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.. default-domain:: cpp
.. cpp:namespace:: ceres
.. _chapter-solving:
=======
Solving
=======
Introduction
============
Effective use of Ceres requires some familiarity with the basic
components of a non-linear least squares solver, so before we describe
how to configure and use the solver, we will take a brief look at how
some of the core optimization algorithms in Ceres work.
Let :math:`x \in \mathbb{R}^n` be an :math:`n`-dimensional vector of
variables, and
:math:`F(x) = \left[f_1(x), ... , f_{m}(x) \right]^{\top}` be a
:math:`m`-dimensional function of :math:`x`. We are interested in
solving the optimization problem [#f1]_
.. math:: \arg \min_x \frac{1}{2}\|F(x)\|^2\ . \\
L \le x \le U
:label: nonlinsq
Where, :math:`L` and :math:`U` are lower and upper bounds on the
parameter vector :math:`x`.
Since the efficient global minimization of :eq:`nonlinsq` for
general :math:`F(x)` is an intractable problem, we will have to settle
for finding a local minimum.
In the following, the Jacobian :math:`J(x)` of :math:`F(x)` is an
:math:`m\times n` matrix, where :math:`J_{ij}(x) = \partial_j f_i(x)`
and the gradient vector is :math:`g(x) = \nabla \frac{1}{2}\|F(x)\|^2
= J(x)^\top F(x)`.
The general strategy when solving non-linear optimization problems is
to solve a sequence of approximations to the original problem
[NocedalWright]_. At each iteration, the approximation is solved to
determine a correction :math:`\Delta x` to the vector :math:`x`. For
non-linear least squares, an approximation can be constructed by using
the linearization :math:`F(x+\Delta x) \approx F(x) + J(x)\Delta x`,
which leads to the following linear least squares problem:
.. math:: \min_{\Delta x} \frac{1}{2}\|J(x)\Delta x + F(x)\|^2
:label: linearapprox
Unfortunately, naively solving a sequence of these problems and
updating :math:`x \leftarrow x+ \Delta x` leads to an algorithm that
may not converge. To get a convergent algorithm, we need to control
the size of the step :math:`\Delta x`. Depending on how the size of
the step :math:`\Delta x` is controlled, non-linear optimization
algorithms can be divided into two major categories [NocedalWright]_.
1. **Trust Region** The trust region approach approximates the
objective function using using a model function (often a quadratic)
over a subset of the search space known as the trust region. If the
model function succeeds in minimizing the true objective function
the trust region is expanded; conversely, otherwise it is
contracted and the model optimization problem is solved again.
2. **Line Search** The line search approach first finds a descent
direction along which the objective function will be reduced and
then computes a step size that decides how far should move along
that direction. The descent direction can be computed by various
methods, such as gradient descent, Newton's method and Quasi-Newton
method. The step size can be determined either exactly or
inexactly.
Trust region methods are in some sense dual to line search methods:
trust region methods first choose a step size (the size of the trust
region) and then a step direction while line search methods first
choose a step direction and then a step size. Ceres implements
multiple algorithms in both categories.
.. _section-trust-region-methods:
Trust Region Methods
====================
The basic trust region algorithm looks something like this.
1. Given an initial point :math:`x` and a trust region radius :math:`\mu`.
2. Solve
.. math::
\arg \min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 \\
\text{such that} &\|D(x)\Delta x\|^2 \le \mu\\
&L \le x + \Delta x \le U.
3. :math:`\rho = \frac{\displaystyle \|F(x + \Delta x)\|^2 -
\|F(x)\|^2}{\displaystyle \|J(x)\Delta x + F(x)\|^2 -
\|F(x)\|^2}`
4. if :math:`\rho > \epsilon` then :math:`x = x + \Delta x`.
5. if :math:`\rho > \eta_1` then :math:`\rho = 2 \rho`
6. else if :math:`\rho < \eta_2` then :math:`\rho = 0.5 * \rho`
7. Go to 2.
Here, :math:`\mu` is the trust region radius, :math:`D(x)` is some
matrix used to define a metric on the domain of :math:`F(x)` and
:math:`\rho` measures the quality of the step :math:`\Delta x`, i.e.,
how well did the linear model predict the decrease in the value of the
non-linear objective. The idea is to increase or decrease the radius
of the trust region depending on how well the linearization predicts
the behavior of the non-linear objective, which in turn is reflected
in the value of :math:`\rho`.
The key computational step in a trust-region algorithm is the solution
of the constrained optimization problem
.. math::
\arg \min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 \\
\text{such that} &\|D(x)\Delta x\|^2 \le \mu\\
&L \le x + \Delta x \le U.
:label: trp
There are a number of different ways of solving this problem, each
giving rise to a different concrete trust-region algorithm. Currently,
Ceres implements two trust-region algorithms - Levenberg-Marquardt
and Dogleg, each of which is augmented with a line search if bounds
constraints are present [Kanzow]_. The user can choose between them by
setting :member:`Solver::Options::trust_region_strategy_type`.
.. rubric:: Footnotes
.. [#f1] At the level of the non-linear solver, the block structure is
not relevant, therefore our discussion here is in terms of an
optimization problem defined over a state vector of size
:math:`n`. Similarly the presence of loss functions is also
ignored as the problem is internally converted into a pure
non-linear least squares problem.
.. _section-levenberg-marquardt:
Levenberg-Marquardt
-------------------
The Levenberg-Marquardt algorithm [Levenberg]_ [Marquardt]_ is the
most popular algorithm for solving non-linear least squares problems.
It was also the first trust region algorithm to be developed
[Levenberg]_ [Marquardt]_. Ceres implements an exact step [Madsen]_
and an inexact step variant of the Levenberg-Marquardt algorithm
[WrightHolt]_ [NashSofer]_.
It can be shown, that the solution to :eq:`trp` can be obtained by
solving an unconstrained optimization of the form
.. math:: \arg\min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 +\lambda \|D(x)\Delta x\|^2
Where, :math:`\lambda` is a Lagrange multiplier that is inverse
related to :math:`\mu`. In Ceres, we solve for
.. math:: \arg\min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 + \frac{1}{\mu} \|D(x)\Delta x\|^2
:label: lsqr
The matrix :math:`D(x)` is a non-negative diagonal matrix, typically
the square root of the diagonal of the matrix :math:`J(x)^\top J(x)`.
Before going further, let us make some notational simplifications. We
will assume that the matrix :math:`\sqrt{\mu} D` has been concatenated
at the bottom of the matrix :math:`J` and similarly a vector of zeros
has been added to the bottom of the vector :math:`f` and the rest of
our discussion will be in terms of :math:`J` and :math:`f`, i.e, the
linear least squares problem.
.. math:: \min_{\Delta x} \frac{1}{2} \|J(x)\Delta x + f(x)\|^2 .
:label: simple
For all but the smallest problems the solution of :eq:`simple` in
each iteration of the Levenberg-Marquardt algorithm is the dominant
computational cost in Ceres. Ceres provides a number of different
options for solving :eq:`simple`. There are two major classes of
methods - factorization and iterative.
The factorization methods are based on computing an exact solution of
:eq:`lsqr` using a Cholesky or a QR factorization and lead to an exact
step Levenberg-Marquardt algorithm. But it is not clear if an exact
solution of :eq:`lsqr` is necessary at each step of the LM algorithm
to solve :eq:`nonlinsq`. In fact, we have already seen evidence
that this may not be the case, as :eq:`lsqr` is itself a regularized
version of :eq:`linearapprox`. Indeed, it is possible to
construct non-linear optimization algorithms in which the linearized
problem is solved approximately. These algorithms are known as inexact
Newton or truncated Newton methods [NocedalWright]_.
An inexact Newton method requires two ingredients. First, a cheap
method for approximately solving systems of linear
equations. Typically an iterative linear solver like the Conjugate
Gradients method is used for this
purpose [NocedalWright]_. Second, a termination rule for
the iterative solver. A typical termination rule is of the form
.. math:: \|H(x) \Delta x + g(x)\| \leq \eta_k \|g(x)\|.
:label: inexact
Here, :math:`k` indicates the Levenberg-Marquardt iteration number and
:math:`0 < \eta_k <1` is known as the forcing sequence. [WrightHolt]_
prove that a truncated Levenberg-Marquardt algorithm that uses an
inexact Newton step based on :eq:`inexact` converges for any
sequence :math:`\eta_k \leq \eta_0 < 1` and the rate of convergence
depends on the choice of the forcing sequence :math:`\eta_k`.
Ceres supports both exact and inexact step solution strategies. When
the user chooses a factorization based linear solver, the exact step
Levenberg-Marquardt algorithm is used. When the user chooses an
iterative linear solver, the inexact step Levenberg-Marquardt
algorithm is used.
.. _section-dogleg:
Dogleg
------
Another strategy for solving the trust region problem :eq:`trp` was
introduced by M. J. D. Powell. The key idea there is to compute two
vectors
.. math::
\Delta x^{\text{Gauss-Newton}} &= \arg \min_{\Delta x}\frac{1}{2} \|J(x)\Delta x + f(x)\|^2.\\
\Delta x^{\text{Cauchy}} &= -\frac{\|g(x)\|^2}{\|J(x)g(x)\|^2}g(x).
Note that the vector :math:`\Delta x^{\text{Gauss-Newton}}` is the
solution to :eq:`linearapprox` and :math:`\Delta
x^{\text{Cauchy}}` is the vector that minimizes the linear
approximation if we restrict ourselves to moving along the direction
of the gradient. Dogleg methods finds a vector :math:`\Delta x`
defined by :math:`\Delta x^{\text{Gauss-Newton}}` and :math:`\Delta
x^{\text{Cauchy}}` that solves the trust region problem. Ceres
supports two variants that can be chose by setting
:member:`Solver::Options::dogleg_type`.
``TRADITIONAL_DOGLEG`` as described by Powell, constructs two line
segments using the Gauss-Newton and Cauchy vectors and finds the point
farthest along this line shaped like a dogleg (hence the name) that is
contained in the trust-region. For more details on the exact reasoning
and computations, please see Madsen et al [Madsen]_.
``SUBSPACE_DOGLEG`` is a more sophisticated method that considers the
entire two dimensional subspace spanned by these two vectors and finds
the point that minimizes the trust region problem in this subspace
[ByrdSchnabel]_.
