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Sameer Agarwal
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.. default-domain:: cpp
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.. cpp:namespace:: ceres
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.. _chapter-nnls_covariance:
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=====================
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Covariance Estimation
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=====================
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Introduction
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============
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One way to assess the quality of the solution returned by a non-linear
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least squares solver is to analyze the covariance of the solution.
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Let us consider the non-linear regression problem
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.. math:: y = f(x) + N(0, I)
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i.e., the observation :math:`y` is a random non-linear function of the
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independent variable :math:`x` with mean :math:`f(x)` and identity
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covariance. Then the maximum likelihood estimate of :math:`x` given
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observations :math:`y` is the solution to the non-linear least squares
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problem:
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.. math:: x^* = \arg \min_x \|f(x)\|^2
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And the covariance of :math:`x^*` is given by
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.. math:: C(x^*) = \left(J'(x^*)J(x^*)\right)^{-1}
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Here :math:`J(x^*)` is the Jacobian of :math:`f` at :math:`x^*`. The
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above formula assumes that :math:`J(x^*)` has full column rank.
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If :math:`J(x^*)` is rank deficient, then the covariance matrix :math:`C(x^*)`
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is also rank deficient and is given by the Moore-Penrose pseudo inverse.
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.. math:: C(x^*) = \left(J'(x^*)J(x^*)\right)^{\dagger}
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Note that in the above, we assumed that the covariance matrix for
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:math:`y` was identity. This is an important assumption. If this is
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not the case and we have
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.. math:: y = f(x) + N(0, S)
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Where :math:`S` is a positive semi-definite matrix denoting the
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covariance of :math:`y`, then the maximum likelihood problem to be
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solved is
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.. math:: x^* = \arg \min_x f'(x) S^{-1} f(x)
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and the corresponding covariance estimate of :math:`x^*` is given by
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.. math:: C(x^*) = \left(J'(x^*) S^{-1} J(x^*)\right)^{-1}
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So, if it is the case that the observations being fitted to have a
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covariance matrix not equal to identity, then it is the user's
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responsibility that the corresponding cost functions are correctly
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scaled, e.g. in the above case the cost function for this problem
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should evaluate :math:`S^{-1/2} f(x)` instead of just :math:`f(x)`,
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where :math:`S^{-1/2}` is the inverse square root of the covariance
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matrix :math:`S`.
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Gauge Invariance
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================
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In structure from motion (3D reconstruction) problems, the
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reconstruction is ambiguous upto a similarity transform. This is
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known as a *Gauge Ambiguity*. Handling Gauges correctly requires the
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use of SVD or custom inversion algorithms. For small problems the
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user can use the dense algorithm. For more details see the work of
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Kanatani & Morris [KanataniMorris]_.
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:class:`Covariance`
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===================
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:class:`Covariance` allows the user to evaluate the covariance for a
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non-linear least squares problem and provides random access to its
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blocks. The computation assumes that the cost functions compute
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residuals such that their covariance is identity.
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Since the computation of the covariance matrix requires computing the
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inverse of a potentially large matrix, this can involve a rather large
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amount of time and memory. However, it is usually the case that the
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user is only interested in a small part of the covariance
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matrix. Quite often just the block diagonal. :class:`Covariance`
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allows the user to specify the parts of the covariance matrix that she
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is interested in and then uses this information to only compute and
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store those parts of the covariance matrix.
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Rank of the Jacobian
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====================
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As we noted above, if the Jacobian is rank deficient, then the inverse
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of :math:`J'J` is not defined and instead a pseudo inverse needs to be
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computed.
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The rank deficiency in :math:`J` can be *structural* -- columns
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which are always known to be zero or *numerical* -- depending on the
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exact values in the Jacobian.
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Structural rank deficiency occurs when the problem contains parameter
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blocks that are constant. This class correctly handles structural rank
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deficiency like that.
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Numerical rank deficiency, where the rank of the matrix cannot be
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predicted by its sparsity structure and requires looking at its
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numerical values is more complicated. Here again there are two
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cases.
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a. The rank deficiency arises from overparameterization. e.g., a
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four dimensional quaternion used to parameterize :math:`SO(3)`,
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which is a three dimensional manifold. In cases like this, the
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user should use an appropriate
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:class:`LocalParameterization`. Not only will this lead to better
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numerical behaviour of the Solver, it will also expose the rank
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deficiency to the :class:`Covariance` object so that it can
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handle it correctly.
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b. More general numerical rank deficiency in the Jacobian requires
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the computation of the so called Singular Value Decomposition
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(SVD) of :math:`J'J`. We do not know how to do this for large
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sparse matrices efficiently. For small and moderate sized
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problems this is done using dense linear algebra.
