
.. default-domain:: cpp

.. cpp:namespace:: ceres

.. _chapter-solving:

=======
Solving
=======


Introduction
============

Effective use of Ceres requires some familiarity with the basic
components of a nonlinear 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 following optimization problem [#f1]_ .

.. math:: \arg \min_x \frac{1}{2}\|F(x)\|^2\ .
  :label: nonlinsq

Here, 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 :math:`g(x) = \nabla \frac{1}{2}\|F(x)\|^2 = J(x)^\top
F(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.

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. :math:`\arg \min_{\Delta x} \frac{1}{2}\|J(x)\Delta
      x + F(x)\|^2` s.t. :math:`\|D(x)\Delta x\|^2 \le \mu`
   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. Goto 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\quad \text{such that}\quad  \|D(x)\Delta x\|^2 \le \mu
   :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. 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`.


.. _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. e.g., 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 `a_1` and `a_2` optimization problems will do. The only constraint
on `a_1` and `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
Algorithm~\ref{alg:trust-region} is a descent algorithm in that they
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 implementation of line search algorithms in Ceres Solver is
fairly new and not very well tested, so for now this part of the
solver should be considered beta quality. We welcome reports of your
experiences both good and bad on the mailinglist.**

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 -`\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_RIBIRERE`` and ``HESTENES_STIEFEL``
   directions.

3. ``LBFGS`` In this method, a limited memory approximation to the
   inverse Hessian is maintained and used to compute a quasi-Newton
   step [Nocedal]_, [ByrdNocedal]_.

Currently Ceres Solver uses a backtracking and interpolation based
Armijo line search algorithm.

.. _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]_.

.. _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)`.  Cseres 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. e.g. 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 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.


.. 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``
   and ``LBFGS``.

.. member:: LineSearchType Solver::Options::line_search_type

   Default: ``ARMIJO``

   ``ARMIJO`` is the only choice right now.

.. member:: NonlinearConjugateGradientType Solver::Options::nonlinear_conjugate_gradient_type

   Default: ``FLETCHER_REEVES``

   Choices are ``FLETCHER_REEVES``, ``POLAK_RIBIRERE`` and
   ``HESTENES_STIEFEL``.

.. member:: int Solver::Options::max_lbfs_rank

   Default: 20

   The LBFGS 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:: 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

   If during the line search, the step size falls below this value, it
   is truncated to zero.

.. member:: double Solver::Options::armijo_sufficient_decrease

   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}]

.. member:: double Solver::Options::min_armijo_relative_step_size_change

   In each iteration of the Armijo line search,

    .. math:: \text{new_step_size} \ge \text{min_relative_step_size_change} * \text{step_size}

.. member:: double Solver::Options::max_armijo_relative_step_size_change

   In each iteration of the Armijo line search,

   .. math:: \text{new_step_size} \le \text{max_relative_step_size_change} * \text{step_size}

.. 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 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 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:: \frac{\|g(x)\|_\infty}{\|g(x_0)\|_\infty} < \text{gradient_tolerance}

   where :math:`\|\cdot\|_\infty` refers to the max norm, and :math:`x_0` is
   the vector of initial parameter values.

.. 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 build with ``SuiteSparse`` linked in then
   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:: SparseLinearAlgebraLibrary Solver::Options::sparse_linear_algebra_library

   Default:``SUITE_SPARSE``

   Ceres supports the use of two sparse linear algebra libraries,
   ``SuiteSparse``, which is enabled by setting this parameter to
   ``SUITE_SPARSE`` and ``CXSparse``, which can be selected by setting
   this parameter to ```CX_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``.

.. member:: int Solver::Options::num_linear_solver_threads

   Default: ``1``

   Number of threads used by the linear solver.

.. member:: 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:: int Solver::Options::min_linear_solver_iterations

   Default: ``1``

   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_itearation_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:: 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

      0: f: 1.250000e+01 d: 0.00e+00 g: 5.00e+00 h: 0.00e+00 rho: 0.00e+00 mu: 1.00e+04 li:  0 it: 6.91e-06 tt: 1.91e-03
      1: f: 1.249750e-07 d: 1.25e+01 g: 5.00e-04 h: 5.00e+00 rho: 1.00e+00 mu: 3.00e+04 li:  1 it: 2.81e-05 tt: 1.99e-03
      2: f: 1.388518e-16 d: 1.25e-07 g: 1.67e-08 h: 5.00e-04 rho: 1.00e+00 mu: 9.00e+04 li:  1 it: 1.00e-05 tt: 2.01e-03

   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.
   #. ``rho`` is the ratio of the actual change in the objective
      function value to the change in the the value of the trust
      region model.
   #. ``mu`` is the size of the trust region radius.
   #. ``li`` 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.
   #. ``it`` is the time take by the current iteration.
   #. ``tt`` is the 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 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`.