The key advantage of the Dogleg over Levenberg-Marquardt is that if
the step computation for a particular choice of :math:`\mu` does not
result in sufficient decrease in the value of the objective function,
Levenberg-Marquardt solves the linear approximation from scratch with
a smaller value of :math:`\mu`. Dogleg on the other hand, only needs
to compute the interpolation between the Gauss-Newton and the Cauchy
vectors, as neither of them depend on the value of :math:`\mu`.
The Dogleg method can only be used with the exact factorization based
linear solvers.
.. _section-inner-iterations:
Inner Iterations
----------------
Some non-linear least squares problems have additional structure in
the way the parameter blocks interact that it is beneficial to modify
the way the trust region step is computed. For example, consider the
following regression problem
.. math:: y = a_1 e^{b_1 x} + a_2 e^{b_3 x^2 + c_1}
Given a set of pairs :math:`\{(x_i, y_i)\}`, the user wishes to estimate
:math:`a_1, a_2, b_1, b_2`, and :math:`c_1`.
Notice that the expression on the left is linear in :math:`a_1` and
:math:`a_2`, and given any value for :math:`b_1, b_2` and :math:`c_1`,
it is possible to use linear regression to estimate the optimal values
of :math:`a_1` and :math:`a_2`. It's possible to analytically
eliminate the variables :math:`a_1` and :math:`a_2` from the problem
entirely. Problems like these are known as separable least squares
problem and the most famous algorithm for solving them is the Variable
Projection algorithm invented by Golub & Pereyra [GolubPereyra]_.
Similar structure can be found in the matrix factorization with
missing data problem. There the corresponding algorithm is known as
Wiberg's algorithm [Wiberg]_.
Ruhe & Wedin present an analysis of various algorithms for solving
separable non-linear least squares problems and refer to *Variable
Projection* as Algorithm I in their paper [RuheWedin]_.
Implementing Variable Projection is tedious and expensive. Ruhe &
Wedin present a simpler algorithm with comparable convergence
properties, which they call Algorithm II. Algorithm II performs an
additional optimization step to estimate :math:`a_1` and :math:`a_2`
exactly after computing a successful Newton step.
This idea can be generalized to cases where the residual is not
linear in :math:`a_1` and :math:`a_2`, i.e.,
.. math:: y = f_1(a_1, e^{b_1 x}) + f_2(a_2, e^{b_3 x^2 + c_1})
In this case, we solve for the trust region step for the full problem,
and then use it as the starting point to further optimize just `a_1`
and `a_2`. For the linear case, this amounts to doing a single linear
least squares solve. For non-linear problems, any method for solving
the :math:`a_1` and :math:`a_2` optimization problems will do. The
only constraint on :math:`a_1` and :math:`a_2` (if they are two
different parameter block) is that they do not co-occur in a residual
block.
This idea can be further generalized, by not just optimizing
:math:`(a_1, a_2)`, but decomposing the graph corresponding to the
Hessian matrix's sparsity structure into a collection of
non-overlapping independent sets and optimizing each of them.
Setting :member:`Solver::Options::use_inner_iterations` to ``true``
enables the use of this non-linear generalization of Ruhe & Wedin's
Algorithm II. This version of Ceres has a higher iteration
complexity, but also displays better convergence behavior per
iteration.
Setting :member:`Solver::Options::num_threads` to the maximum number
possible is highly recommended.
.. _section-non-monotonic-steps:
Non-monotonic Steps
-------------------
Note that the basic trust-region algorithm described in
:ref:`section-trust-region-methods` is a descent algorithm in that it
only accepts a point if it strictly reduces the value of the objective
function.
Relaxing this requirement allows the algorithm to be more efficient in
the long term at the cost of some local increase in the value of the
objective function.
This is because allowing for non-decreasing objective function values
in a principled manner allows the algorithm to *jump over boulders* as
the method is not restricted to move into narrow valleys while
preserving its convergence properties.
Setting :member:`Solver::Options::use_nonmonotonic_steps` to ``true``
enables the non-monotonic trust region algorithm as described by Conn,
Gould & Toint in [Conn]_.
Even though the value of the objective function may be larger
than the minimum value encountered over the course of the
optimization, the final parameters returned to the user are the
ones corresponding to the minimum cost over all iterations.
The option to take non-monotonic steps is available for all trust
region strategies.
.. _section-line-search-methods:
Line Search Methods
===================
The line search method in Ceres Solver cannot handle bounds
constraints right now, so it can only be used for solving
unconstrained problems.
Line search algorithms
1. Given an initial point :math:`x`
2. :math:`\Delta x = -H^{-1}(x) g(x)`
3. :math:`\arg \min_\mu \frac{1}{2} \| F(x + \mu \Delta x) \|^2`
4. :math:`x = x + \mu \Delta x`
5. Goto 2.
Here :math:`H(x)` is some approximation to the Hessian of the
objective function, and :math:`g(x)` is the gradient at
:math:`x`. Depending on the choice of :math:`H(x)` we get a variety of
different search directions :math:`\Delta x`.
Step 4, which is a one dimensional optimization or `Line Search` along
:math:`\Delta x` is what gives this class of methods its name.
Different line search algorithms differ in their choice of the search
direction :math:`\Delta x` and the method used for one dimensional
optimization along :math:`\Delta x`. The choice of :math:`H(x)` is the
primary source of computational complexity in these
methods. Currently, Ceres Solver supports three choices of search
directions, all aimed at large scale problems.
1. ``STEEPEST_DESCENT`` This corresponds to choosing :math:`H(x)` to
be the identity matrix. This is not a good search direction for
anything but the simplest of the problems. It is only included here
for completeness.
2. ``NONLINEAR_CONJUGATE_GRADIENT`` A generalization of the Conjugate
Gradient method to non-linear functions. The generalization can be
performed in a number of different ways, resulting in a variety of
search directions. Ceres Solver currently supports
``FLETCHER_REEVES``, ``POLAK_RIBIERE`` and ``HESTENES_STIEFEL``
directions.
3. ``BFGS`` A generalization of the Secant method to multiple
dimensions in which a full, dense approximation to the inverse
Hessian is maintained and used to compute a quasi-Newton step
[NocedalWright]_. BFGS is currently the best known general
quasi-Newton algorithm.
4. ``LBFGS`` A limited memory approximation to the full ``BFGS``
method in which the last `M` iterations are used to approximate the
inverse Hessian used to compute a quasi-Newton step [Nocedal]_,
[ByrdNocedal]_.
Currently Ceres Solver supports both a backtracking and interpolation
based Armijo line search algorithm, and a sectioning / zoom
interpolation (strong) Wolfe condition line search algorithm.
However, note that in order for the assumptions underlying the
``BFGS`` and ``LBFGS`` methods to be guaranteed to be satisfied the
Wolfe line search algorithm should be used.
.. _section-linear-solver:
LinearSolver
============
Recall that in both of the trust-region methods described above, the
key computational cost is the solution of a linear least squares
problem of the form
.. math:: \min_{\Delta x} \frac{1}{2} \|J(x)\Delta x + f(x)\|^2 .
:label: simple2
Let :math:`H(x)= J(x)^\top J(x)` and :math:`g(x) = -J(x)^\top
f(x)`. For notational convenience let us also drop the dependence on
:math:`x`. Then it is easy to see that solving :eq:`simple2` is
equivalent to solving the *normal equations*.
.. math:: H \Delta x = g
:label: normal
Ceres provides a number of different options for solving :eq:`normal`.
.. _section-qr:
``DENSE_QR``
------------
For small problems (a couple of hundred parameters and a few thousand
residuals) with relatively dense Jacobians, ``DENSE_QR`` is the method
of choice [Bjorck]_. Let :math:`J = QR` be the QR-decomposition of
:math:`J`, where :math:`Q` is an orthonormal matrix and :math:`R` is
an upper triangular matrix [TrefethenBau]_. Then it can be shown that
the solution to :eq:`normal` is given by
.. math:: \Delta x^* = -R^{-1}Q^\top f
Ceres uses ``Eigen`` 's dense QR factorization routines.
.. _section-cholesky:
``DENSE_NORMAL_CHOLESKY`` & ``SPARSE_NORMAL_CHOLESKY``
------------------------------------------------------
Large non-linear least square problems are usually sparse. In such
cases, using a dense QR factorization is inefficient. Let :math:`H =
R^\top R` be the Cholesky factorization of the normal equations, where
:math:`R` is an upper triangular matrix, then the solution to
:eq:`normal` is given by
.. math::
\Delta x^* = R^{-1} R^{-\top} g.
The observant reader will note that the :math:`R` in the Cholesky
factorization of :math:`H` is the same upper triangular matrix
:math:`R` in the QR factorization of :math:`J`. Since :math:`Q` is an
orthonormal matrix, :math:`J=QR` implies that :math:`J^\top J = R^\top
Q^\top Q R = R^\top R`. There are two variants of Cholesky
factorization -- sparse and dense.
``DENSE_NORMAL_CHOLESKY`` as the name implies performs a dense
Cholesky factorization of the normal equations. Ceres uses
``Eigen`` 's dense LDLT factorization routines.
``SPARSE_NORMAL_CHOLESKY``, as the name implies performs a sparse
Cholesky factorization of the normal equations. This leads to
substantial savings in time and memory for large sparse
problems. Ceres uses the sparse Cholesky factorization routines in
Professor Tim Davis' ``SuiteSparse`` or ``CXSparse`` packages [Chen]_
or the sparse Cholesky factorization algorithm in ``Eigen`` (which
incidently is a port of the algorithm implemented inside ``CXSparse``)
.. _section-schur:
``DENSE_SCHUR`` & ``SPARSE_SCHUR``
----------------------------------
While it is possible to use ``SPARSE_NORMAL_CHOLESKY`` to solve bundle
adjustment problems, bundle adjustment problem have a special
structure, and a more efficient scheme for solving :eq:`normal`
can be constructed.
Suppose that the SfM problem consists of :math:`p` cameras and
:math:`q` points and the variable vector :math:`x` has the block
structure :math:`x = [y_{1}, ... ,y_{p},z_{1}, ... ,z_{q}]`. Where,
:math:`y` and :math:`z` correspond to camera and point parameters,
respectively. Further, let the camera blocks be of size :math:`c` and
the point blocks be of size :math:`s` (for most problems :math:`c` =
:math:`6`--`9` and :math:`s = 3`). Ceres does not impose any constancy
requirement on these block sizes, but choosing them to be constant
simplifies the exposition.