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:class:`Covariance::Options`
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.. class:: Covariance::Options
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.. member:: int Covariance::Options::num_threads
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Default: ``1``
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Number of threads to be used for evaluating the Jacobian and
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estimation of covariance.
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.. member:: SparseLinearAlgebraLibraryType Covariance::Options::sparse_linear_algebra_library_type
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Default: ``SUITE_SPARSE`` Ceres Solver is built with support for
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`SuiteSparse <http://faculty.cse.tamu.edu/davis/suitesparse.html>`_
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and ``EIGEN_SPARSE`` otherwise. Note that ``EIGEN_SPARSE`` is
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always available.
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.. member:: CovarianceAlgorithmType Covariance::Options::algorithm_type
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Default: ``SPARSE_QR``
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Ceres supports two different algorithms for covariance estimation,
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which represent different tradeoffs in speed, accuracy and
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reliability.
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1. ``SPARSE_QR`` uses the sparse QR factorization algorithm to
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compute the decomposition
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.. math::
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QR &= J\\
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\left(J^\top J\right)^{-1} &= \left(R^\top R\right)^{-1}
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The speed of this algorithm depends on the sparse linear algebra
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library being used. ``Eigen``'s sparse QR factorization is a
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moderately fast algorithm suitable for small to medium sized
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matrices. For best performance we recommend using
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``SuiteSparseQR`` which is enabled by setting
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:member:`Covaraince::Options::sparse_linear_algebra_library_type`
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to ``SUITE_SPARSE``.
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Neither ``SPARSE_QR`` cannot compute the covariance if the
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Jacobian is rank deficient.
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2. ``DENSE_SVD`` uses ``Eigen``'s ``JacobiSVD`` to perform the
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computations. It computes the singular value decomposition
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.. math:: U S V^\top = J
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and then uses it to compute the pseudo inverse of J'J as
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.. math:: (J'J)^{\dagger} = V S^{\dagger} V^\top
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It is an accurate but slow method and should only be used for
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small to moderate sized problems. It can handle full-rank as
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well as rank deficient Jacobians.
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.. member:: int Covariance::Options::min_reciprocal_condition_number
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Default: :math:`10^{-14}`
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If the Jacobian matrix is near singular, then inverting :math:`J'J`
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will result in unreliable results, e.g, if
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.. math::
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J = \begin{bmatrix}
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1.0& 1.0 \\
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1.0& 1.0000001
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\end{bmatrix}
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which is essentially a rank deficient matrix, we have
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.. math::
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(J'J)^{-1} = \begin{bmatrix}
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2.0471e+14& -2.0471e+14 \\
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-2.0471e+14 2.0471e+14
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\end{bmatrix}
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This is not a useful result. Therefore, by default
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:func:`Covariance::Compute` will return ``false`` if a rank
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deficient Jacobian is encountered. How rank deficiency is detected
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depends on the algorithm being used.
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1. ``DENSE_SVD``
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.. math:: \frac{\sigma_{\text{min}}}{\sigma_{\text{max}}} < \sqrt{\text{min_reciprocal_condition_number}}
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where :math:`\sigma_{\text{min}}` and
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:math:`\sigma_{\text{max}}` are the minimum and maxiumum
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singular values of :math:`J` respectively.
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2. ``SPARSE_QR``
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.. math:: \operatorname{rank}(J) < \operatorname{num\_col}(J)
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Here :math:`\operatorname{rank}(J)` is the estimate of the rank
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of :math:`J` returned by the sparse QR factorization
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algorithm. It is a fairly reliable indication of rank
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deficiency.
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.. member:: int Covariance::Options::null_space_rank
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When using ``DENSE_SVD``, the user has more control in dealing
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with singular and near singular covariance matrices.
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As mentioned above, when the covariance matrix is near singular,
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instead of computing the inverse of :math:`J'J`, the Moore-Penrose
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pseudoinverse of :math:`J'J` should be computed.
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If :math:`J'J` has the eigen decomposition :math:`(\lambda_i,
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e_i)`, where :math:`\lambda_i` is the :math:`i^\textrm{th}`
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eigenvalue and :math:`e_i` is the corresponding eigenvector, then
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the inverse of :math:`J'J` is
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.. math:: (J'J)^{-1} = \sum_i \frac{1}{\lambda_i} e_i e_i'
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and computing the pseudo inverse involves dropping terms from this
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sum that correspond to small eigenvalues.
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How terms are dropped is controlled by
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`min_reciprocal_condition_number` and `null_space_rank`.
