS
};
using Scalar = typename Function::Scalar;
using Parameters = typename Eigen::Matrix<Scalar, NUM_PARAMETERS, 1>;
enum Status {
// max_norm |J'(x) * f(x)| < gradient_tolerance
GRADIENT_TOO_SMALL,
// ||dx|| <= parameter_tolerance * (||x|| + parameter_tolerance)
RELATIVE_STEP_SIZE_TOO_SMALL,
// cost_threshold > ||f(x)||^2 / 2
COST_TOO_SMALL,
// num_iterations >= max_num_iterations
HIT_MAX_ITERATIONS,
// (new_cost - old_cost) < function_tolerance * old_cost
COST_CHANGE_TOO_SMALL,
// TODO(sameeragarwal): Deal with numerical failures.
};
struct Options {
int max_num_iterations = 50;
// max_norm |J'(x) * f(x)| < gradient_tolerance
Scalar gradient_tolerance = 1e-10;
// ||dx|| <= parameter_tolerance * (||x|| + parameter_tolerance)
Scalar parameter_tolerance = 1e-8;
// (new_cost - old_cost) < function_tolerance * old_cost
Scalar function_tolerance = 1e-6;
// cost_threshold > ||f(x)||^2 / 2
Scalar cost_threshold = std::numeric_limits<Scalar>::epsilon();
Scalar initial_trust_region_radius = 1e4;
};
struct Summary {
// 1/2 ||f(x_0)||^2
Scalar initial_cost = -1;
// 1/2 ||f(x)||^2
Scalar final_cost = -1;
// max_norm(J'f(x))
Scalar gradient_max_norm = -1;
int iterations = -1;
Status status = HIT_MAX_ITERATIONS;
};
bool Update(const Function& function, const Parameters& x) {
if (!function(x.data(), residuals_.data(), jacobian_.data())) {
return false;
}
residuals_ = -residuals_;
// On the first iteration, compute a diagonal (Jacobi) scaling
// matrix, which we store as a vector.
if (summary.iterations == 0) {
// jacobi_scaling = 1 / (1 + diagonal(J'J))
//
// 1 is added to the denominator to regularize small diagonal
// entries.
jacobi_scaling_ = 1.0 / (1.0 + jacobian_.colwise().norm().array());
}
// This explicitly computes the normal equations, which is numerically
// unstable. Nevertheless, it is often good enough and is fast.
//
// TODO(sameeragarwal): Refactor this to allow for DenseQR
// factorization.
jacobian_ = jacobian_ * jacobi_scaling_.asDiagonal();
jtj_ = jacobian_.transpose() * jacobian_;
g_ = jacobian_.transpose() * residuals_;
summary.gradient_max_norm = g_.array().abs().maxCoeff();
cost_ = residuals_.squaredNorm() / 2;
return true;
}
const Summary& Solve(const Function& function, Parameters* x_and_min) {
Initialize<NUM_RESIDUALS, NUM_PARAMETERS>(function);
assert(x_and_min);
Parameters& x = *x_and_min;
summary = Summary();
summary.iterations = 0;
// TODO(sameeragarwal): Deal with failure here.
Update(function, x);
summary.initial_cost = cost_;
summary.final_cost = cost_;
if (summary.gradient_max_norm < options.gradient_tolerance) {
summary.status = GRADIENT_TOO_SMALL;
return summary;
}
if (cost_ < options.cost_threshold) {
summary.status = COST_TOO_SMALL;
return summary;
}
Scalar u = 1.0 / options.initial_trust_region_radius;
Scalar v = 2;
for (summary.iterations = 1;
summary.iterations < options.max_num_iterations;
summary.iterations++) {
jtj_regularized_ = jtj_;
const Scalar min_diagonal = 1e-6;
const Scalar max_diagonal = 1e32;
for (int i = 0; i < lm_diagonal_.rows(); ++i) {
lm_diagonal_[i] = std::sqrt(
u * (std::min)((std::max)(jtj_(i, i), min_diagonal), max_diagonal));
jtj_regularized_(i, i) += lm_diagonal_[i] * lm_diagonal_[i];
}
// TODO(sameeragarwal): Check for failure and deal with it.
linear_solver_.compute(jtj_regularized_);
lm_step_ = linear_solver_.solve(g_);
dx_ = jacobi_scaling_.asDiagonal() * lm_step_;
// Adding parameter_tolerance to x.norm() ensures that this
// works if x is near zero.