A key characteristic of the bundle adjustment problem is that there is
no term :math:`f_{i}` that includes two or more point blocks. This in
turn implies that the matrix :math:`H` is of the form
.. math:: H = \left[ \begin{matrix} B & E\\ E^\top & C \end{matrix} \right]\ ,
:label: hblock
where :math:`B \in \mathbb{R}^{pc\times pc}` is a block sparse matrix
with :math:`p` blocks of size :math:`c\times c` and :math:`C \in
\mathbb{R}^{qs\times qs}` is a block diagonal matrix with :math:`q` blocks
of size :math:`s\times s`. :math:`E \in \mathbb{R}^{pc\times qs}` is a
general block sparse matrix, with a block of size :math:`c\times s`
for each observation. Let us now block partition :math:`\Delta x =
[\Delta y,\Delta z]` and :math:`g=[v,w]` to restate :eq:`normal`
as the block structured linear system
.. math:: \left[ \begin{matrix} B & E\\ E^\top & C \end{matrix}
\right]\left[ \begin{matrix} \Delta y \\ \Delta z
\end{matrix} \right] = \left[ \begin{matrix} v\\ w
\end{matrix} \right]\ ,
:label: linear2
and apply Gaussian elimination to it. As we noted above, :math:`C` is
a block diagonal matrix, with small diagonal blocks of size
:math:`s\times s`. Thus, calculating the inverse of :math:`C` by
inverting each of these blocks is cheap. This allows us to eliminate
:math:`\Delta z` by observing that :math:`\Delta z = C^{-1}(w - E^\top
\Delta y)`, giving us
.. math:: \left[B - EC^{-1}E^\top\right] \Delta y = v - EC^{-1}w\ .
:label: schur
The matrix
.. math:: S = B - EC^{-1}E^\top
is the Schur complement of :math:`C` in :math:`H`. It is also known as
the *reduced camera matrix*, because the only variables
participating in :eq:`schur` are the ones corresponding to the
cameras. :math:`S \in \mathbb{R}^{pc\times pc}` is a block structured
symmetric positive definite matrix, with blocks of size :math:`c\times
c`. The block :math:`S_{ij}` corresponding to the pair of images
:math:`i` and :math:`j` is non-zero if and only if the two images
observe at least one common point.
Now, :eq:`linear2` can be solved by first forming :math:`S`, solving for
:math:`\Delta y`, and then back-substituting :math:`\Delta y` to
obtain the value of :math:`\Delta z`. Thus, the solution of what was
an :math:`n\times n`, :math:`n=pc+qs` linear system is reduced to the
inversion of the block diagonal matrix :math:`C`, a few matrix-matrix
and matrix-vector multiplies, and the solution of block sparse
:math:`pc\times pc` linear system :eq:`schur`. For almost all
problems, the number of cameras is much smaller than the number of
points, :math:`p \ll q`, thus solving :eq:`schur` is
significantly cheaper than solving :eq:`linear2`. This is the
*Schur complement trick* [Brown]_.
This still leaves open the question of solving :eq:`schur`. The
method of choice for solving symmetric positive definite systems
exactly is via the Cholesky factorization [TrefethenBau]_ and
depending upon the structure of the matrix, there are, in general, two
options. The first is direct factorization, where we store and factor
:math:`S` as a dense matrix [TrefethenBau]_. This method has
:math:`O(p^2)` space complexity and :math:`O(p^3)` time complexity and
is only practical for problems with up to a few hundred cameras. Ceres
implements this strategy as the ``DENSE_SCHUR`` solver.
But, :math:`S` is typically a fairly sparse matrix, as most images
only see a small fraction of the scene. This leads us to the second
option: Sparse Direct Methods. These methods store :math:`S` as a
sparse matrix, use row and column re-ordering algorithms to maximize
the sparsity of the Cholesky decomposition, and focus their compute
effort on the non-zero part of the factorization [Chen]_. Sparse
direct methods, depending on the exact sparsity structure of the Schur
complement, allow bundle adjustment algorithms to significantly scale
up over those based on dense factorization. Ceres implements this
strategy as the ``SPARSE_SCHUR`` solver.
.. _section-cgnr:
``CGNR``
--------
For general sparse problems, if the problem is too large for
``CHOLMOD`` or a sparse linear algebra library is not linked into
Ceres, another option is the ``CGNR`` solver. This solver uses the
Conjugate Gradients solver on the *normal equations*, but without
forming the normal equations explicitly. It exploits the relation
.. math::
H x = J^\top J x = J^\top(J x)
When the user chooses ``ITERATIVE_SCHUR`` as the linear solver, Ceres
automatically switches from the exact step algorithm to an inexact
step algorithm.
.. _section-iterative_schur:
``ITERATIVE_SCHUR``
-------------------
Another option for bundle adjustment problems is to apply PCG to the
reduced camera matrix :math:`S` instead of :math:`H`. One reason to do
this is that :math:`S` is a much smaller matrix than :math:`H`, but
more importantly, it can be shown that :math:`\kappa(S)\leq
\kappa(H)`. Ceres implements PCG on :math:`S` as the
``ITERATIVE_SCHUR`` solver. When the user chooses ``ITERATIVE_SCHUR``
as the linear solver, Ceres automatically switches from the exact step
algorithm to an inexact step algorithm.
The cost of forming and storing the Schur complement :math:`S` can be
prohibitive for large problems. Indeed, for an inexact Newton solver
that computes :math:`S` and runs PCG on it, almost all of its time is
spent in constructing :math:`S`; the time spent inside the PCG
algorithm is negligible in comparison. Because PCG only needs access
to :math:`S` via its product with a vector, one way to evaluate
:math:`Sx` is to observe that
.. math:: x_1 &= E^\top x
.. math:: x_2 &= C^{-1} x_1
.. math:: x_3 &= Ex_2\\
.. math:: x_4 &= Bx\\
.. math:: Sx &= x_4 - x_3
:label: schurtrick1
Thus, we can run PCG on :math:`S` with the same computational effort
per iteration as PCG on :math:`H`, while reaping the benefits of a
more powerful preconditioner. In fact, we do not even need to compute
:math:`H`, :eq:`schurtrick1` can be implemented using just the columns
of :math:`J`.
Equation :eq:`schurtrick1` is closely related to *Domain
Decomposition methods* for solving large linear systems that arise in
structural engineering and partial differential equations. In the
language of Domain Decomposition, each point in a bundle adjustment
problem is a domain, and the cameras form the interface between these
domains. The iterative solution of the Schur complement then falls
within the sub-category of techniques known as Iterative
Sub-structuring [Saad]_ [Mathew]_.
.. _section-preconditioner:
Preconditioner
--------------
The convergence rate of Conjugate Gradients for
solving :eq:`normal` depends on the distribution of eigenvalues
of :math:`H` [Saad]_. A useful upper bound is
:math:`\sqrt{\kappa(H)}`, where, :math:`\kappa(H)` is the condition
number of the matrix :math:`H`. For most bundle adjustment problems,
:math:`\kappa(H)` is high and a direct application of Conjugate
Gradients to :eq:`normal` results in extremely poor performance.
The solution to this problem is to replace :eq:`normal` with a
*preconditioned* system. Given a linear system, :math:`Ax =b` and a
preconditioner :math:`M` the preconditioned system is given by
:math:`M^{-1}Ax = M^{-1}b`. The resulting algorithm is known as
Preconditioned Conjugate Gradients algorithm (PCG) and its worst case
complexity now depends on the condition number of the *preconditioned*
matrix :math:`\kappa(M^{-1}A)`.
The computational cost of using a preconditioner :math:`M` is the cost
of computing :math:`M` and evaluating the product :math:`M^{-1}y` for
arbitrary vectors :math:`y`. Thus, there are two competing factors to
consider: How much of :math:`H`'s structure is captured by :math:`M`
so that the condition number :math:`\kappa(HM^{-1})` is low, and the
computational cost of constructing and using :math:`M`. The ideal
preconditioner would be one for which :math:`\kappa(M^{-1}A)
=1`. :math:`M=A` achieves this, but it is not a practical choice, as
applying this preconditioner would require solving a linear system
equivalent to the unpreconditioned problem. It is usually the case
that the more information :math:`M` has about :math:`H`, the more
expensive it is use. For example, Incomplete Cholesky factorization
based preconditioners have much better convergence behavior than the
Jacobi preconditioner, but are also much more expensive.
The simplest of all preconditioners is the diagonal or Jacobi
preconditioner, i.e., :math:`M=\operatorname{diag}(A)`, which for
block structured matrices like :math:`H` can be generalized to the
block Jacobi preconditioner.
For ``ITERATIVE_SCHUR`` there are two obvious choices for block
diagonal preconditioners for :math:`S`. The block diagonal of the
matrix :math:`B` [Mandel]_ and the block diagonal :math:`S`, i.e, the
block Jacobi preconditioner for :math:`S`. Ceres's implements both of
these preconditioners and refers to them as ``JACOBI`` and
``SCHUR_JACOBI`` respectively.
For bundle adjustment problems arising in reconstruction from
community photo collections, more effective preconditioners can be
constructed by analyzing and exploiting the camera-point visibility
structure of the scene [KushalAgarwal]_. Ceres implements the two
visibility based preconditioners described by Kushal & Agarwal as
``CLUSTER_JACOBI`` and ``CLUSTER_TRIDIAGONAL``. These are fairly new
preconditioners and Ceres' implementation of them is in its early
stages and is not as mature as the other preconditioners described
above.
.. _section-ordering:
Ordering
--------
The order in which variables are eliminated in a linear solver can
have a significant of impact on the efficiency and accuracy of the
method. For example when doing sparse Cholesky factorization, there
are matrices for which a good ordering will give a Cholesky factor
with :math:`O(n)` storage, where as a bad ordering will result in an
completely dense factor.
Ceres allows the user to provide varying amounts of hints to the
solver about the variable elimination ordering to use. This can range
from no hints, where the solver is free to decide the best ordering
based on the user's choices like the linear solver being used, to an
exact order in which the variables should be eliminated, and a variety
of possibilities in between.
Instances of the :class:`ParameterBlockOrdering` class are used to
communicate this information to Ceres.
