Solving Nonlinear Least Squares¶
Introduction¶
Effective use of Ceres requires some familiarity with the basic components of a nonlinear least squares solver, so before we describe how to configure and use the solver, we will take a brief look at how some of the core optimization algorithms in Ceres work.
Let \(x \in \mathbb{R}^n\) be an \(n\)dimensional vector of variables, and \(F(x) = \left[f_1(x), ... , f_{m}(x) \right]^{\top}\) be a \(m\)dimensional function of \(x\). We are interested in solving the optimization problem [1]
Where, \(L\) and \(U\) are lower and upper bounds on the parameter vector \(x\).
Since the efficient global minimization of (1) for general \(F(x)\) is an intractable problem, we will have to settle for finding a local minimum.
In the following, the Jacobian \(J(x)\) of \(F(x)\) is an \(m\times n\) matrix, where \(J_{ij}(x) = \partial_j f_i(x)\) and the gradient vector is \(g(x) = \nabla \frac{1}{2}\F(x)\^2 = J(x)^\top F(x)\).
The general strategy when solving nonlinear optimization problems is to solve a sequence of approximations to the original problem [NocedalWright]. At each iteration, the approximation is solved to determine a correction \(\Delta x\) to the vector \(x\). For nonlinear least squares, an approximation can be constructed by using the linearization \(F(x+\Delta x) \approx F(x) + J(x)\Delta x\), which leads to the following linear least squares problem:
Unfortunately, naively solving a sequence of these problems and updating \(x \leftarrow x+ \Delta x\) leads to an algorithm that may not converge. To get a convergent algorithm, we need to control the size of the step \(\Delta x\). Depending on how the size of the step \(\Delta x\) is controlled, nonlinear optimization algorithms can be divided into two major categories [NocedalWright].
 Trust Region The trust region approach approximates the objective function using using a model function (often a quadratic) over a subset of the search space known as the trust region. If the model function succeeds in minimizing the true objective function the trust region is expanded; conversely, otherwise it is contracted and the model optimization problem is solved again.
 Line Search The line search approach first finds a descent direction along which the objective function will be reduced and then computes a step size that decides how far should move along that direction. The descent direction can be computed by various methods, such as gradient descent, Newton’s method and QuasiNewton method. The step size can be determined either exactly or inexactly.
Trust region methods are in some sense dual to line search methods: trust region methods first choose a step size (the size of the trust region) and then a step direction while line search methods first choose a step direction and then a step size. Ceres implements multiple algorithms in both categories.
Trust Region Methods¶
The basic trust region algorithm looks something like this.
Given an initial point \(x\) and a trust region radius \(\mu\).
Solve
\[\begin{split}\arg \min_{\Delta x}& \frac{1}{2}\J(x)\Delta x + F(x)\^2 \\ \text{such that} &\D(x)\Delta x\^2 \le \mu\\ &L \le x + \Delta x \le U.\end{split}\]\(\rho = \frac{\displaystyle \F(x + \Delta x)\^2  \F(x)\^2}{\displaystyle \J(x)\Delta x + F(x)\^2  \F(x)\^2}\)
if \(\rho > \epsilon\) then \(x = x + \Delta x\).
if \(\rho > \eta_1\) then \(\mu = 2 \mu\)
else if \(\rho < \eta_2\) then \(\mu = 0.5 * \mu\)
Go to 2.
Here, \(\mu\) is the trust region radius, \(D(x)\) is some matrix used to define a metric on the domain of \(F(x)\) and \(\rho\) measures the quality of the step \(\Delta x\), i.e., how well did the linear model predict the decrease in the value of the nonlinear objective. The idea is to increase or decrease the radius of the trust region depending on how well the linearization predicts the behavior of the nonlinear objective, which in turn is reflected in the value of \(\rho\).
The key computational step in a trustregion algorithm is the solution of the constrained optimization problem
There are a number of different ways of solving this problem, each
giving rise to a different concrete trustregion algorithm. Currently,
Ceres implements two trustregion algorithms  LevenbergMarquardt
and Dogleg, each of which is augmented with a line search if bounds
constraints are present [Kanzow]. The user can choose between them by
setting Solver::Options::trust_region_strategy_type
.
Footnotes
[1]  At the level of the nonlinear solver, the block structure is not relevant, therefore our discussion here is in terms of an optimization problem defined over a state vector of size \(n\). Similarly the presence of loss functions is also ignored as the problem is internally converted into a pure nonlinear least squares problem. 
LevenbergMarquardt¶
The LevenbergMarquardt algorithm [Levenberg] [Marquardt] is the most popular algorithm for solving nonlinear least squares problems. It was also the first trust region algorithm to be developed [Levenberg] [Marquardt]. Ceres implements an exact step [Madsen] and an inexact step variant of the LevenbergMarquardt algorithm [WrightHolt] [NashSofer].
It can be shown, that the solution to (3) can be obtained by solving an unconstrained optimization of the form