.. member:: string Solver::Options::solver_log

   Default: ``empty``

   If non-empty, a summary of the execution of the solver is recorded
   to this file.  This file is used for recording and Ceres'
   performance. Currently, only the iteration number, total time and
   the objective function value are logged. The format of this file is
   expected to change over time as the performance evaluation
   framework is fleshed out.

: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 optimizer
   after each iteration of the minimization. Note that all times are
   wall times.

   .. code-block:: c++

     struct IterationSummary {
       // Current iteration number.
       int32 iteration;

       // Step was numerically valid, i.e., all values are finite and the
       // step reduces the value of the linearized model.
       //
       // Note: step_is_valid is false when iteration = 0.
       bool step_is_valid;

       // 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: step_is_nonmonotonic is false when iteration = 0;
       bool step_is_nonmonotonic;

       // 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 Solver::Options::min_relative_decrease. However, if the
       // non-monotonic trust region algorithm is used
       // (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: step_is_successful is false when iteration = 0.
       bool step_is_successful;

       // Value of the objective function.
       double cost;

       // Change in the value of the objective function in this
       // iteration. This can be positive or negative.
       double cost_change;

       // Infinity norm of the gradient vector.
       double gradient_max_norm;

       // 2-norm of the size of the step computed by the optimization
       // algorithm.
       double step_norm;

       // For trust region algorithms, the ratio of the actual change in
       // cost and the change in the cost of the linearized approximation.
       double relative_decrease;

       // Size of the trust region at the end of the current iteration. For
       // the Levenberg-Marquardt algorithm, the regularization parameter
       // mu = 1.0 / trust_region_radius.
       double trust_region_radius;

       // For the inexact step Levenberg-Marquardt algorithm, this is the
       // relative accuracy with which the Newton(LM) step is solved. This
       // number affects only the iterative solvers capable of solving
       // linear systems inexactly. Factorization-based exact solvers
       // ignore it.
       double eta;

       // Step sized computed by the line search algorithm.
       double step_size;

       // Number of function evaluations used by the line search algorithm.
       int line_search_function_evaluations;

       // Number of iterations taken by the linear solver to solve for the
       // Newton step.
       int linear_solver_iterations;

       // Time (in seconds) spent inside the minimizer loop in the current
       // iteration.
       double iteration_time_in_seconds;

       // Time (in seconds) spent inside the trust region step solver.
       double step_solver_time_in_seconds;

       // Time (in seconds) since the user called Solve().
       double cumulative_time_in_seconds;
    };

.. class:: IterationCallback

   .. code-block:: c++

      class IterationCallback {
       public:
        virtual ~IterationCallback() {}
        virtual CallbackReturnType operator()(const IterationSummary& summary) = 0;
      };

  Interface for specifying callbacks that are executed at the end of
  each iteration of the Minimizer. 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_ABORT``.

  #. ``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 ``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

   .. code-block:: c++

      struct CRSMatrix {
        int num_rows;
        int num_cols;
        vector<int> cols;
        vector<int> rows;
        vector<double> values;
      };

   A compressed row sparse matrix used primarily for communicating the
   Jacobian matrix to the user.

   A compressed row matrix stores its contents in three arrays,
   ``rows``, ``cols`` and ``values``.

   ``rows`` is a ``num_rows + 1`` sized array that points into the ``cols`` and
   ``values`` array. For each row ``i``:

   ``cols[rows[i]]`` ... ``cols[rows[i + 1] - 1]`` are the indices of the
   non-zero columns of row ``i``.

   ``values[rows[i]]`` ... ``values[rows[i + 1] - 1]`` are the values of the
   corresponding entries.

   ``cols`` and ``values`` contain as many entries as there are
   non-zeros in the matrix.

   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

  Note that all times reported in this struct are wall times.

  .. code-block:: c++

     struct Summary {
       // A brief one line description of the state of the solver after
       // termination.
       string BriefReport() const;

       // A full multiline description of the state of the solver after
       // termination.
       string FullReport() const;

       // Minimizer summary -------------------------------------------------
       MinimizerType minimizer_type;

       SolverTerminationType termination_type;

       // If the solver did not run, or there was a failure, a
       // description of the error.
       string error;

       // Cost of the problem before and after the optimization. See
       // problem.h for definition of the cost of a problem.
       double initial_cost;
       double final_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.
       double fixed_cost;

       vector<IterationSummary> iterations;

       int num_successful_steps;
       int num_unsuccessful_steps;
       int num_inner_iteration_steps;