Formally an ordering is an ordered partitioning of the parameter
blocks. Each parameter block belongs to exactly one group, and each
group has a unique integer associated with it, that determines its
order in the set of groups. We call these groups *Elimination Groups*
Given such an ordering, Ceres ensures that the parameter blocks in the
lowest numbered elimination group are eliminated first, and then the
parameter blocks in the next lowest numbered elimination group and so
on. Within each elimination group, Ceres is free to order the
parameter blocks as it chooses. For example, consider the linear system
.. math::
x + y &= 3\\
2x + 3y &= 7
There are two ways in which it can be solved. First eliminating
:math:`x` from the two equations, solving for :math:`y` and then back
substituting for :math:`x`, or first eliminating :math:`y`, solving
for :math:`x` and back substituting for :math:`y`. The user can
construct three orderings here.
1. :math:`\{0: x\}, \{1: y\}` : Eliminate :math:`x` first.
2. :math:`\{0: y\}, \{1: x\}` : Eliminate :math:`y` first.
3. :math:`\{0: x, y\}` : Solver gets to decide the elimination order.
Thus, to have Ceres determine the ordering automatically using
heuristics, put all the variables in the same elimination group. The
identity of the group does not matter. This is the same as not
specifying an ordering at all. To control the ordering for every
variable, create an elimination group per variable, ordering them in
the desired order.
If the user is using one of the Schur solvers (``DENSE_SCHUR``,
``SPARSE_SCHUR``, ``ITERATIVE_SCHUR``) and chooses to specify an
ordering, it must have one important property. The lowest numbered
elimination group must form an independent set in the graph
corresponding to the Hessian, or in other words, no two parameter
blocks in in the first elimination group should co-occur in the same
residual block. For the best performance, this elimination group
should be as large as possible. For standard bundle adjustment
problems, this corresponds to the first elimination group containing
all the 3d points, and the second containing the all the cameras
parameter blocks.
If the user leaves the choice to Ceres, then the solver uses an
approximate maximum independent set algorithm to identify the first
elimination group [LiSaad]_.
.. _section-solver-options:
:class:`Solver::Options`
------------------------
.. class:: Solver::Options
:class:`Solver::Options` controls the overall behavior of the
solver. We list the various settings and their default values below.
.. function:: bool Solver::Options::IsValid(string* error) const
Validate the values in the options struct and returns true on
success. If there is a problem, the method returns false with
``error`` containing a textual description of the cause.
.. member:: MinimizerType Solver::Options::minimizer_type
Default: ``TRUST_REGION``
Choose between ``LINE_SEARCH`` and ``TRUST_REGION`` algorithms. See
:ref:`section-trust-region-methods` and
:ref:`section-line-search-methods` for more details.
.. member:: LineSearchDirectionType Solver::Options::line_search_direction_type
Default: ``LBFGS``
Choices are ``STEEPEST_DESCENT``, ``NONLINEAR_CONJUGATE_GRADIENT``,
``BFGS`` and ``LBFGS``.
.. member:: LineSearchType Solver::Options::line_search_type
Default: ``WOLFE``
Choices are ``ARMIJO`` and ``WOLFE`` (strong Wolfe conditions).
Note that in order for the assumptions underlying the ``BFGS`` and
``LBFGS`` line search direction algorithms to be guaranteed to be
satisifed, the ``WOLFE`` line search should be used.
.. member:: NonlinearConjugateGradientType Solver::Options::nonlinear_conjugate_gradient_type
Default: ``FLETCHER_REEVES``
Choices are ``FLETCHER_REEVES``, ``POLAK_RIBIERE`` and
``HESTENES_STIEFEL``.
.. member:: int Solver::Options::max_lbfs_rank
Default: 20
The L-BFGS hessian approximation is a low rank approximation to the
inverse of the Hessian matrix. The rank of the approximation
determines (linearly) the space and time complexity of using the
approximation. Higher the rank, the better is the quality of the
approximation. The increase in quality is however is bounded for a
number of reasons.
1. The method only uses secant information and not actual
derivatives.
2. The Hessian approximation is constrained to be positive
definite.
So increasing this rank to a large number will cost time and space
complexity without the corresponding increase in solution
quality. There are no hard and fast rules for choosing the maximum
rank. The best choice usually requires some problem specific
experimentation.
.. member:: bool Solver::Options::use_approximate_eigenvalue_bfgs_scaling
Default: ``false``
As part of the ``BFGS`` update step / ``LBFGS`` right-multiply
step, the initial inverse Hessian approximation is taken to be the
Identity. However, [Oren]_ showed that using instead :math:`I *
\gamma`, where :math:`\gamma` is a scalar chosen to approximate an
eigenvalue of the true inverse Hessian can result in improved
convergence in a wide variety of cases. Setting
``use_approximate_eigenvalue_bfgs_scaling`` to true enables this
scaling in ``BFGS`` (before first iteration) and ``LBFGS`` (at each
iteration).
Precisely, approximate eigenvalue scaling equates to
.. math:: \gamma = \frac{y_k' s_k}{y_k' y_k}
With:
.. math:: y_k = \nabla f_{k+1} - \nabla f_k
.. math:: s_k = x_{k+1} - x_k
Where :math:`f()` is the line search objective and :math:`x` the
vector of parameter values [NocedalWright]_.
It is important to note that approximate eigenvalue scaling does
**not** *always* improve convergence, and that it can in fact
*significantly* degrade performance for certain classes of problem,
which is why it is disabled by default. In particular it can
degrade performance when the sensitivity of the problem to different
parameters varies significantly, as in this case a single scalar
factor fails to capture this variation and detrimentally downscales
parts of the Jacobian approximation which correspond to
low-sensitivity parameters. It can also reduce the robustness of the
solution to errors in the Jacobians.
.. member:: LineSearchIterpolationType Solver::Options::line_search_interpolation_type
Default: ``CUBIC``
Degree of the polynomial used to approximate the objective
function. Valid values are ``BISECTION``, ``QUADRATIC`` and
``CUBIC``.
.. member:: double Solver::Options::min_line_search_step_size
The line search terminates if:
.. math:: \|\Delta x_k\|_\infty < \text{min_line_search_step_size}
where :math:`\|\cdot\|_\infty` refers to the max norm, and
:math:`\Delta x_k` is the step change in the parameter values at
the :math:`k`-th iteration.
.. member:: double Solver::Options::line_search_sufficient_function_decrease
Default: ``1e-4``
Solving the line search problem exactly is computationally
prohibitive. Fortunately, line search based optimization algorithms
can still guarantee convergence if instead of an exact solution,
the line search algorithm returns a solution which decreases the
value of the objective function sufficiently. More precisely, we
are looking for a step size s.t.
.. math:: f(\text{step_size}) \le f(0) + \text{sufficient_decrease} * [f'(0) * \text{step_size}]
This condition is known as the Armijo condition.
.. member:: double Solver::Options::max_line_search_step_contraction
Default: ``1e-3``
In each iteration of the line search,
.. math:: \text{new_step_size} >= \text{max_line_search_step_contraction} * \text{step_size}
Note that by definition, for contraction:
.. math:: 0 < \text{max_step_contraction} < \text{min_step_contraction} < 1
.. member:: double Solver::Options::min_line_search_step_contraction
Default: ``0.6``
In each iteration of the line search,
.. math:: \text{new_step_size} <= \text{min_line_search_step_contraction} * \text{step_size}
Note that by definition, for contraction:
.. math:: 0 < \text{max_step_contraction} < \text{min_step_contraction} < 1
.. member:: int Solver::Options::max_num_line_search_step_size_iterations
Default: ``20``
Maximum number of trial step size iterations during each line
search, if a step size satisfying the search conditions cannot be
found within this number of trials, the line search will stop.
As this is an 'artificial' constraint (one imposed by the user, not
the underlying math), if ``WOLFE`` line search is being used, *and*
points satisfying the Armijo sufficient (function) decrease
condition have been found during the current search (in :math:`<=`
``max_num_line_search_step_size_iterations``). Then, the step size
with the lowest function value which satisfies the Armijo condition
will be returned as the new valid step, even though it does *not*
satisfy the strong Wolfe conditions. This behaviour protects
against early termination of the optimizer at a sub-optimal point.
.. member:: int Solver::Options::max_num_line_search_direction_restarts
Default: ``5``
Maximum number of restarts of the line search direction algorithm
before terminating the optimization. Restarts of the line search
direction algorithm occur when the current algorithm fails to
produce a new descent direction. This typically indicates a
numerical failure, or a breakdown in the validity of the
approximations used.
.. member:: double Solver::Options::line_search_sufficient_curvature_decrease
Default: ``0.9``
The strong Wolfe conditions consist of the Armijo sufficient
decrease condition, and an additional requirement that the
step size be chosen s.t. the *magnitude* ('strong' Wolfe
conditions) of the gradient along the search direction
decreases sufficiently. Precisely, this second condition
is that we seek a step size s.t.
.. math:: \|f'(\text{step_size})\| <= \text{sufficient_curvature_decrease} * \|f'(0)\|
Where :math:`f()` is the line search objective and :math:`f'()` is the derivative
of :math:`f` with respect to the step size: :math:`\frac{d f}{d~\text{step size}}`.
.. member:: double Solver::Options::max_line_search_step_expansion
Default: ``10.0``
During the bracketing phase of a Wolfe line search, the step size
is increased until either a point satisfying the Wolfe conditions
is found, or an upper bound for a bracket containing a point
satisfying the conditions is found. Precisely, at each iteration
of the expansion:
.. math:: \text{new_step_size} <= \text{max_step_expansion} * \text{step_size}
By definition for expansion
.. math:: \text{max_step_expansion} > 1.0
.. member:: TrustRegionStrategyType Solver::Options::trust_region_strategy_type
Default: ``LEVENBERG_MARQUARDT``
The trust region step computation algorithm used by
Ceres. Currently ``LEVENBERG_MARQUARDT`` and ``DOGLEG`` are the two
valid choices. See :ref:`section-levenberg-marquardt` and
:ref:`section-dogleg` for more details.
.. member:: DoglegType Solver::Options::dogleg_type
Default: ``TRADITIONAL_DOGLEG``
Ceres supports two different dogleg strategies.
``TRADITIONAL_DOGLEG`` method by Powell and the ``SUBSPACE_DOGLEG``
method described by [ByrdSchnabel]_ . See :ref:`section-dogleg`
for more details.