Where, \(\lambda\) is a Lagrange multiplier that is inverse related to \(\mu\). In Ceres, we solve for
The matrix \(D(x)\) is a nonnegative diagonal matrix, typically the square root of the diagonal of the matrix \(J(x)^\top J(x)\).
Before going further, let us make some notational simplifications. We will assume that the matrix \(\sqrt{\mu} D\) has been concatenated at the bottom of the matrix \(J\) and similarly a vector of zeros has been added to the bottom of the vector \(f\) and the rest of our discussion will be in terms of \(J\) and \(f\), i.e, the linear least squares problem.
For all but the smallest problems the solution of (5) in each iteration of the LevenbergMarquardt algorithm is the dominant computational cost in Ceres. Ceres provides a number of different options for solving (5). There are two major classes of methods  factorization and iterative.
The factorization methods are based on computing an exact solution of (4) using a Cholesky or a QR factorization and lead to an exact step LevenbergMarquardt algorithm. But it is not clear if an exact solution of (4) is necessary at each step of the LM algorithm to solve (1). In fact, we have already seen evidence that this may not be the case, as (4) is itself a regularized version of (2). Indeed, it is possible to construct nonlinear optimization algorithms in which the linearized problem is solved approximately. These algorithms are known as inexact Newton or truncated Newton methods [NocedalWright].
An inexact Newton method requires two ingredients. First, a cheap method for approximately solving systems of linear equations. Typically an iterative linear solver like the Conjugate Gradients method is used for this purpose [NocedalWright]. Second, a termination rule for the iterative solver. A typical termination rule is of the form
Here, \(k\) indicates the LevenbergMarquardt iteration number and \(0 < \eta_k <1\) is known as the forcing sequence. [WrightHolt] prove that a truncated LevenbergMarquardt algorithm that uses an inexact Newton step based on (6) converges for any sequence \(\eta_k \leq \eta_0 < 1\) and the rate of convergence depends on the choice of the forcing sequence \(\eta_k\).
Ceres supports both exact and inexact step solution strategies. When the user chooses a factorization based linear solver, the exact step LevenbergMarquardt algorithm is used. When the user chooses an iterative linear solver, the inexact step LevenbergMarquardt algorithm is used.
Dogleg¶
Another strategy for solving the trust region problem (3) was introduced by M. J. D. Powell. The key idea there is to compute two vectors
Note that the vector \(\Delta x^{\text{GaussNewton}}\) is the
solution to (2) and \(\Delta
x^{\text{Cauchy}}\) is the vector that minimizes the linear
approximation if we restrict ourselves to moving along the direction
of the gradient. Dogleg methods finds a vector \(\Delta x\)
defined by \(\Delta x^{\text{GaussNewton}}\) and \(\Delta
x^{\text{Cauchy}}\) that solves the trust region problem. Ceres
supports two variants that can be chose by setting
Solver::Options::dogleg_type
.
TRADITIONAL_DOGLEG
as described by Powell, constructs two line
segments using the GaussNewton and Cauchy vectors and finds the point
farthest along this line shaped like a dogleg (hence the name) that is
contained in the trustregion. For more details on the exact reasoning
and computations, please see Madsen et al [Madsen].
SUBSPACE_DOGLEG
is a more sophisticated method that considers the
entire two dimensional subspace spanned by these two vectors and finds
the point that minimizes the trust region problem in this subspace
[ByrdSchnabel].
The key advantage of the Dogleg over LevenbergMarquardt is that if the step computation for a particular choice of \(\mu\) does not result in sufficient decrease in the value of the objective function, LevenbergMarquardt solves the linear approximation from scratch with a smaller value of \(\mu\). Dogleg on the other hand, only needs to compute the interpolation between the GaussNewton and the Cauchy vectors, as neither of them depend on the value of \(\mu\).
The Dogleg method can only be used with the exact factorization based linear solvers.
Inner Iterations¶
Some nonlinear least squares problems have additional structure in the way the parameter blocks interact that it is beneficial to modify the way the trust region step is computed. For example, consider the following regression problem
Given a set of pairs \(\{(x_i, y_i)\}\), the user wishes to estimate \(a_1, a_2, b_1, b_2\), and \(c_1\).
Notice that the expression on the left is linear in \(a_1\) and \(a_2\), and given any value for \(b_1, b_2\) and \(c_1\), it is possible to use linear regression to estimate the optimal values of \(a_1\) and \(a_2\). It’s possible to analytically eliminate the variables \(a_1\) and \(a_2\) from the problem entirely. Problems like these are known as separable least squares problem and the most famous algorithm for solving them is the Variable Projection algorithm invented by Golub & Pereyra [GolubPereyra].
Similar structure can be found in the matrix factorization with missing data problem. There the corresponding algorithm is known as Wiberg’s algorithm [Wiberg].