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If `null_space_rank` is non-negative, then the smallest
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`null_space_rank` eigenvalue/eigenvectors are dropped irrespective
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of the magnitude of :math:`\lambda_i`. If the ratio of the
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smallest non-zero eigenvalue to the largest eigenvalue in the
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truncated matrix is still below min_reciprocal_condition_number,
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then the `Covariance::Compute()` will fail and return `false`.
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Setting `null_space_rank = -1` drops all terms for which
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.. math:: \frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}
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This option has no effect on ``SPARSE_QR``.
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.. member:: bool Covariance::Options::apply_loss_function
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Default: `true`
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Even though the residual blocks in the problem may contain loss
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functions, setting ``apply_loss_function`` to false will turn off
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the application of the loss function to the output of the cost
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function and in turn its effect on the covariance.
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.. class:: Covariance
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:class:`Covariance::Options` as the name implies is used to control
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the covariance estimation algorithm. Covariance estimation is a
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complicated and numerically sensitive procedure. Please read the
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entire documentation for :class:`Covariance::Options` before using
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:class:`Covariance`.
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.. function:: bool Covariance::Compute(const vector<pair<const double*, const double*> >& covariance_blocks, Problem* problem)
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Compute a part of the covariance matrix.
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The vector ``covariance_blocks``, indexes into the covariance
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matrix block-wise using pairs of parameter blocks. This allows the
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covariance estimation algorithm to only compute and store these
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blocks.
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Since the covariance matrix is symmetric, if the user passes
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``<block1, block2>``, then ``GetCovarianceBlock`` can be called with
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``block1``, ``block2`` as well as ``block2``, ``block1``.
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``covariance_blocks`` cannot contain duplicates. Bad things will
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happen if they do.
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Note that the list of ``covariance_blocks`` is only used to
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determine what parts of the covariance matrix are computed. The
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full Jacobian is used to do the computation, i.e. they do not have
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an impact on what part of the Jacobian is used for computation.
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The return value indicates the success or failure of the covariance
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computation. Please see the documentation for
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:class:`Covariance::Options` for more on the conditions under which
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this function returns ``false``.
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.. function:: bool GetCovarianceBlock(const double* parameter_block1, const double* parameter_block2, double* covariance_block) const
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Return the block of the cross-covariance matrix corresponding to
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``parameter_block1`` and ``parameter_block2``.
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Compute must be called before the first call to ``GetCovarianceBlock``
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and the pair ``<parameter_block1, parameter_block2>`` OR the pair
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``<parameter_block2, parameter_block1>`` must have been present in the
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vector covariance_blocks when ``Compute`` was called. Otherwise
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``GetCovarianceBlock`` will return false.
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``covariance_block`` must point to a memory location that can store
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a ``parameter_block1_size x parameter_block2_size`` matrix. The
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returned covariance will be a row-major matrix.
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.. function:: bool GetCovarianceBlockInTangentSpace(const double* parameter_block1, const double* parameter_block2, double* covariance_block) const
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Return the block of the cross-covariance matrix corresponding to
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``parameter_block1`` and ``parameter_block2``.
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Returns cross-covariance in the tangent space if a local
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parameterization is associated with either parameter block;
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else returns cross-covariance in the ambient space.
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Compute must be called before the first call to ``GetCovarianceBlock``
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and the pair ``<parameter_block1, parameter_block2>`` OR the pair
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``<parameter_block2, parameter_block1>`` must have been present in the
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vector covariance_blocks when ``Compute`` was called. Otherwise
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``GetCovarianceBlock`` will return false.
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``covariance_block`` must point to a memory location that can store
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a ``parameter_block1_local_size x parameter_block2_local_size`` matrix. The
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returned covariance will be a row-major matrix.
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Example Usage
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=============
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.. code-block:: c++
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double x[3];
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double y[2];
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Problem problem;
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problem.AddParameterBlock(x, 3);
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problem.AddParameterBlock(y, 2);
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<Build Problem>
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<Solve Problem>
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Covariance::Options options;
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Covariance covariance(options);
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vector<pair<const double*, const double*> > covariance_blocks;
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covariance_blocks.push_back(make_pair(x, x));
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covariance_blocks.push_back(make_pair(y, y));
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covariance_blocks.push_back(make_pair(x, y));
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CHECK(covariance.Compute(covariance_blocks, &problem));
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double covariance_xx[3 * 3];
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double covariance_yy[2 * 2];
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double covariance_xy[3 * 2];
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covariance.GetCovarianceBlock(x, x, covariance_xx)
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covariance.GetCovarianceBlock(y, y, covariance_yy)
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covariance.GetCovarianceBlock(x, y, covariance_xy)
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