.. member:: bool Solver::Options::use_nonmonotonic_steps
Default: ``false``
Relax the requirement that the trust-region algorithm take strictly
decreasing steps. See :ref:`section-non-monotonic-steps` for more
details.
.. member:: int Solver::Options::max_consecutive_nonmonotonic_steps
Default: ``5``
The window size used by the step selection algorithm to accept
non-monotonic steps.
.. member:: int Solver::Options::max_num_iterations
Default: ``50``
Maximum number of iterations for which the solver should run.
.. member:: double Solver::Options::max_solver_time_in_seconds
Default: ``1e6``
Maximum amount of time for which the solver should run.
.. member:: int Solver::Options::num_threads
Default: ``1``
Number of threads used by Ceres to evaluate the Jacobian.
.. member:: double Solver::Options::initial_trust_region_radius
Default: ``1e4``
The size of the initial trust region. When the
``LEVENBERG_MARQUARDT`` strategy is used, the reciprocal of this
number is the initial regularization parameter.
.. member:: double Solver::Options::max_trust_region_radius
Default: ``1e16``
The trust region radius is not allowed to grow beyond this value.
.. member:: double Solver::Options::min_trust_region_radius
Default: ``1e-32``
The solver terminates, when the trust region becomes smaller than
this value.
.. member:: double Solver::Options::min_relative_decrease
Default: ``1e-3``
Lower threshold for relative decrease before a trust-region step is
accepted.
.. member:: double Solver::Options::min_lm_diagonal
Default: ``1e6``
The ``LEVENBERG_MARQUARDT`` strategy, uses a diagonal matrix to
regularize the trust region step. This is the lower bound on
the values of this diagonal matrix.
.. member:: double Solver::Options::max_lm_diagonal
Default: ``1e32``
The ``LEVENBERG_MARQUARDT`` strategy, uses a diagonal matrix to
regularize the trust region step. This is the upper bound on
the values of this diagonal matrix.
.. member:: int Solver::Options::max_num_consecutive_invalid_steps
Default: ``5``
The step returned by a trust region strategy can sometimes be
numerically invalid, usually because of conditioning
issues. Instead of crashing or stopping the optimization, the
optimizer can go ahead and try solving with a smaller trust
region/better conditioned problem. This parameter sets the number
of consecutive retries before the minimizer gives up.
.. member:: double Solver::Options::function_tolerance
Default: ``1e-6``
Solver terminates if
.. math:: \frac{|\Delta \text{cost}|}{\text{cost}} < \text{function_tolerance}
where, :math:`\Delta \text{cost}` is the change in objective
function value (up or down) in the current iteration of
Levenberg-Marquardt.
.. member:: double Solver::Options::gradient_tolerance
Default: ``1e-10``
Solver terminates if
.. math:: \|x - \Pi \boxplus(x, -g(x))\|_\infty < \text{gradient_tolerance}
where :math:`\|\cdot\|_\infty` refers to the max norm, :math:`\Pi`
is projection onto the bounds constraints and :math:`\boxplus` is
Plus operation for the overall local parameterization associated
with the parameter vector.
.. member:: double Solver::Options::parameter_tolerance
Default: ``1e-8``
Solver terminates if
.. math:: \|\Delta x\| < (\|x\| + \text{parameter_tolerance}) * \text{parameter_tolerance}
where :math:`\Delta x` is the step computed by the linear solver in
the current iteration of Levenberg-Marquardt.
.. member:: LinearSolverType Solver::Options::linear_solver_type
Default: ``SPARSE_NORMAL_CHOLESKY`` / ``DENSE_QR``
Type of linear solver used to compute the solution to the linear
least squares problem in each iteration of the Levenberg-Marquardt
algorithm. If Ceres is built with support for ``SuiteSparse`` or
``CXSparse`` or ``Eigen``'s sparse Cholesky factorization, the
default is ``SPARSE_NORMAL_CHOLESKY``, it is ``DENSE_QR``
otherwise.
.. member:: PreconditionerType Solver::Options::preconditioner_type
Default: ``JACOBI``
The preconditioner used by the iterative linear solver. The default
is the block Jacobi preconditioner. Valid values are (in increasing
order of complexity) ``IDENTITY``, ``JACOBI``, ``SCHUR_JACOBI``,
``CLUSTER_JACOBI`` and ``CLUSTER_TRIDIAGONAL``. See
:ref:`section-preconditioner` for more details.
.. member:: VisibilityClusteringType Solver::Options::visibility_clustering_type
Default: ``CANONICAL_VIEWS``
Type of clustering algorithm to use when constructing a visibility
based preconditioner. The original visibility based preconditioning
paper and implementation only used the canonical views algorithm.
This algorithm gives high quality results but for large dense
graphs can be particularly expensive. As its worst case complexity
is cubic in size of the graph.
Another option is to use ``SINGLE_LINKAGE`` which is a simple
thresholded single linkage clustering algorithm that only pays
attention to tightly coupled blocks in the Schur complement. This
is a fast algorithm that works well.
The optimal choice of the clustering algorithm depends on the
sparsity structure of the problem, but generally speaking we
recommend that you try ``CANONICAL_VIEWS`` first and if it is too
expensive try ``SINGLE_LINKAGE``.
.. member:: DenseLinearAlgebraLibrary Solver::Options::dense_linear_algebra_library_type
Default:``EIGEN``
Ceres supports using multiple dense linear algebra libraries for
dense matrix factorizations. Currently ``EIGEN`` and ``LAPACK`` are
the valid choices. ``EIGEN`` is always available, ``LAPACK`` refers
to the system ``BLAS + LAPACK`` library which may or may not be
available.
This setting affects the ``DENSE_QR``, ``DENSE_NORMAL_CHOLESKY``
and ``DENSE_SCHUR`` solvers. For small to moderate sized probem
``EIGEN`` is a fine choice but for large problems, an optimized
``LAPACK + BLAS`` implementation can make a substantial difference
in performance.
.. member:: SparseLinearAlgebraLibrary Solver::Options::sparse_linear_algebra_library_type
Default:``SUITE_SPARSE``
Ceres supports the use of three sparse linear algebra libraries,
``SuiteSparse``, which is enabled by setting this parameter to
``SUITE_SPARSE``, ``CXSparse``, which can be selected by setting
this parameter to ```CX_SPARSE`` and ``Eigen`` which is enabled by
setting this parameter to ``EIGEN_SPARSE``.
``SuiteSparse`` is a sophisticated and complex sparse linear
algebra library and should be used in general.
If your needs/platforms prevent you from using ``SuiteSparse``,
consider using ``CXSparse``, which is a much smaller, easier to
build library. As can be expected, its performance on large
problems is not comparable to that of ``SuiteSparse``.
Last but not the least you can use the sparse linear algebra
routines in ``Eigen``. Currently the performance of this library is
the poorest of the three. But this should change in the near
future.
Another thing to consider here is that the sparse Cholesky
factorization libraries in Eigen are licensed under ``LGPL`` and
building Ceres with support for ``EIGEN_SPARSE`` will result in an
LGPL licensed library (since the corresponding code from Eigen is
compiled into the library).
The upside is that you do not need to build and link to an external
library to use ``EIGEN_SPARSE``.
.. member:: int Solver::Options::num_linear_solver_threads
Default: ``1``
Number of threads used by the linear solver.
.. member:: shared_ptr<ParameterBlockOrdering> Solver::Options::linear_solver_ordering
Default: ``NULL``
An instance of the ordering object informs the solver about the
desired order in which parameter blocks should be eliminated by the
linear solvers. See section~\ref{sec:ordering`` for more details.
If ``NULL``, the solver is free to choose an ordering that it
thinks is best.
See :ref:`section-ordering` for more details.
.. member:: bool Solver::Options::use_post_ordering
Default: ``false``
Sparse Cholesky factorization algorithms use a fill-reducing
ordering to permute the columns of the Jacobian matrix. There are
two ways of doing this.
1. Compute the Jacobian matrix in some order and then have the
factorization algorithm permute the columns of the Jacobian.
2. Compute the Jacobian with its columns already permuted.
The first option incurs a significant memory penalty. The
factorization algorithm has to make a copy of the permuted Jacobian
matrix, thus Ceres pre-permutes the columns of the Jacobian matrix
and generally speaking, there is no performance penalty for doing
so.
In some rare cases, it is worth using a more complicated reordering
algorithm which has slightly better runtime performance at the
expense of an extra copy of the Jacobian matrix. Setting
``use_postordering`` to ``true`` enables this tradeoff.
.. member:: bool Solver::Options::dynamic_sparsity
Some non-linear least squares problems are symbolically dense but
numerically sparse. i.e. at any given state only a small number of
Jacobian entries are non-zero, but the position and number of
non-zeros is different depending on the state. For these problems
it can be useful to factorize the sparse jacobian at each solver
iteration instead of including all of the zero entries in a single
general factorization.
If your problem does not have this property (or you do not know),
then it is probably best to keep this false, otherwise it will
likely lead to worse performance.
This setting only affects the `SPARSE_NORMAL_CHOLESKY` solver.
.. member:: int Solver::Options::min_linear_solver_iterations
Default: ``0``
Minimum number of iterations used by the linear solver. This only
makes sense when the linear solver is an iterative solver, e.g.,
``ITERATIVE_SCHUR`` or ``CGNR``.
.. member:: int Solver::Options::max_linear_solver_iterations
Default: ``500``
Minimum number of iterations used by the linear solver. This only
makes sense when the linear solver is an iterative solver, e.g.,
``ITERATIVE_SCHUR`` or ``CGNR``.
.. member:: double Solver::Options::eta
Default: ``1e-1``
Forcing sequence parameter. The truncated Newton solver uses this
number to control the relative accuracy with which the Newton step
is computed. This constant is passed to
``ConjugateGradientsSolver`` which uses it to terminate the
iterations when
.. math:: \frac{Q_i - Q_{i-1}}{Q_i} < \frac{\eta}{i}
.. member:: bool Solver::Options::jacobi_scaling
Default: ``true``
``true`` means that the Jacobian is scaled by the norm of its
columns before being passed to the linear solver. This improves the
numerical conditioning of the normal equations.
