ceres-solver / ceres-solver / 00a05cf70fcc573c34662188465ea5729baf845d / . / docs / source / nnls_covariance.rst

.. 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) - y\|^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 up to 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:`Covariance::Options::sparse_linear_algebra_library_type` | |

to ``SUITE_SPARSE``. | |

``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 D V^\top = J | |

and then uses it to compute the pseudo inverse of J'J as | |

.. math:: (J'J)^{\dagger} = V D^{2\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:: double Covariance::Options::column_pivot_threshold | |

Default: :math:`-1` | |

During QR factorization, if a column with Euclidean norm less than | |

``column_pivot_threshold`` is encountered it is treated as zero. | |

If ``column_pivot_threshold < 0``, then an automatic default value | |

of `20*(m+n)*eps*sqrt(max(diag(J’*J)))` is used. Here `m` and `n` | |

are the number of rows and columns of the Jacobian (`J`) | |

respectively. | |

This is an advanced option meant for users who know enough about | |

their Jacobian matrices that they can determine a value better | |

than the default. | |

.. 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 maximum | |

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) |