Ruhe & Wedin present an analysis of various algorithms for solving separable nonlinear least squares problems and refer to Variable Projection as Algorithm I in their paper [RuheWedin].
Implementing Variable Projection is tedious and expensive. Ruhe & Wedin present a simpler algorithm with comparable convergence properties, which they call Algorithm II. Algorithm II performs an additional optimization step to estimate \(a_1\) and \(a_2\) exactly after computing a successful Newton step.
This idea can be generalized to cases where the residual is not linear in \(a_1\) and \(a_2\), i.e.,
In this case, we solve for the trust region step for the full problem, and then use it as the starting point to further optimize just a_1 and a_2. For the linear case, this amounts to doing a single linear least squares solve. For nonlinear problems, any method for solving the \(a_1\) and \(a_2\) optimization problems will do. The only constraint on \(a_1\) and \(a_2\) (if they are two different parameter block) is that they do not cooccur in a residual block.
This idea can be further generalized, by not just optimizing \((a_1, a_2)\), but decomposing the graph corresponding to the Hessian matrix’s sparsity structure into a collection of nonoverlapping independent sets and optimizing each of them.
Setting Solver::Options::use_inner_iterations
to true
enables the use of this nonlinear generalization of Ruhe & Wedin’s
Algorithm II. This version of Ceres has a higher iteration
complexity, but also displays better convergence behavior per
iteration.
Setting Solver::Options::num_threads
to the maximum number
possible is highly recommended.
Nonmonotonic Steps¶
Note that the basic trustregion algorithm described in Trust Region Methods is a descent algorithm in that it only accepts a point if it strictly reduces the value of the objective function.
Relaxing this requirement allows the algorithm to be more efficient in the long term at the cost of some local increase in the value of the objective function.
This is because allowing for nondecreasing objective function values in a principled manner allows the algorithm to jump over boulders as the method is not restricted to move into narrow valleys while preserving its convergence properties.
Setting Solver::Options::use_nonmonotonic_steps
to true
enables the nonmonotonic trust region algorithm as described by Conn,
Gould & Toint in [Conn].
Even though the value of the objective function may be larger than the minimum value encountered over the course of the optimization, the final parameters returned to the user are the ones corresponding to the minimum cost over all iterations.
The option to take nonmonotonic steps is available for all trust region strategies.
Line Search Methods¶
The line search method in Ceres Solver cannot handle bounds constraints right now, so it can only be used for solving unconstrained problems.
Line search algorithms
 Given an initial point \(x\)
 \(\Delta x = H^{1}(x) g(x)\)
 \(\arg \min_\mu \frac{1}{2} \ F(x + \mu \Delta x) \^2\)
 \(x = x + \mu \Delta x\)
 Goto 2.
Here \(H(x)\) is some approximation to the Hessian of the objective function, and \(g(x)\) is the gradient at \(x\). Depending on the choice of \(H(x)\) we get a variety of different search directions \(\Delta x\).
Step 4, which is a one dimensional optimization or Line Search along \(\Delta x\) is what gives this class of methods its name.
Different line search algorithms differ in their choice of the search direction \(\Delta x\) and the method used for one dimensional optimization along \(\Delta x\). The choice of \(H(x)\) is the primary source of computational complexity in these methods. Currently, Ceres Solver supports three choices of search directions, all aimed at large scale problems.
STEEPEST_DESCENT
This corresponds to choosing \(H(x)\) to be the identity matrix. This is not a good search direction for anything but the simplest of the problems. It is only included here for completeness.NONLINEAR_CONJUGATE_GRADIENT
A generalization of the Conjugate Gradient method to nonlinear functions. The generalization can be performed in a number of different ways, resulting in a variety of search directions. Ceres Solver currently supportsFLETCHER_REEVES
,POLAK_RIBIERE
andHESTENES_STIEFEL
directions.BFGS
A generalization of the Secant method to multiple dimensions in which a full, dense approximation to the inverse Hessian is maintained and used to compute a quasiNewton step [NocedalWright]. BFGS is currently the best known general quasiNewton algorithm.LBFGS
A limited memory approximation to the fullBFGS
method in which the last M iterations are used to approximate the inverse Hessian used to compute a quasiNewton step [Nocedal], [ByrdNocedal].
Currently Ceres Solver supports both a backtracking and interpolation
based Armijo line search algorithm, and a sectioning / zoom
interpolation (strong) Wolfe condition line search algorithm.
However, note that in order for the assumptions underlying the
BFGS
and LBFGS
methods to be guaranteed to be satisfied the
Wolfe line search algorithm should be used.
LinearSolver¶
Recall that in both of the trustregion methods described above, the key computational cost is the solution of a linear least squares problem of the form