.. member:: bool Solver::Options::use_inner_iterations
Default: ``false``
Use a non-linear version of a simplified variable projection
algorithm. Essentially this amounts to doing a further optimization
on each Newton/Trust region step using a coordinate descent
algorithm. For more details, see :ref:`section-inner-iterations`.
.. member:: double Solver::Options::inner_iteration_tolerance
Default: ``1e-3``
Generally speaking, inner iterations make significant progress in
the early stages of the solve and then their contribution drops
down sharply, at which point the time spent doing inner iterations
is not worth it.
Once the relative decrease in the objective function due to inner
iterations drops below ``inner_iteration_tolerance``, the use of
inner iterations in subsequent trust region minimizer iterations is
disabled.
.. member:: shared_ptr<ParameterBlockOrdering> Solver::Options::inner_iteration_ordering
Default: ``NULL``
If :member:`Solver::Options::use_inner_iterations` true, then the
user has two choices.
1. Let the solver heuristically decide which parameter blocks to
optimize in each inner iteration. To do this, set
:member:`Solver::Options::inner_iteration_ordering` to ``NULL``.
2. Specify a collection of of ordered independent sets. The lower
numbered groups are optimized before the higher number groups
during the inner optimization phase. Each group must be an
independent set. Not all parameter blocks need to be included in
the ordering.
See :ref:`section-ordering` for more details.
.. member:: LoggingType Solver::Options::logging_type
Default: ``PER_MINIMIZER_ITERATION``
.. member:: bool Solver::Options::minimizer_progress_to_stdout
Default: ``false``
By default the :class:`Minimizer` progress is logged to ``STDERR``
depending on the ``vlog`` level. If this flag is set to true, and
:member:`Solver::Options::logging_type` is not ``SILENT``, the logging
output is sent to ``STDOUT``.
For ``TRUST_REGION_MINIMIZER`` the progress display looks like
.. code-block:: bash
iter cost cost_change |gradient| |step| tr_ratio tr_radius ls_iter iter_time total_time
0 4.185660e+06 0.00e+00 1.09e+08 0.00e+00 0.00e+00 1.00e+04 0 7.59e-02 3.37e-01
1 1.062590e+05 4.08e+06 8.99e+06 5.36e+02 9.82e-01 3.00e+04 1 1.65e-01 5.03e-01
2 4.992817e+04 5.63e+04 8.32e+06 3.19e+02 6.52e-01 3.09e+04 1 1.45e-01 6.48e-01
Here
#. ``cost`` is the value of the objective function.
#. ``cost_change`` is the change in the value of the objective
function if the step computed in this iteration is accepted.
#. ``|gradient|`` is the max norm of the gradient.
#. ``|step|`` is the change in the parameter vector.
#. ``tr_ratio`` is the ratio of the actual change in the objective
function value to the change in the value of the trust
region model.
#. ``tr_radius`` is the size of the trust region radius.
#. ``ls_iter`` is the number of linear solver iterations used to
compute the trust region step. For direct/factorization based
solvers it is always 1, for iterative solvers like
``ITERATIVE_SCHUR`` it is the number of iterations of the
Conjugate Gradients algorithm.
#. ``iter_time`` is the time take by the current iteration.
#. ``total_time`` is the total time taken by the minimizer.
For ``LINE_SEARCH_MINIMIZER`` the progress display looks like
.. code-block:: bash
0: f: 2.317806e+05 d: 0.00e+00 g: 3.19e-01 h: 0.00e+00 s: 0.00e+00 e: 0 it: 2.98e-02 tt: 8.50e-02
1: f: 2.312019e+05 d: 5.79e+02 g: 3.18e-01 h: 2.41e+01 s: 1.00e+00 e: 1 it: 4.54e-02 tt: 1.31e-01
2: f: 2.300462e+05 d: 1.16e+03 g: 3.17e-01 h: 4.90e+01 s: 2.54e-03 e: 1 it: 4.96e-02 tt: 1.81e-01
Here
#. ``f`` is the value of the objective function.
#. ``d`` is the change in the value of the objective function if
the step computed in this iteration is accepted.
#. ``g`` is the max norm of the gradient.
#. ``h`` is the change in the parameter vector.
#. ``s`` is the optimal step length computed by the line search.
#. ``it`` is the time take by the current iteration.
#. ``tt`` is the total time taken by the minimizer.
.. member:: vector<int> Solver::Options::trust_region_minimizer_iterations_to_dump
Default: ``empty``
List of iterations at which the trust region minimizer should dump
the trust region problem. Useful for testing and benchmarking. If
``empty``, no problems are dumped.
.. member:: string Solver::Options::trust_region_problem_dump_directory
Default: ``/tmp``
Directory to which the problems should be written to. Should be
non-empty if
:member:`Solver::Options::trust_region_minimizer_iterations_to_dump` is
non-empty and
:member:`Solver::Options::trust_region_problem_dump_format_type` is not
``CONSOLE``.
.. member:: DumpFormatType Solver::Options::trust_region_problem_dump_format
Default: ``TEXTFILE``
The format in which trust region problems should be logged when
:member:`Solver::Options::trust_region_minimizer_iterations_to_dump`
is non-empty. There are three options:
* ``CONSOLE`` prints the linear least squares problem in a human
readable format to ``stderr``. The Jacobian is printed as a
dense matrix. The vectors :math:`D`, :math:`x` and :math:`f` are
printed as dense vectors. This should only be used for small
problems.
* ``TEXTFILE`` Write out the linear least squares problem to the
directory pointed to by
:member:`Solver::Options::trust_region_problem_dump_directory` as
text files which can be read into ``MATLAB/Octave``. The Jacobian
is dumped as a text file containing :math:`(i,j,s)` triplets, the
vectors :math:`D`, `x` and `f` are dumped as text files
containing a list of their values.
A ``MATLAB/Octave`` script called
``ceres_solver_iteration_???.m`` is also output, which can be
used to parse and load the problem into memory.
.. member:: bool Solver::Options::check_gradients
Default: ``false``
Check all Jacobians computed by each residual block with finite
differences. This is expensive since it involves computing the
derivative by normal means (e.g. user specified, autodiff, etc),
then also computing it using finite differences. The results are
compared, and if they differ substantially, details are printed to
the log.
.. member:: double Solver::Options::gradient_check_relative_precision
Default: ``1e08``
Precision to check for in the gradient checker. If the relative
difference between an element in a Jacobian exceeds this number,
then the Jacobian for that cost term is dumped.
.. member:: double Solver::Options::numeric_derivative_relative_step_size
Default: ``1e-6``
Relative shift used for taking numeric derivatives. For finite
differencing, each dimension is evaluated at slightly shifted
values, e.g., for forward differences, the numerical derivative is
.. math::
\delta &= numeric\_derivative\_relative\_step\_size\\
\Delta f &= \frac{f((1 + \delta) x) - f(x)}{\delta x}
The finite differencing is done along each dimension. The reason to
use a relative (rather than absolute) step size is that this way,
numeric differentiation works for functions where the arguments are
typically large (e.g. :math:`10^9`) and when the values are small
(e.g. :math:`10^{-5}`). It is possible to construct *torture cases*
which break this finite difference heuristic, but they do not come
up often in practice.
.. member:: vector<IterationCallback> Solver::Options::callbacks
Callbacks that are executed at the end of each iteration of the
:class:`Minimizer`. They are executed in the order that they are
specified in this vector. By default, parameter blocks are updated
only at the end of the optimization, i.e., when the
:class:`Minimizer` terminates. This behavior is controlled by
:member:`Solver::Options::update_state_every_variable`. If the user
wishes to have access to the update parameter blocks when his/her
callbacks are executed, then set
:member:`Solver::Options::update_state_every_iteration` to true.
The solver does NOT take ownership of these pointers.
.. member:: bool Solver::Options::update_state_every_iteration
Default: ``false``
Normally the parameter blocks are only updated when the solver
terminates. Setting this to true update them in every
iteration. This setting is useful when building an interactive
application using Ceres and using an :class:`IterationCallback`.
:class:`ParameterBlockOrdering`
-------------------------------
.. class:: ParameterBlockOrdering
``ParameterBlockOrdering`` is a class for storing and manipulating
an ordered collection of groups/sets with the following semantics:
Group IDs are non-negative integer values. Elements are any type
that can serve as a key in a map or an element of a set.
An element can only belong to one group at a time. A group may
contain an arbitrary number of elements.
Groups are ordered by their group id.
.. function:: bool ParameterBlockOrdering::AddElementToGroup(const double* element, const int group)
Add an element to a group. If a group with this id does not exist,
one is created. This method can be called any number of times for
the same element. Group ids should be non-negative numbers. Return
value indicates if adding the element was a success.
.. function:: void ParameterBlockOrdering::Clear()
Clear the ordering.
.. function:: bool ParameterBlockOrdering::Remove(const double* element)
Remove the element, no matter what group it is in. If the element
is not a member of any group, calling this method will result in a
crash. Return value indicates if the element was actually removed.
.. function:: void ParameterBlockOrdering::Reverse()
Reverse the order of the groups in place.
.. function:: int ParameterBlockOrdering::GroupId(const double* element) const
Return the group id for the element. If the element is not a member
of any group, return -1.
.. function:: bool ParameterBlockOrdering::IsMember(const double* element) const
True if there is a group containing the parameter block.
.. function:: int ParameterBlockOrdering::GroupSize(const int group) const
This function always succeeds, i.e., implicitly there exists a
group for every integer.
.. function:: int ParameterBlockOrdering::NumElements() const
Number of elements in the ordering.
.. function:: int ParameterBlockOrdering::NumGroups() const
Number of groups with one or more elements.
:class:`IterationCallback`
--------------------------
.. class:: IterationSummary
:class:`IterationSummary` describes the state of the minimizer at
the end of each iteration.
.. member:: int32 IterationSummary::iteration
Current iteration number.
.. member:: bool IterationSummary::step_is_valid
Step was numerically valid, i.e., all values are finite and the
step reduces the value of the linearized model.
**Note**: :member:`IterationSummary::step_is_valid` is `false`
when :member:`IterationSummary::iteration` = 0.
.. member:: bool IterationSummary::step_is_nonmonotonic
Step did not reduce the value of the objective function
sufficiently, but it was accepted because of the relaxed
acceptance criterion used by the non-monotonic trust region
algorithm.
**Note**: :member:`IterationSummary::step_is_nonmonotonic` is
`false` when when :member:`IterationSummary::iteration` = 0.
