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// 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/internal/eigen.h"
#include "ceres/low_rank_inverse_hessian.h"
#include "glog/logging.h"
namespace ceres {
namespace internal {
LowRankInverseHessian::LowRankInverseHessian(
int num_parameters,
int max_num_corrections,
bool use_approximate_eigenvalue_scaling)
: num_parameters_(num_parameters),
max_num_corrections_(max_num_corrections),
use_approximate_eigenvalue_scaling_(use_approximate_eigenvalue_scaling),
num_corrections_(0),
approximate_eigenvalue_scale_(1.0),
delta_x_history_(num_parameters, max_num_corrections),
delta_gradient_history_(num_parameters, max_num_corrections),
delta_x_dot_delta_gradient_(max_num_corrections) {
}
bool LowRankInverseHessian::Update(const Vector& delta_x,
const Vector& delta_gradient) {
const double delta_x_dot_delta_gradient = delta_x.dot(delta_gradient);
if (delta_x_dot_delta_gradient <= 1e-10) {
VLOG(2) << "Skipping LBFGS Update, delta_x_dot_delta_gradient too small: "
<< delta_x_dot_delta_gradient;
return false;
}
if (num_corrections_ == max_num_corrections_) {
// TODO(sameeragarwal): This can be done more efficiently using
// a circular buffer/indexing scheme, but for simplicity we will
// do the expensive copy for now.
delta_x_history_.block(0, 0, num_parameters_, max_num_corrections_ - 1) =
delta_x_history_
.block(0, 1, num_parameters_, max_num_corrections_ - 1);
delta_gradient_history_
.block(0, 0, num_parameters_, max_num_corrections_ - 1) =
delta_gradient_history_
.block(0, 1, num_parameters_, max_num_corrections_ - 1);
delta_x_dot_delta_gradient_.head(num_corrections_ - 1) =
delta_x_dot_delta_gradient_.tail(num_corrections_ - 1);
} else {
++num_corrections_;
}
delta_x_history_.col(num_corrections_ - 1) = delta_x;
delta_gradient_history_.col(num_corrections_ - 1) = delta_gradient;
delta_x_dot_delta_gradient_(num_corrections_ - 1) =
delta_x_dot_delta_gradient;
approximate_eigenvalue_scale_ =
delta_x_dot_delta_gradient / delta_gradient.squaredNorm();
return true;
}
void LowRankInverseHessian::RightMultiply(const double* x_ptr,
double* y_ptr) const {
ConstVectorRef gradient(x_ptr, num_parameters_);
VectorRef search_direction(y_ptr, num_parameters_);
search_direction = gradient;
Vector alpha(num_corrections_);
for (int i = num_corrections_ - 1; i >= 0; --i) {
alpha(i) = delta_x_history_.col(i).dot(search_direction) /
delta_x_dot_delta_gradient_(i);
search_direction -= alpha(i) * delta_gradient_history_.col(i);
}
if (use_approximate_eigenvalue_scaling_) {
// Rescale the initial inverse Hessian approximation (H_0) to be iteratively
// updated so that it is of similar 'size' to the true inverse Hessian along
// the most recent search direction. As shown in [1]:
//
// \gamma_k = (delta_gradient_{k-1}' * delta_x_{k-1}) /
// (delta_gradient_{k-1}' * delta_gradient_{k-1})
//
// Satisfies:
//
// (1 / \lambda_m) <= \gamma_k <= (1 / \lambda_1)
//
// Where \lambda_1 & \lambda_m are the smallest and largest eigenvalues of
// the true Hessian (not the inverse) along the most recent search direction
// respectively. Thus \gamma is an approximate eigenvalue of the true
// inverse Hessian, and choosing: H_0 = I * \gamma will yield a starting
// point that has a similar scale to the true inverse Hessian. This
// technique is widely reported to often improve convergence, however this
// is not universally true, particularly if there are errors in the initial
// jacobians, or if there are significant differences in the sensitivity
// of the problem to the parameters (i.e. the range of the magnitudes of
// the components of the gradient is large).
//
// The original origin of this rescaling trick is somewhat unclear, the
// earliest reference appears to be Oren [1], however it is widely discussed
// without specific attributation in various texts including [2] (p143/178).
//
// [1] Oren S.S., Self-scaling variable metric (SSVM) algorithms Part II:
// Implementation and experiments, Management Science,
// 20(5), 863-874, 1974.
// [2] Nocedal J., Wright S., Numerical Optimization, Springer, 1999.
search_direction *= approximate_eigenvalue_scale_;
}
for (int i = 0; i < num_corrections_; ++i) {
const double beta = delta_gradient_history_.col(i).dot(search_direction) /
delta_x_dot_delta_gradient_(i);
search_direction += delta_x_history_.col(i) * (alpha(i) - beta);
}
}
} // namespace internal
} // namespace ceres
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