Let \(H(x)= J(x)^\top J(x)\) and \(g(x) = J(x)^\top f(x)\). For notational convenience let us also drop the dependence on \(x\). Then it is easy to see that solving (7) is equivalent to solving the normal equations.
Ceres provides a number of different options for solving (8).
DENSE_QR
¶
For small problems (a couple of hundred parameters and a few thousand
residuals) with relatively dense Jacobians, DENSE_QR
is the method
of choice [Bjorck]. Let \(J = QR\) be the QRdecomposition of
\(J\), where \(Q\) is an orthonormal matrix and \(R\) is
an upper triangular matrix [TrefethenBau]. Then it can be shown that
the solution to (8) is given by
Ceres uses Eigen
‘s dense QR factorization routines.
DENSE_NORMAL_CHOLESKY
& SPARSE_NORMAL_CHOLESKY
¶
Large nonlinear least square problems are usually sparse. In such cases, using a dense QR factorization is inefficient. Let \(H = R^\top R\) be the Cholesky factorization of the normal equations, where \(R\) is an upper triangular matrix, then the solution to (8) is given by
The observant reader will note that the \(R\) in the Cholesky factorization of \(H\) is the same upper triangular matrix \(R\) in the QR factorization of \(J\). Since \(Q\) is an orthonormal matrix, \(J=QR\) implies that \(J^\top J = R^\top Q^\top Q R = R^\top R\). There are two variants of Cholesky factorization – sparse and dense.
DENSE_NORMAL_CHOLESKY
as the name implies performs a dense
Cholesky factorization of the normal equations. Ceres uses
Eigen
‘s dense LDLT factorization routines.
SPARSE_NORMAL_CHOLESKY
, as the name implies performs a sparse
Cholesky factorization of the normal equations. This leads to
substantial savings in time and memory for large sparse
problems. Ceres uses the sparse Cholesky factorization routines in
Professor Tim Davis’ SuiteSparse
or CXSparse
packages [Chen]
or the sparse Cholesky factorization algorithm in Eigen
(which
incidently is a port of the algorithm implemented inside CXSparse
)
CGNR
¶
For general sparse problems, if the problem is too large for
CHOLMOD
or a sparse linear algebra library is not linked into
Ceres, another option is the CGNR
solver. This solver uses the
Conjugate Gradients solver on the normal equations, but without
forming the normal equations explicitly. It exploits the relation
The convergence of Conjugate Gradients depends on the conditioner
number \(\kappa(H)\). Usually \(H\) is poorly conditioned and
a Preconditioner must be used to get reasonable
performance. Currently only the JACOBI
preconditioner is available
for use with CGNR
. It uses the block diagonal of \(H\) to
precondition the normal equations.
When the user chooses CGNR
as the linear solver, Ceres
automatically switches from the exact step algorithm to an inexact
step algorithm.
DENSE_SCHUR
& SPARSE_SCHUR
¶
While it is possible to use SPARSE_NORMAL_CHOLESKY
to solve bundle
adjustment problems, bundle adjustment problem have a special
structure, and a more efficient scheme for solving (8)
can be constructed.
Suppose that the SfM problem consists of \(p\) cameras and \(q\) points and the variable vector \(x\) has the block structure \(x = [y_{1}, ... ,y_{p},z_{1}, ... ,z_{q}]\). Where, \(y\) and \(z\) correspond to camera and point parameters, respectively. Further, let the camera blocks be of size \(c\) and the point blocks be of size \(s\) (for most problems \(c\) = \(6\)–9 and \(s = 3\)). Ceres does not impose any constancy requirement on these block sizes, but choosing them to be constant simplifies the exposition.
A key characteristic of the bundle adjustment problem is that there is no term \(f_{i}\) that includes two or more point blocks. This in turn implies that the matrix \(H\) is of the form
where \(B \in \mathbb{R}^{pc\times pc}\) is a block sparse matrix with \(p\) blocks of size \(c\times c\) and \(C \in \mathbb{R}^{qs\times qs}\) is a block diagonal matrix with \(q\) blocks of size \(s\times s\). \(E \in \mathbb{R}^{pc\times qs}\) is a general block sparse matrix, with a block of size \(c\times s\) for each observation. Let us now block partition \(\Delta x = [\Delta y,\Delta z]\) and \(g=[v,w]\) to restate (8) as the block structured linear system
and apply Gaussian elimination to it. As we noted above, \(C\) is a block diagonal matrix, with small diagonal blocks of size \(s\times s\). Thus, calculating the inverse of \(C\) by inverting each of these blocks is cheap. This allows us to eliminate \(\Delta z\) by observing that \(\Delta z = C^{1}(w  E^\top \Delta y)\), giving us
The matrix
is the Schur complement of \(C\) in \(H\). It is also known as the reduced camera matrix, because the only variables participating in (11) are the ones corresponding to the cameras. \(S \in \mathbb{R}^{pc\times pc}\) is a block structured symmetric positive definite matrix, with blocks of size \(c\times c\). The block \(S_{ij}\) corresponding to the pair of images \(i\) and \(j\) is nonzero if and only if the two images observe at least one common point.
