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+
+.. 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)
