.. default-domain:: cpp

.. cpp:namespace:: ceres

.. _chapter-nnls_covariance:

=====================

Covariance Estimation

=====================

Introduction

============

One way to assess the quality of the solution returned by a non-linear

least squares solver 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:: SparseLinearAlgebraLibraryType Covariance::Options::sparse_linear_algebra_library_type

   Default: ``SUITE_SPARSE`` Ceres Solver is built with support for

   `SuiteSparse <http://faculty.cse.tamu.edu/davis/suitesparse.html>`_

   and ``EIGEN_SPARSE`` otherwise. Note that ``EIGEN_SPARSE`` is

   always available.

.. member:: CovarianceAlgorithmType Covariance::Options::algorithm_type

   Default: ``SPARSE_QR``

   Ceres supports two different algorithms for covariance estimation,

   which represent different tradeoffs in speed, accuracy and

   reliability.

   1. ``SPARSE_QR`` uses the sparse QR factorization algorithm to

      compute the decomposition

       .. math::

          QR &= J\\

          \left(J^\top J\right)^{-1} &= \left(R^\top R\right)^{-1}

      The speed of this algorithm depends on the sparse linear algebra

      library being used. ``Eigen``'s sparse QR factorization is a

      moderately fast algorithm suitable for small to medium sized

      matrices. For best performance we recommend using

      ``SuiteSparseQR`` which is enabled by setting

      :member:`Covaraince::Options::sparse_linear_algebra_library_type`

      to ``SUITE_SPARSE``.

      Neither ``SPARSE_QR`` cannot compute the covariance if the

      Jacobian is rank deficient.

   2. ``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.

.. 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. ``SPARSE_QR``

       .. math:: \operatorname{rank}(J) < \operatorname{num\_col}(J)

       Here :math:`\operatorname{rank}(J)` is the estimate of the rank

       of :math:`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 ``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 cross-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.

.. function:: bool GetCovarianceBlockInTangentSpace(const double* parameter_block1, const double* parameter_block2, double* covariance_block) const

   Return the block of the cross-covariance matrix corresponding to

   ``parameter_block1`` and ``parameter_block2``.

   Returns cross-covariance in the tangent space if a local

   parameterization is associated with either parameter block;

   else returns cross-covariance in the ambient space.

   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_local_size x parameter_block2_local_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)