Now, (10) can be solved by first forming \(S\), solving for \(\Delta y\), and then backsubstituting \(\Delta y\) to obtain the value of \(\Delta z\). Thus, the solution of what was an \(n\times n\), \(n=pc+qs\) linear system is reduced to the inversion of the block diagonal matrix \(C\), a few matrixmatrix and matrixvector multiplies, and the solution of block sparse \(pc\times pc\) linear system (11). For almost all problems, the number of cameras is much smaller than the number of points, \(p \ll q\), thus solving (11) is significantly cheaper than solving (10). This is the Schur complement trick [Brown].
This still leaves open the question of solving (11). The
method of choice for solving symmetric positive definite systems
exactly is via the Cholesky factorization [TrefethenBau] and
depending upon the structure of the matrix, there are, in general, two
options. The first is direct factorization, where we store and factor
\(S\) as a dense matrix [TrefethenBau]. This method has
\(O(p^2)\) space complexity and \(O(p^3)\) time complexity and
is only practical for problems with up to a few hundred cameras. Ceres
implements this strategy as the DENSE_SCHUR
solver.
But, \(S\) is typically a fairly sparse matrix, as most images
only see a small fraction of the scene. This leads us to the second
option: Sparse Direct Methods. These methods store \(S\) as a
sparse matrix, use row and column reordering algorithms to maximize
the sparsity of the Cholesky decomposition, and focus their compute
effort on the nonzero part of the factorization [Chen]. Sparse
direct methods, depending on the exact sparsity structure of the Schur
complement, allow bundle adjustment algorithms to significantly scale
up over those based on dense factorization. Ceres implements this
strategy as the SPARSE_SCHUR
solver.
ITERATIVE_SCHUR
¶
Another option for bundle adjustment problems is to apply
Preconditioned Conjugate Gradients to the reduced camera matrix
\(S\) instead of \(H\). One reason to do this is that
\(S\) is a much smaller matrix than \(H\), but more
importantly, it can be shown that \(\kappa(S)\leq \kappa(H)\).
Ceres implements Conjugate Gradients on \(S\) as the
ITERATIVE_SCHUR
solver. When the user chooses ITERATIVE_SCHUR
as the linear solver, Ceres automatically switches from the exact step
algorithm to an inexact step algorithm.
The key computational operation when using Conjuagate Gradients is the
evaluation of the matrix vector product \(Sx\) for an arbitrary
vector \(x\). There are two ways in which this product can be
evaluated, and this can be controlled using
Solver::Options::use_explicit_schur_complement
. Depending on the
problem at hand, the performance difference between these two methods
can be quite substantial.
Implicit This is default. Implicit evaluation is suitable for large problems where the cost of computing and storing the Schur Complement \(S\) is prohibitive. Because PCG only needs access to \(S\) via its product with a vector, one way to evaluate \(Sx\) is to observe that
\[x_1 &= E^\top x\]\[x_2 &= C^{1} x_1\]\[\begin{split}x_3 &= Ex_2\\\end{split}\]\[\begin{split}x_4 &= Bx\\\end{split}\](12)\[Sx &= x_4  x_3\]Thus, we can run PCG on \(S\) with the same computational effort per iteration as PCG on \(H\), while reaping the benefits of a more powerful preconditioner. In fact, we do not even need to compute \(H\), (12) can be implemented using just the columns of \(J\).
Equation (12) is closely related to Domain Decomposition methods for solving large linear systems that arise in structural engineering and partial differential equations. In the language of Domain Decomposition, each point in a bundle adjustment problem is a domain, and the cameras form the interface between these domains. The iterative solution of the Schur complement then falls within the subcategory of techniques known as Iterative Substructuring [Saad] [Mathew].
Explicit The complexity of implicit matrixvector product evaluation scales with the number of nonzeros in the Jacobian. For small to medium sized problems, the cost of constructing the Schur Complement is small enough that it is better to construct it explicitly in memory and use it to evaluate the product \(Sx\).
When the user chooses ITERATIVE_SCHUR
as the linear solver, Ceres
automatically switches from the exact step algorithm to an inexact
step algorithm.
Note
In exact arithmetic, the choice of implicit versus explicit Schur complement would have no impact on solution quality. However, in practice if the Jacobian is poorly conditioned, one may observe (usually small) differences in solution quality. This is a natural consequence of performing computations in finite arithmetic.
Preconditioner¶
The convergence rate of Conjugate Gradients for solving (8) depends on the distribution of eigenvalues of \(H\) [Saad]. A useful upper bound is \(\sqrt{\kappa(H)}\), where, \(\kappa(H)\) is the condition number of the matrix \(H\). For most bundle adjustment problems, \(\kappa(H)\) is high and a direct application of Conjugate Gradients to (8) results in extremely poor performance.
The solution to this problem is to replace (8) with a preconditioned system. Given a linear system, \(Ax =b\) and a preconditioner \(M\) the preconditioned system is given by \(M^{1}Ax = M^{1}b\). The resulting algorithm is known as Preconditioned Conjugate Gradients algorithm (PCG) and its worst case complexity now depends on the condition number of the preconditioned matrix \(\kappa(M^{1}A)\).