const Scalar parameter_tolerance =
options.parameter_tolerance *
(x.norm() + options.parameter_tolerance);
if (dx_.norm() < parameter_tolerance) {
summary.status = RELATIVE_STEP_SIZE_TOO_SMALL;
break;
}
x_new_ = x + dx_;
// TODO(keir): Add proper handling of errors from user eval of cost
// functions.
function(&x_new_[0], &f_x_new_[0], nullptr);
const Scalar cost_change = (2 * cost_ - f_x_new_.squaredNorm());
// TODO(sameeragarwal): Better more numerically stable evaluation.
const Scalar model_cost_change = lm_step_.dot(2 * g_ - jtj_ * lm_step_);
// rho is the ratio of the actual reduction in error to the reduction
// in error that would be obtained if the problem was linear. See [1]
// for details.
Scalar rho(cost_change / model_cost_change);
if (rho > 0) {
// Accept the Levenberg-Marquardt step because the linear
// model fits well.
x = x_new_;
if (std::abs(cost_change) < options.function_tolerance) {
cost_ = f_x_new_.squaredNorm() / 2;
summary.status = COST_CHANGE_TOO_SMALL;
break;
}
// TODO(sameeragarwal): Deal with failure.
Update(function, x);
if (summary.gradient_max_norm < options.gradient_tolerance) {
summary.status = GRADIENT_TOO_SMALL;
break;
}
if (cost_ < options.cost_threshold) {
summary.status = COST_TOO_SMALL;
break;
}
Scalar tmp = Scalar(2 * rho - 1);
u = u * (std::max)(Scalar(1 / 3.), Scalar(1) - tmp * tmp * tmp);
v = 2;
} else {
// Reject the update because either the normal equations failed to solve
// or the local linear model was not good (rho < 0).
// Additionally if the cost change is too small, then terminate.
if (std::abs(cost_change) < options.function_tolerance) {
// Terminate
summary.status = COST_CHANGE_TOO_SMALL;
break;
}
// Reduce the size of the trust region.
u *= v;
v *= 2;
}
}
summary.final_cost = cost_;
return summary;
}
Options options;
Summary summary;
private:
// Preallocate everything, including temporary storage needed for solving the
// linear system. This allows reusing the intermediate storage across solves.
LinearSolver linear_solver_;
Scalar cost_;
Parameters dx_, x_new_, g_, jacobi_scaling_, lm_diagonal_, lm_step_;
Eigen::Matrix<Scalar, NUM_RESIDUALS, 1> residuals_, f_x_new_;
Eigen::Matrix<Scalar, NUM_RESIDUALS, NUM_PARAMETERS> jacobian_;
Eigen::Matrix<Scalar, NUM_PARAMETERS, NUM_PARAMETERS> jtj_, jtj_regularized_;
// The following definitions are needed for template metaprogramming.
template <bool Condition, typename T>
struct enable_if;
template <typename T>
struct enable_if<true, T> {
using type = T;
};
// The number of parameters and residuals are dynamically sized.
template <int R, int P>
typename enable_if<(R == Eigen::Dynamic && P == Eigen::Dynamic), void>::type
Initialize(const Function& function) {
Initialize(function.NumResiduals(), function.NumParameters());
}
// The number of parameters is dynamically sized and the number of
// residuals is statically sized.
template <int R, int P>
typename enable_if<(R == Eigen::Dynamic && P != Eigen::Dynamic), void>::type
Initialize(const Function& function) {
Initialize(function.NumResiduals(), P);
}
// The number of parameters is statically sized and the number of
// residuals is dynamically sized.
template <int R, int P>
typename enable_if<(R != Eigen::Dynamic && P == Eigen::Dynamic), void>::type
Initialize(const Function& function) {
Initialize(R, function.NumParameters());
}
// The number of parameters and residuals are statically sized.
template <int R, int P>
typename enable_if<(R != Eigen::Dynamic && P != Eigen::Dynamic), void>::type
Initialize(const Function& /* function */) {}
void Initialize(int num_residuals, int num_parameters) {
dx_.resize(num_parameters);
x_new_.resize(num_parameters);
g_.resize(num_parameters);
jacobi_scaling_.resize(num_parameters);
lm_diagonal_.resize(num_parameters);
lm_step_.resize(num_parameters);
residuals_.resize(num_residuals);
f_x_new_.resize(num_residuals);
jacobian_.resize(num_residuals, num_parameters);
jtj_.resize(num_parameters, num_parameters);
jtj_regularized_.resize(num_parameters, num_parameters);
}
};
} // namespace ceres
#endif // CERES_PUBLIC_TINY_SOLVER_H_