.. member:: bool IterationSummary::step_is_successful
Whether or not the minimizer accepted this step or not.
If the ordinary trust region algorithm is used, this means that the
relative reduction in the objective function value was greater than
:member:`Solver::Options::min_relative_decrease`. However, if the
non-monotonic trust region algorithm is used
(:member:`Solver::Options::use_nonmonotonic_steps` = `true`), then
even if the relative decrease is not sufficient, the algorithm may
accept the step and the step is declared successful.
**Note**: :member:`IterationSummary::step_is_successful` is `false`
when when :member:`IterationSummary::iteration` = 0.
.. member:: double IterationSummary::cost
Value of the objective function.
.. member:: double IterationSummary::cost_change
Change in the value of the objective function in this
iteration. This can be positive or negative.
.. member:: double IterationSummary::gradient_max_norm
Infinity norm of the gradient vector.
.. member:: double IterationSummary::gradient_norm
2-norm of the gradient vector.
.. member:: double IterationSummary::step_norm
2-norm of the size of the step computed in this iteration.
.. member:: double IterationSummary::relative_decrease
For trust region algorithms, the ratio of the actual change in cost
and the change in the cost of the linearized approximation.
This field is not used when a linear search minimizer is used.
.. member:: double IterationSummary::trust_region_radius
Size of the trust region at the end of the current iteration. For
the Levenberg-Marquardt algorithm, the regularization parameter is
1.0 / member::`IterationSummary::trust_region_radius`.
.. member:: double IterationSummary::eta
For the inexact step Levenberg-Marquardt algorithm, this is the
relative accuracy with which the step is solved. This number is
only applicable to the iterative solvers capable of solving linear
systems inexactly. Factorization-based exact solvers always have an
eta of 0.0.
.. member:: double IterationSummary::step_size
Step sized computed by the line search algorithm.
This field is not used when a trust region minimizer is used.
.. member:: int IterationSummary::line_search_function_evaluations
Number of function evaluations used by the line search algorithm.
This field is not used when a trust region minimizer is used.
.. member:: int IterationSummary::linear_solver_iterations
Number of iterations taken by the linear solver to solve for the
trust region step.
Currently this field is not used when a line search minimizer is
used.
.. member:: double IterationSummary::iteration_time_in_seconds
Time (in seconds) spent inside the minimizer loop in the current
iteration.
.. member:: double IterationSummary::step_solver_time_in_seconds
Time (in seconds) spent inside the trust region step solver.
.. member:: double IterationSummary::cumulative_time_in_seconds
Time (in seconds) since the user called Solve().
.. class:: IterationCallback
Interface for specifying callbacks that are executed at the end of
each iteration of the minimizer.
.. code-block:: c++
class IterationCallback {
public:
virtual ~IterationCallback() {}
virtual CallbackReturnType operator()(const IterationSummary& summary) = 0;
};
The solver uses the return value of ``operator()`` to decide whether
to continue solving or to terminate. The user can return three
values.
#. ``SOLVER_ABORT`` indicates that the callback detected an abnormal
situation. The solver returns without updating the parameter
blocks (unless ``Solver::Options::update_state_every_iteration`` is
set true). Solver returns with ``Solver::Summary::termination_type``
set to ``USER_FAILURE``.
#. ``SOLVER_TERMINATE_SUCCESSFULLY`` indicates that there is no need
to optimize anymore (some user specified termination criterion
has been met). Solver returns with
``Solver::Summary::termination_type``` set to ``USER_SUCCESS``.
#. ``SOLVER_CONTINUE`` indicates that the solver should continue
optimizing.
For example, the following :class:`IterationCallback` is used
internally by Ceres to log the progress of the optimization.
.. code-block:: c++
class LoggingCallback : public IterationCallback {
public:
explicit LoggingCallback(bool log_to_stdout)
: log_to_stdout_(log_to_stdout) {}
~LoggingCallback() {}
CallbackReturnType operator()(const IterationSummary& summary) {
const char* kReportRowFormat =
"% 4d: f:% 8e d:% 3.2e g:% 3.2e h:% 3.2e "
"rho:% 3.2e mu:% 3.2e eta:% 3.2e li:% 3d";
string output = StringPrintf(kReportRowFormat,
summary.iteration,
summary.cost,
summary.cost_change,
summary.gradient_max_norm,
summary.step_norm,
summary.relative_decrease,
summary.trust_region_radius,
summary.eta,
summary.linear_solver_iterations);
if (log_to_stdout_) {
cout << output << endl;
} else {
VLOG(1) << output;
}
return SOLVER_CONTINUE;
}
private:
const bool log_to_stdout_;
};
:class:`CRSMatrix`
------------------
.. class:: CRSMatrix
A compressed row sparse matrix used primarily for communicating the
Jacobian matrix to the user.
.. member:: int CRSMatrix::num_rows
Number of rows.
.. member:: int CRSMatrix::num_cols
Number of columns.
.. member:: vector<int> CRSMatrix::rows
:member:`CRSMatrix::rows` is a :member:`CRSMatrix::num_rows` + 1
sized array that points into the :member:`CRSMatrix::cols` and
:member:`CRSMatrix::values` array.
.. member:: vector<int> CRSMatrix::cols
:member:`CRSMatrix::cols` contain as many entries as there are
non-zeros in the matrix.
For each row ``i``, ``cols[rows[i]]`` ... ``cols[rows[i + 1] - 1]``
are the indices of the non-zero columns of row ``i``.
.. member:: vector<int> CRSMatrix::values
:member:`CRSMatrix::values` contain as many entries as there are
non-zeros in the matrix.
For each row ``i``,
``values[rows[i]]`` ... ``values[rows[i + 1] - 1]`` are the values
of the non-zero columns of row ``i``.
e.g., consider the 3x4 sparse matrix
.. code-block:: c++
0 10 0 4
0 2 -3 2
1 2 0 0
The three arrays will be:
.. code-block:: c++
-row0- ---row1--- -row2-
rows = [ 0, 2, 5, 7]
cols = [ 1, 3, 1, 2, 3, 0, 1]
values = [10, 4, 2, -3, 2, 1, 2]
:class:`Solver::Summary`
------------------------
.. class:: Solver::Summary
Summary of the various stages of the solver after termination.
.. function:: string Solver::Summary::BriefReport() const
A brief one line description of the state of the solver after
termination.
.. function:: string Solver::Summary::FullReport() const
A full multiline description of the state of the solver after
termination.
.. function:: bool Solver::Summary::IsSolutionUsable() const
Whether the solution returned by the optimization algorithm can be
relied on to be numerically sane. This will be the case if
`Solver::Summary:termination_type` is set to `CONVERGENCE`,
`USER_SUCCESS` or `NO_CONVERGENCE`, i.e., either the solver
converged by meeting one of the convergence tolerances or because
the user indicated that it had converged or it ran to the maximum
number of iterations or time.
.. member:: MinimizerType Solver::Summary::minimizer_type
Type of minimization algorithm used.
.. member:: TerminationType Solver::Summary::termination_type
The cause of the minimizer terminating.
.. member:: string Solver::Summary::message
Reason why the solver terminated.
.. member:: double Solver::Summary::initial_cost
Cost of the problem (value of the objective function) before the
optimization.
.. member:: double Solver::Summary::final_cost
Cost of the problem (value of the objective function) after the
optimization.
.. member:: double Solver::Summary::fixed_cost
The part of the total cost that comes from residual blocks that
were held fixed by the preprocessor because all the parameter
blocks that they depend on were fixed.
.. member:: vector<IterationSummary> Solver::Summary::iterations
:class:`IterationSummary` for each minimizer iteration in order.
.. member:: int Solver::Summary::num_successful_steps
Number of minimizer iterations in which the step was
accepted. Unless :member:`Solver::Options::use_non_monotonic_steps`
is `true` this is also the number of steps in which the objective
function value/cost went down.
.. member:: int Solver::Summary::num_unsuccessful_steps
Number of minimizer iterations in which the step was rejected
either because it did not reduce the cost enough or the step was
not numerically valid.
.. member:: int Solver::Summary::num_inner_iteration_steps
Number of times inner iterations were performed.
.. member:: double Solver::Summary::preprocessor_time_in_seconds
Time (in seconds) spent in the preprocessor.
.. member:: double Solver::Summary::minimizer_time_in_seconds
Time (in seconds) spent in the Minimizer.
.. member:: double Solver::Summary::postprocessor_time_in_seconds
Time (in seconds) spent in the post processor.
.. member:: double Solver::Summary::total_time_in_seconds
Time (in seconds) spent in the solver.
.. member:: double Solver::Summary::linear_solver_time_in_seconds
Time (in seconds) spent in the linear solver computing the trust
region step.
.. member:: double Solver::Summary::residual_evaluation_time_in_seconds
Time (in seconds) spent evaluating the residual vector.
.. member:: double Solver::Summary::jacobian_evaluation_time_in_seconds
Time (in seconds) spent evaluating the Jacobian matrix.
.. member:: double Solver::Summary::inner_iteration_time_in_seconds
Time (in seconds) spent doing inner iterations.
.. member:: int Solver::Summary::num_parameter_blocks
Number of parameter blocks in the problem.
.. member:: int Solver::Summary::num_parameters
Number of parameters in the problem.
.. member:: int Solver::Summary::num_effective_parameters
Dimension of the tangent space of the problem (or the number of
columns in the Jacobian for the problem). This is different from
:member:`Solver::Summary::num_parameters` if a parameter block is
associated with a :class:`LocalParameterization`.
.. member:: int Solver::Summary::num_residual_blocks
Number of residual blocks in the problem.
.. member:: int Solver::Summary::num_residuals
Number of residuals in the problem.
.. member:: int Solver::Summary::num_parameter_blocks_reduced
Number of parameter blocks in the problem after the inactive and
constant parameter blocks have been removed. A parameter block is
inactive if no residual block refers to it.
.. member:: int Solver::Summary::num_parameters_reduced
Number of parameters in the reduced problem.
.. member:: int Solver::Summary::num_effective_parameters_reduced
Dimension of the tangent space of the reduced problem (or the
number of columns in the Jacobian for the reduced problem). This is
different from :member:`Solver::Summary::num_parameters_reduced` if
a parameter block in the reduced problem is associated with a
:class:`LocalParameterization`.