The computational cost of using a preconditioner \(M\) is the cost of computing \(M\) and evaluating the product \(M^{1}y\) for arbitrary vectors \(y\). Thus, there are two competing factors to consider: How much of \(H\)‘s structure is captured by \(M\) so that the condition number \(\kappa(HM^{1})\) is low, and the computational cost of constructing and using \(M\). The ideal preconditioner would be one for which \(\kappa(M^{1}A) =1\). \(M=A\) achieves this, but it is not a practical choice, as applying this preconditioner would require solving a linear system equivalent to the unpreconditioned problem. It is usually the case that the more information \(M\) has about \(H\), the more expensive it is use. For example, Incomplete Cholesky factorization based preconditioners have much better convergence behavior than the Jacobi preconditioner, but are also much more expensive.
The simplest of all preconditioners is the diagonal or Jacobi
preconditioner, i.e., \(M=\operatorname{diag}(A)\), which for
block structured matrices like \(H\) can be generalized to the
block Jacobi preconditioner. Ceres implements the block Jacobi
preconditioner and refers to it as JACOBI
. When used with
CGNR it refers to the block diagonal of \(H\) and
when used with ITERATIVE_SCHUR it refers to the block
diagonal of \(B\) [Mandel].
Another obvious choice for ITERATIVE_SCHUR is the block
diagonal of the Schur complement matrix \(S\), i.e, the block
Jacobi preconditioner for \(S\). Ceres implements it and refers to
is as the SCHUR_JACOBI
preconditioner.
For bundle adjustment problems arising in reconstruction from
community photo collections, more effective preconditioners can be
constructed by analyzing and exploiting the camerapoint visibility
structure of the scene [KushalAgarwal]. Ceres implements the two
visibility based preconditioners described by Kushal & Agarwal as
CLUSTER_JACOBI
and CLUSTER_TRIDIAGONAL
. These are fairly new
preconditioners and Ceres’ implementation of them is in its early
stages and is not as mature as the other preconditioners described
above.
Ordering¶
The order in which variables are eliminated in a linear solver can have a significant of impact on the efficiency and accuracy of the method. For example when doing sparse Cholesky factorization, there are matrices for which a good ordering will give a Cholesky factor with \(O(n)\) storage, where as a bad ordering will result in an completely dense factor.
Ceres allows the user to provide varying amounts of hints to the solver about the variable elimination ordering to use. This can range from no hints, where the solver is free to decide the best ordering based on the user’s choices like the linear solver being used, to an exact order in which the variables should be eliminated, and a variety of possibilities in between.
Instances of the ParameterBlockOrdering
class are used to
communicate this information to Ceres.
Formally an ordering is an ordered partitioning of the parameter blocks. Each parameter block belongs to exactly one group, and each group has a unique integer associated with it, that determines its order in the set of groups. We call these groups Elimination Groups
Given such an ordering, Ceres ensures that the parameter blocks in the lowest numbered elimination group are eliminated first, and then the parameter blocks in the next lowest numbered elimination group and so on. Within each elimination group, Ceres is free to order the parameter blocks as it chooses. For example, consider the linear system
There are two ways in which it can be solved. First eliminating \(x\) from the two equations, solving for \(y\) and then back substituting for \(x\), or first eliminating \(y\), solving for \(x\) and back substituting for \(y\). The user can construct three orderings here.
 \(\{0: x\}, \{1: y\}\) : Eliminate \(x\) first.
 \(\{0: y\}, \{1: x\}\) : Eliminate \(y\) first.
 \(\{0: x, y\}\) : Solver gets to decide the elimination order.
Thus, to have Ceres determine the ordering automatically using heuristics, put all the variables in the same elimination group. The identity of the group does not matter. This is the same as not specifying an ordering at all. To control the ordering for every variable, create an elimination group per variable, ordering them in the desired order.
If the user is using one of the Schur solvers (DENSE_SCHUR
,
SPARSE_SCHUR
, ITERATIVE_SCHUR
) and chooses to specify an
ordering, it must have one important property. The lowest numbered
elimination group must form an independent set in the graph
corresponding to the Hessian, or in other words, no two parameter
blocks in in the first elimination group should cooccur in the same
residual block. For the best performance, this elimination group
should be as large as possible. For standard bundle adjustment
problems, this corresponds to the first elimination group containing
all the 3d points, and the second containing the all the cameras
parameter blocks.
If the user leaves the choice to Ceres, then the solver uses an approximate maximum independent set algorithm to identify the first elimination group [LiSaad].
Solver::Options
¶

