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Add Covariance documentation to html docs.
Change-Id: I11ddc9f7069964596760c6ea4d85c44312c0a67a
This commit is contained in:
Sameer Agarwal
@@ -39,6 +39,11 @@ Bibliography
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.. [HartleyZisserman] R.I. Hartley & A. Zisserman, **Multiview
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Geometry in Computer Vision**, Cambridge University Press, 2004.
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.. [KanataniMorris] K. Kanatani and D. D. Morris, **Gauges and gauge
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transformations for uncertainty description of geometric structure
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with indeterminacy**, *IEEE Transactions on Information Theory*
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47(5):2017-2028, 2001.
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.. [KushalAgarwal] A. Kushal and S. Agarwal, **Visibility based
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preconditioning for bundle adjustment**, *In Proceedings of the
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IEEE Conference on Computer Vision and Pattern Recognition*, 2012.
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+345
-3
@@ -1678,8 +1678,350 @@ elimination group [LiSaad]_.
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};
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Covariance Estimation
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=====================
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:class:`GradientChecker`
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------------------------
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Background
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----------
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One way to assess the quality of the solution returned by a
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non-linear least squares solve is to analyze the covariance of the
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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:: bool Covariance::Options::use_dense_linear_algebra
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Default: ``false``
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When ``true``, ``Eigen``'s ``JacobiSVD`` algorithm is used to
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perform the computations. It is an accurate but slow method and
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should only be used for small to moderate sized problems.
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When ``false``, ``SuiteSparse/CHOLMOD`` is used to perform the
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computation. Recent versions of ``SuiteSparse`` (>= 4.2.0) provide
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a much more efficient method for solving for rows of the covariance
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matrix. Therefore, if you are doing large scale covariance
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estimation, we strongly recommend using a recent version of
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``SuiteSparse``.
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This setting also has an effect on how the following two options
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are interpreted.
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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.
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The reciprocal condition number of a matrix is a measure of
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ill-conditioning or how close the matrix is to being singular/rank
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deficient. It is defined as the ratio of the smallest eigenvalue of
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the matrix to the largest eigenvalue. In the above case the
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reciprocal condition number is about :math:`10^{-16}`. Which is
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close to machine precision and even though the inverse exists, it
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is meaningless, and care should be taken to interpet the results of
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such an inversion.
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Matrices with condition number lower than
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``min_reciprocal_condition_number`` are considered rank deficient
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and by default Covariance::Compute will return false if it
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encounters such a matrix.
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a. ``use_dense_linear_algebra = false``
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When performing large scale sparse covariance estimation,
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computing the exact value of the reciprocal condition number is
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not possible as it would require computing the eigenvalues of
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:math:`J'J`.
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In this case we use cholmod_rcond, which uses the ratio of the
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smallest to the largest diagonal entries of the Cholesky
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factorization as an approximation to the reciprocal condition
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number.
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However, care must be taken as this is a heuristic and can
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sometimes be a very crude estimate. The default value of
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``min_reciprocal_condition_number`` has been set to a conservative
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value, and sometimes the ``Covariance::Compute`` may return false
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even if it is possible to estimate the covariance reliably. In
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such cases, the user should exercise their judgement before
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lowering the value of ``min_reciprocal_condition_number``.
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b. ``use_dense_linear_algebra = true``
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When using dense linear algebra, the user has more control in
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dealing 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
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Moore-Penrose 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,
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then 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
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this 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
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irrespective of the magnitude of :math:`\lambda_i`. If the ratio
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of the smallest non-zero eigenvalue to the largest eigenvalue in
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the truncated matrix is still below
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min_reciprocal_condition_number, then the
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`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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.. member:: int Covariance::Options::null_space_rank
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Truncate the smallest ``null_space_rank`` eigenvectors when
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computing the pseudo inverse of :math:`J'J`.
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If ``null_space_rank = -1``, then all eigenvectors with eigenvalues
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s.t.
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:math:: \frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}
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are dropped. See the documentation for
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``min_reciprocal_condition_number`` for more details.
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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 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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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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.. class:: GradientChecker
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@@ -41,6 +41,7 @@
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#include "ceres/autodiff_local_parameterization.h"
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#include "ceres/cost_function.h"
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#include "ceres/cost_function_to_functor.h"
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#include "ceres/covariance.h"
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#include "ceres/crs_matrix.h"
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#include "ceres/iteration_callback.h"
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#include "ceres/jet.h"
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+14
-20
@@ -129,7 +129,9 @@ class CovarianceImpl;
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//
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// The rank deficiency in J can be structural -- columns which are
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// always known to be zero or numerical -- depending on the exact
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// values in the Jacobian. This happens when the problem contains
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// values in the Jacobian.
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//
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// Structural rank deficiency occurs when the problem contains
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// parameter blocks that are constant. This class correctly handles
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// structural rank deficiency like that.
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//
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@@ -165,21 +167,6 @@ class CovarianceImpl;
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// with indeterminacy. IEEE Transactions on Information Theory 47(5):
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// 2017-2028 (2001)
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//
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// Speed
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// -----
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//
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// When use_dense_linear_algebra = true, Eigen's JacobiSVD algorithm
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// is used to perform the computations. It is an accurate but slow
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// method and should only be used for small to moderate sized
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// problems.
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//
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// When use_dense_linear_algebra = false, SuiteSparse/CHOLMOD is used
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// to perform the computation. Recent versions of SuiteSparse (>=
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// 4.2.0) provide a much more efficient method for solving for rows of
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// the covariance matrix. Therefore, if you are doing large scale
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// covariance estimation, we strongly recommend using a recent version
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// of SuiteSparse.
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//
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// Example Usage
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// =============
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//
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@@ -228,11 +215,18 @@ class Covariance {
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// estimation of covariance.
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int num_threads;
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// Use Eigen's JacobiSVD algorithm to compute the covariance
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// instead of SuiteSparse. This is a very accurate but slow
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// algorithm. The up side is that it can handle numerically rank
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// deficient jacobians. This option only makes sense for small to
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// When use_dense_linear_algebra = true, Eigen's JacobiSVD
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// algorithm is used to perform the computations. It is an
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// accurate but slow method and should only be used for small to
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// moderate sized problems.
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//
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// When use_dense_linear_algebra = false, SuiteSparse/CHOLMOD is
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// used to perform the computation. Recent versions of SuiteSparse
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// (>= 4.2.0) provide a much more efficient method for solving for
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// rows of the covariance matrix. Therefore, if you are doing
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// large scale covariance estimation, we strongly recommend using
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// a recent version of SuiteSparse.
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bool use_dense_linear_algebra;
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// If the Jacobian matrix is near singular, then inverting J'J
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