       // When the user calls Solve, before the actual optimization
       // occurs, Ceres performs a number of preprocessing steps. These
       // include error checks, memory allocations, and reorderings. This
       // time is accounted for as preprocessing time.
       double preprocessor_time_in_seconds;

       // Time spent in the TrustRegionMinimizer.
       double minimizer_time_in_seconds;

       // After the Minimizer is finished, some time is spent in
       // re-evaluating residuals etc. This time is accounted for in the
       // postprocessor time.
       double postprocessor_time_in_seconds;

       // Some total of all time spent inside Ceres when Solve is called.
       double total_time_in_seconds;

       double linear_solver_time_in_seconds;
       double residual_evaluation_time_in_seconds;
       double jacobian_evaluation_time_in_seconds;
       double inner_iteration_time_in_seconds;

       // Preprocessor summary.
       int num_parameter_blocks;
       int num_parameters;
       int num_effective_parameters;
       int num_residual_blocks;
       int num_residuals;

       int num_parameter_blocks_reduced;
       int num_parameters_reduced;
       int num_effective_parameters_reduced;
       int num_residual_blocks_reduced;
       int num_residuals_reduced;

       int num_eliminate_blocks_given;
       int num_eliminate_blocks_used;

       int num_threads_given;
       int num_threads_used;

       int num_linear_solver_threads_given;
       int num_linear_solver_threads_used;

       LinearSolverType linear_solver_type_given;
       LinearSolverType linear_solver_type_used;

       vector<int> linear_solver_ordering_given;
       vector<int> linear_solver_ordering_used;

       bool inner_iterations_given;
       bool inner_iterations_used;

       vector<int> inner_iteration_ordering_given;
       vector<int> inner_iteration_ordering_used;

       PreconditionerType preconditioner_type;

       TrustRegionStrategyType trust_region_strategy_type;
       DoglegType dogleg_type;

       SparseLinearAlgebraLibraryType sparse_linear_algebra_library;

       LineSearchDirectionType line_search_direction_type;
       LineSearchType line_search_type;
       int max_lbfgs_rank;
    };


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:: bool Covariance::Options::use_dense_linear_algebra

   Default: ``false``

   When ``true``, ``Eigen``'s ``JacobiSVD`` algorithm is used to
   perform the computations. It is an accurate but slow method and
   should only be used for small to moderate sized problems.

   When ``false``, ``SuiteSparse/CHOLMOD`` is used to perform the
   computation. Recent versions of ``SuiteSparse`` (>= 4.2.0) provide
   a much more efficient method for solving for rows of the covariance
   matrix. Therefore, if you are doing large scale covariance
   estimation, we strongly recommend using a recent version of
   ``SuiteSparse``.

   This setting also has an effect on how the following two options
   are interpreted.

.. 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.

   The reciprocal condition number of a matrix is a measure of
   ill-conditioning or how close the matrix is to being singular/rank
   deficient. It is defined as the ratio of the smallest eigenvalue of
   the matrix to the largest eigenvalue. In the above case the
   reciprocal condition number is about :math:`10^{-16}`. Which is
   close to machine precision and even though the inverse exists, it
   is meaningless, and care should be taken to interpet the results of
   such an inversion.

   Matrices with condition number lower than
   ``min_reciprocal_condition_number`` are considered rank deficient
   and by default Covariance::Compute will return false if it
   encounters such a matrix.

   a. ``use_dense_linear_algebra = false``

      When performing large scale sparse covariance estimation,
      computing the exact value of the reciprocal condition number is
      not possible as it would require computing the eigenvalues of
      :math:`J'J`.

      In this case we use cholmod_rcond, which uses the ratio of the
      smallest to the largest diagonal entries of the Cholesky
      factorization as an approximation to the reciprocal condition
      number.


      However, care must be taken as this is a heuristic and can
      sometimes be a very crude estimate. The default value of
      ``min_reciprocal_condition_number`` has been set to a conservative
      value, and sometimes the ``Covariance::Compute`` may return false
      even if it is possible to estimate the covariance reliably. In
      such cases, the user should exercise their judgement before
      lowering the value of ``min_reciprocal_condition_number``.

   b. ``use_dense_linear_algebra = true``

      When using dense linear algebra, 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}

.. member:: int Covariance::Options::null_space_rank

   Truncate the smallest ``null_space_rank`` eigenvectors when
   computing the pseudo inverse of :math:`J'J`.

   If ``null_space_rank = -1``, then all eigenvectors with eigenvalues
   s.t.

   :math:: \frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}

   are dropped. See the documentation for
   ``min_reciprocal_condition_number`` for more details.

.. 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)