class
Solver::
Options
¶ Solver::Options
controls the overall behavior of the solver. We list the various settings and their default values below.

bool
Solver::Options::
IsValid
(string *error) const¶ Validate the values in the options struct and returns true on success. If there is a problem, the method returns false with
error
containing a textual description of the cause.

MinimizerType
Solver::Options::
minimizer_type
¶ Default:
TRUST_REGION
Choose between
LINE_SEARCH
andTRUST_REGION
algorithms. See Trust Region Methods and Line Search Methods for more details.

LineSearchDirectionType
Solver::Options::
line_search_direction_type
¶ Default:
LBFGS
Choices are
STEEPEST_DESCENT
,NONLINEAR_CONJUGATE_GRADIENT
,BFGS
andLBFGS
.

LineSearchType
Solver::Options::
line_search_type
¶ Default:
WOLFE
Choices are
ARMIJO
andWOLFE
(strong Wolfe conditions). Note that in order for the assumptions underlying theBFGS
andLBFGS
line search direction algorithms to be guaranteed to be satisifed, theWOLFE
line search should be used.

NonlinearConjugateGradientType
Solver::Options::
nonlinear_conjugate_gradient_type
¶ Default:
FLETCHER_REEVES
Choices are
FLETCHER_REEVES
,POLAK_RIBIERE
andHESTENES_STIEFEL
.

int
Solver::Options::
max_lbfs_rank
¶ Default: 20
The LBFGS hessian approximation is a low rank approximation to the inverse of the Hessian matrix. The rank of the approximation determines (linearly) the space and time complexity of using the approximation. Higher the rank, the better is the quality of the approximation. The increase in quality is however is bounded for a number of reasons.
 The method only uses secant information and not actual derivatives.
 The Hessian approximation is constrained to be positive definite.
So increasing this rank to a large number will cost time and space complexity without the corresponding increase in solution quality. There are no hard and fast rules for choosing the maximum rank. The best choice usually requires some problem specific experimentation.

bool
Solver::Options::
use_approximate_eigenvalue_bfgs_scaling
¶ Default:
false
As part of the
BFGS
update step /LBFGS
rightmultiply step, the initial inverse Hessian approximation is taken to be the Identity. However, [Oren] showed that using instead \(I * \gamma\), where \(\gamma\) is a scalar chosen to approximate an eigenvalue of the true inverse Hessian can result in improved convergence in a wide variety of cases. Settinguse_approximate_eigenvalue_bfgs_scaling
to true enables this scaling inBFGS
(before first iteration) andLBFGS
(at each iteration).Precisely, approximate eigenvalue scaling equates to
\[\gamma = \frac{y_k' s_k}{y_k' y_k}\]With:
\[y_k = \nabla f_{k+1}  \nabla f_k\]\[s_k = x_{k+1}  x_k\]Where \(f()\) is the line search objective and \(x\) the vector of parameter values [NocedalWright].
It is important to note that approximate eigenvalue scaling does not always improve convergence, and that it can in fact significantly degrade performance for certain classes of problem, which is why it is disabled by default. In particular it can degrade performance when the sensitivity of the problem to different parameters varies significantly, as in this case a single scalar factor fails to capture this variation and detrimentally downscales parts of the Jacobian approximation which correspond to lowsensitivity parameters. It can also reduce the robustness of the solution to errors in the Jacobians.

LineSearchIterpolationType
Solver::Options::
line_search_interpolation_type
¶ Default:
CUBIC
Degree of the polynomial used to approximate the objective function. Valid values are
BISECTION
,QUADRATIC
andCUBIC
.

double
Solver::Options::
min_line_search_step_size
¶ The line search terminates if:
\[\\Delta x_k\_\infty < \text{min_line_search_step_size}\]where \(\\cdot\_\infty\) refers to the max norm, and \(\Delta x_k\) is the step change in the parameter values at the \(k\)th iteration.

double
Solver::Options::
line_search_sufficient_function_decrease
¶ Default:
1e4
Solving the line search problem exactly is computationally prohibitive. Fortunately, line search based optimization algorithms can still guarantee convergence if instead of an exact solution, the line search algorithm returns a solution which decreases the value of the objective function sufficiently. More precisely, we are looking for a step size s.t.
\[f(\text{step_size}) \le f(0) + \text{sufficient_decrease} * [f'(0) * \text{step_size}]\]This condition is known as the Armijo condition.

