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Ceres Solver
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1.13
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Why?
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Installation
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Tutorial
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On Derivatives
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Modeling Non-linear Least Squares
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Solving Non-linear Least Squares
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Covariance Estimation
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Introduction
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Gauge Invariance
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Covariance
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Rank of the Jacobian
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Example Usage
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General Unconstrained Minimization
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FAQS, Tips & Tricks
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Users
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Contributing
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Bibliography
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License
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Covariance Estimation
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Covariance Estimation¶
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Introduction¶
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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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\[y = f(x) + N(0, I)\]
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i.e., the observation \(y\) is a random non-linear function of the
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independent variable \(x\) with mean \(f(x)\) and identity
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covariance. Then the maximum likelihood estimate of \(x\) given
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observations \(y\) is the solution to the non-linear least squares
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problem:
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\[x^* = \arg \min_x \|f(x)\|^2\]
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And the covariance of \(x^*\) is given by
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\[C(x^*) = \left(J'(x^*)J(x^*)\right)^{-1}\]
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Here \(J(x^*)\) is the Jacobian of \(f\) at \(x^*\). The
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above formula assumes that \(J(x^*)\) has full column rank.
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If \(J(x^*)\) is rank deficient, then the covariance matrix \(C(x^*)\)
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is also rank deficient and is given by the Moore-Penrose pseudo inverse.
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\[C(x^*) = \left(J'(x^*)J(x^*)\right)^{\dagger}\]
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Note that in the above, we assumed that the covariance matrix for
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\(y\) was identity. This is an important assumption. If this is
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not the case and we have
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\[y = f(x) + N(0, S)\]
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Where \(S\) is a positive semi-definite matrix denoting the
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covariance of \(y\), then the maximum likelihood problem to be
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solved is
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\[x^* = \arg \min_x f'(x) S^{-1} f(x)\]
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and the corresponding covariance estimate of \(x^*\) is given by
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\[C(x^*) = \left(J'(x^*) S^{-1} J(x^*)\right)^{-1}\]
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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 \(S^{-1/2} f(x)\) instead of just \(f(x)\),
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where \(S^{-1/2}\) is the inverse square root of the covariance
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matrix \(S\).
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Gauge Invariance¶
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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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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. 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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As we noted above, if the Jacobian is rank deficient, then the inverse
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of \(J'J\) is not defined and instead a pseudo inverse needs to be
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computed.
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The rank deficiency in \(J\) 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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The rank deficiency arises from overparameterization. e.g., a
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four dimensional quaternion used to parameterize \(SO(3)\),
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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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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 Covariance object so that it can
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handle it correctly.
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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 \(J'J\). 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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Covariance::Options
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class Covariance::Options¶
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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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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
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and EIGEN_SPARSE otherwise. Note that EIGEN_SPARSE is
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always available.
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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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SPARSE_QR uses the sparse QR factorization algorithm to
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compute the decomposition
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\[\begin{split}QR &= J\\
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\left(J^\top J\right)^{-1} &= \left(R^\top R\right)^{-1}\end{split}\]
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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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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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DENSE_SVD uses Eigen‘s JacobiSVD to perform the
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computations. It computes the singular value decomposition
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\[U S V^\top = J\]
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and then uses it to compute the pseudo inverse of J’J as
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\[(J'J)^{\dagger} = V S^{\dagger} V^\top\]
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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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int Covariance::Options::min_reciprocal_condition_number¶
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Default: \(10^{-14}\)
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If the Jacobian matrix is near singular, then inverting \(J'J\)
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will result in unreliable results, e.g, if
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\[\begin{split}J = \begin{bmatrix}
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1.0& 1.0 \\
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1.0& 1.0000001
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\end{bmatrix}\end{split}\]
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which is essentially a rank deficient matrix, we have
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\[\begin{split}(J'J)^{-1} = \begin{bmatrix}
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2.0471e+14& -2.0471e+14 \\
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-2.0471e+14 2.0471e+14
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\end{bmatrix}\end{split}\]
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This is not a useful result. Therefore, by default
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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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DENSE_SVD
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\[\frac{\sigma_{\text{min}}}{\sigma_{\text{max}}} < \sqrt{\text{min_reciprocal_condition_number}}\]
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where \(\sigma_{\text{min}}\) and
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\(\sigma_{\text{max}}\) are the minimum and maxiumum
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singular values of \(J\) respectively.
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SPARSE_QR
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\[\operatorname{rank}(J) < \operatorname{num\_col}(J)\]
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Here \(\operatorname{rank}(J)\) is the estimate of the rank
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of \(J\) 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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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 \(J'J\), the Moore-Penrose
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pseudoinverse of \(J'J\) should be computed.
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If \(J'J\) has the eigen decomposition \((\lambda_i,
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e_i)\), where \(\lambda_i\) is the \(i^\textrm{th}\)
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eigenvalue and \(e_i\) is the corresponding eigenvector, then
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the inverse of \(J'J\) is
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\[(J'J)^{-1} = \sum_i \frac{1}{\lambda_i} e_i e_i'\]
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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 \(\lambda_i\). 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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\[\frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}\]
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This option has no effect on SPARSE_QR.
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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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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 Covariance::Options before using
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Covariance.
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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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Covariance::Options for more on the conditions under which
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this function returns false.
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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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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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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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