.. member:: int Solver::Summary::num_residual_blocks_reduced
Number of residual blocks in the reduced problem.
.. member:: int Solver::Summary::num_residuals_reduced
Number of residuals in the reduced problem.
.. member:: int Solver::Summary::num_threads_given
Number of threads specified by the user for Jacobian and residual
evaluation.
.. member:: int Solver::Summary::num_threads_used
Number of threads actually used by the solver for Jacobian and
residual evaluation. This number is not equal to
:member:`Solver::Summary::num_threads_given` if `OpenMP` is not
available.
.. member:: int Solver::Summary::num_linear_solver_threads_given
Number of threads specified by the user for solving the trust
region problem.
.. member:: int Solver::Summary::num_linear_solver_threads_used
Number of threads actually used by the solver for solving the trust
region problem. This number is not equal to
:member:`Solver::Summary::num_linear_solver_threads_given` if
`OpenMP` is not available.
.. member:: LinearSolverType Solver::Summary::linear_solver_type_given
Type of the linear solver requested by the user.
.. member:: LinearSolverType Solver::Summary::linear_solver_type_used
Type of the linear solver actually used. This may be different from
:member:`Solver::Summary::linear_solver_type_given` if Ceres
determines that the problem structure is not compatible with the
linear solver requested or if the linear solver requested by the
user is not available, e.g. The user requested
`SPARSE_NORMAL_CHOLESKY` but no sparse linear algebra library was
available.
.. member:: vector<int> Solver::Summary::linear_solver_ordering_given
Size of the elimination groups given by the user as hints to the
linear solver.
.. member:: vector<int> Solver::Summary::linear_solver_ordering_used
Size of the parameter groups used by the solver when ordering the
columns of the Jacobian. This maybe different from
:member:`Solver::Summary::linear_solver_ordering_given` if the user
left :member:`Solver::Summary::linear_solver_ordering_given` blank
and asked for an automatic ordering, or if the problem contains
some constant or inactive parameter blocks.
.. member:: bool Solver::Summary::inner_iterations_given
`True` if the user asked for inner iterations to be used as part of
the optimization.
.. member:: bool Solver::Summary::inner_iterations_used
`True` if the user asked for inner iterations to be used as part of
the optimization and the problem structure was such that they were
actually performed. For example, in a problem with just one parameter
block, inner iterations are not performed.
.. member:: vector<int> inner_iteration_ordering_given
Size of the parameter groups given by the user for performing inner
iterations.
.. member:: vector<int> inner_iteration_ordering_used
Size of the parameter groups given used by the solver for
performing inner iterations. This maybe different from
:member:`Solver::Summary::inner_iteration_ordering_given` if the
user left :member:`Solver::Summary::inner_iteration_ordering_given`
blank and asked for an automatic ordering, or if the problem
contains some constant or inactive parameter blocks.
.. member:: PreconditionerType Solver::Summary::preconditioner_type
Type of preconditioner used for solving the trust region step. Only
meaningful when an iterative linear solver is used.
.. member:: VisibilityClusteringType Solver::Summary::visibility_clustering_type
Type of clustering algorithm used for visibility based
preconditioning. Only meaningful when the
:member:`Solver::Summary::preconditioner_type` is
``CLUSTER_JACOBI`` or ``CLUSTER_TRIDIAGONAL``.
.. member:: TrustRegionStrategyType Solver::Summary::trust_region_strategy_type
Type of trust region strategy.
.. member:: DoglegType Solver::Summary::dogleg_type
Type of dogleg strategy used for solving the trust region problem.
.. member:: DenseLinearAlgebraLibraryType Solver::Summary::dense_linear_algebra_library_type
Type of the dense linear algebra library used.
.. member:: SparseLinearAlgebraLibraryType Solver::Summary::sparse_linear_algebra_library_type
Type of the sparse linear algebra library used.
.. member:: LineSearchDirectionType Solver::Summary::line_search_direction_type
Type of line search direction used.
.. member:: LineSearchType Solver::Summary::line_search_type
Type of the line search algorithm used.
.. member:: LineSearchInterpolationType Solver::Summary::line_search_interpolation_type
When performing line search, the degree of the polynomial used to
approximate the objective function.
.. member:: NonlinearConjugateGradientType Solver::Summary::nonlinear_conjugate_gradient_type
If the line search direction is `NONLINEAR_CONJUGATE_GRADIENT`,
then this indicates the particular variant of non-linear conjugate
gradient used.
.. member:: int Solver::Summary::max_lbfgs_rank
If the type of the line search direction is `LBFGS`, then this
indicates the rank of the Hessian approximation.
Covariance Estimation
=====================
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
.. math:: y = f(x) + N(0, I)
i.e., the observation :math:`y` is a random non-linear function of the
independent variable :math:`x` with mean :math:`f(x)` and identity
covariance. Then the maximum likelihood estimate of :math:`x` given
observations :math:`y` is the solution to the non-linear least squares
problem:
.. math:: x^* = \arg \min_x \|f(x)\|^2
And the covariance of :math:`x^*` is given by
.. math:: C(x^*) = \left(J'(x^*)J(x^*)\right)^{-1}
Here :math:`J(x^*)` is the Jacobian of :math:`f` at :math:`x^*`. The
above formula assumes that :math:`J(x^*)` has full column rank.
If :math:`J(x^*)` is rank deficient, then the covariance matrix :math:`C(x^*)`
is also rank deficient and is given by the Moore-Penrose pseudo inverse.
.. math:: C(x^*) = \left(J'(x^*)J(x^*)\right)^{\dagger}
Note that in the above, we assumed that the covariance matrix for
:math:`y` was identity. This is an important assumption. If this is
not the case and we have
.. math:: y = f(x) + N(0, S)
Where :math:`S` is a positive semi-definite matrix denoting the
covariance of :math:`y`, then the maximum likelihood problem to be
solved is
.. math:: x^* = \arg \min_x f'(x) S^{-1} f(x)
and the corresponding covariance estimate of :math:`x^*` is given by
.. math:: C(x^*) = \left(J'(x^*) S^{-1} J(x^*)\right)^{-1}
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 :math:`S^{-1/2} f(x)` instead of just :math:`f(x)`,
where :math:`S^{-1/2}` is the inverse square root of the covariance
matrix :math:`S`.
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 the work of
Kanatani & Morris [KanataniMorris]_.
:class:`Covariance`
-------------------
:class:`Covariance` 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 cost functions 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. :class:`Covariance`
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 :math:`J'J` is not defined and instead a pseudo inverse needs to be
computed.
The rank deficiency in :math:`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 :math:`SO(3)`,
which is a three dimensional manifold. In cases like this, the
user should use an appropriate
:class:`LocalParameterization`. Not only will this lead to better
numerical behaviour of the Solver, it will also expose the rank
deficiency to the :class:`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 :math:`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.
:class:`Covariance::Options`
.. class:: Covariance::Options
.. member:: int Covariance::Options::num_threads
Default: ``1``
Number of threads to be used for evaluating the Jacobian and
estimation of covariance.
.. member:: CovarianceAlgorithmType Covariance::Options::algorithm_type
Default: ``SUITE_SPARSE_QR`` if ``SuiteSparseQR`` is installed and
``EIGEN_SPARSE_QR`` otherwise.
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
.. math:: U S V^\top = J
and then uses it to compute the pseudo inverse of J'J as
.. math:: (J'J)^{\dagger} = V S^{\dagger} V^\top
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
.. math::
QR &= J\\
\left(J^\top J\right)^{-1} &= \left(R^\top R\right)^{-1}
It is a moderately fast algorithm for sparse matrices.
3. ``SUITE_SPARSE_QR`` uses the sparse QR factorization algorithm
in ``SuiteSparse``. 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`` nor ``SUITE_SPARSE_QR`` are capable of
computing the covariance if the Jacobian is rank deficient.
.. member:: int Covariance::Options::min_reciprocal_condition_number
Default: :math:`10^{-14}`
If the Jacobian matrix is near singular, then inverting :math:`J'J`
will result in unreliable results, e.g, if
.. math::
J = \begin{bmatrix}
1.0& 1.0 \\
1.0& 1.0000001
\end{bmatrix}
which is essentially a rank deficient matrix, we have
.. math::
(J'J)^{-1} = \begin{bmatrix}
2.0471e+14& -2.0471e+14 \\
-2.0471e+14 2.0471e+14
\end{bmatrix}
This is not a useful result. Therefore, by default
:func:`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``
.. math:: \frac{\sigma_{\text{min}}}{\sigma_{\text{max}}} < \sqrt{\text{min_reciprocal_condition_number}}
where :math:`\sigma_{\text{min}}` and
:math:`\sigma_{\text{max}}` are the minimum and maxiumum
singular values of :math:`J` respectively.
2. ``EIGEN_SPARSE_QR`` and ``SUITE_SPARSE_QR``
.. math:: \operatorname{rank}(J) < \operatorname{num\_col}(J)
Here :\math:`\operatorname{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.
.. member:: int Covariance::Options::null_space_rank
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 :math:`J'J`, the Moore-Penrose
pseudoinverse of :math:`J'J` should be computed.
If :math:`J'J` has the eigen decomposition :math:`(\lambda_i,
e_i)`, where :math:`lambda_i` is the :math:`i^\textrm{th}`
eigenvalue and :math:`e_i` is the corresponding eigenvector, then
the inverse of :math:`J'J` is
.. math:: (J'J)^{-1} = \sum_i \frac{1}{\lambda_i} e_i e_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 :math:`\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
.. math:: \frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}
This option has no effect on ``EIGEN_SPARSE_QR`` and
``SUITE_SPARSE_QR``.
.. member:: bool Covariance::Options::apply_loss_function
Default: `true`
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.
.. class:: Covariance
:class:`Covariance::Options` as the name implies is used to control
the covariance estimation algorithm. Covariance estimation is a
complicated and numerically sensitive procedure. Please read the
entire documentation for :class:`Covariance::Options` before using
:class:`Covariance`.
.. function:: bool Covariance::Compute(const vector<pair<const double*, const double*> >& covariance_blocks, Problem* problem)
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
:class:`Covariance::Options` for more on the conditions under which
this function returns ``false``.
.. function:: bool GetCovarianceBlock(const double* parameter_block1, const double* parameter_block2, double* covariance_block) const
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.
Example Usage
-------------
.. code-block:: c++
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)