double
Solver::Options::
max_line_search_step_contraction
¶ Default:
1e3
In each iteration of the line search,
\[\text{new_step_size} >= \text{max_line_search_step_contraction} * \text{step_size}\]Note that by definition, for contraction:
\[0 < \text{max_step_contraction} < \text{min_step_contraction} < 1\]

double
Solver::Options::
min_line_search_step_contraction
¶ Default:
0.6
In each iteration of the line search,
\[\text{new_step_size} <= \text{min_line_search_step_contraction} * \text{step_size}\]Note that by definition, for contraction:
\[0 < \text{max_step_contraction} < \text{min_step_contraction} < 1\]

int
Solver::Options::
max_num_line_search_step_size_iterations
¶ Default:
20
Maximum number of trial step size iterations during each line search, if a step size satisfying the search conditions cannot be found within this number of trials, the line search will stop.
As this is an ‘artificial’ constraint (one imposed by the user, not the underlying math), if
WOLFE
line search is being used, and points satisfying the Armijo sufficient (function) decrease condition have been found during the current search (in \(<=\)max_num_line_search_step_size_iterations
). Then, the step size with the lowest function value which satisfies the Armijo condition will be returned as the new valid step, even though it does not satisfy the strong Wolfe conditions. This behaviour protects against early termination of the optimizer at a suboptimal point.

int
Solver::Options::
max_num_line_search_direction_restarts
¶ Default:
5
Maximum number of restarts of the line search direction algorithm before terminating the optimization. Restarts of the line search direction algorithm occur when the current algorithm fails to produce a new descent direction. This typically indicates a numerical failure, or a breakdown in the validity of the approximations used.

double
Solver::Options::
line_search_sufficient_curvature_decrease
¶ Default:
0.9
The strong Wolfe conditions consist of the Armijo sufficient decrease condition, and an additional requirement that the step size be chosen s.t. the magnitude (‘strong’ Wolfe conditions) of the gradient along the search direction decreases sufficiently. Precisely, this second condition is that we seek a step size s.t.
\[\f'(\text{step_size})\ <= \text{sufficient_curvature_decrease} * \f'(0)\\]Where \(f()\) is the line search objective and \(f'()\) is the derivative of \(f\) with respect to the step size: \(\frac{d f}{d~\text{step size}}\).

double
Solver::Options::
max_line_search_step_expansion
¶ Default:
10.0
During the bracketing phase of a Wolfe line search, the step size is increased until either a point satisfying the Wolfe conditions is found, or an upper bound for a bracket containing a point satisfying the conditions is found. Precisely, at each iteration of the expansion:
\[\text{new_step_size} <= \text{max_step_expansion} * \text{step_size}\]By definition for expansion
\[\text{max_step_expansion} > 1.0\]

TrustRegionStrategyType
Solver::Options::
trust_region_strategy_type
¶ Default:
LEVENBERG_MARQUARDT
The trust region step computation algorithm used by Ceres. Currently
LEVENBERG_MARQUARDT
andDOGLEG
are the two valid choices. See LevenbergMarquardt and Dogleg for more details.

DoglegType
Solver::Options::
dogleg_type
¶ Default:
TRADITIONAL_DOGLEG
Ceres supports two different dogleg strategies.
TRADITIONAL_DOGLEG
method by Powell and theSUBSPACE_DOGLEG
method described by [ByrdSchnabel] . See Dogleg for more details.

bool
Solver::Options::
use_nonmonotonic_steps
¶ Default:
false
Relax the requirement that the trustregion algorithm take strictly decreasing steps. See Nonmonotonic Steps for more details.

int
Solver::Options::
max_consecutive_nonmonotonic_steps
¶ Default:
5
The window size used by the step selection algorithm to accept nonmonotonic steps.

int
Solver::Options::
max_num_iterations
¶ Default:
50
Maximum number of iterations for which the solver should run.

double
Solver::Options::
max_solver_time_in_seconds
¶ Default:
1e6
Maximum amount of time for which the solver should run.

int
Solver::Options::
num_threads
¶ Default:
1
Number of threads used by Ceres to evaluate the Jacobian.

double
Solver::Options::
initial_trust_region_radius
¶ Default:
1e4
The size of the initial trust region. When the
LEVENBERG_MARQUARDT
strategy is used, the reciprocal of this number is the initial regularization parameter.

double
Solver::Options::
max_trust_region_radius
¶ Default:
1e16
The trust region radius is not allowed to grow beyond this value.

double
Solver::Options::
min_trust_region_radius
¶ Default:
1e32
The solver terminates, when the trust region becomes smaller than this value.

double
Solver::Options::
min_relative_decrease
¶ Default:
1e3
Lower threshold for relative decrease before a trustregion step is accepted.

double
Solver::Options::
min_lm_diagonal
¶ Default:
1e6
The
LEVENBERG_MARQUARDT
strategy, uses a diagonal matrix to regularize the trust region step. This is the lower bound on the values of this diagonal matrix.

double
Solver::Options::
max_lm_diagonal
¶ Default:
1e32
The
LEVENBERG_MARQUARDT
strategy, uses a diagonal matrix to regularize the trust region step. This is the upper bound on the values of this diagonal matrix.

int
Solver::Options::
max_num_consecutive_invalid_steps
¶ Default:
5
The step returned by a trust region strategy can sometimes be numerically invalid, usually because of conditioning issues. Instead of crashing or stopping the optimization, the optimizer can go ahead and try solving with a smaller trust region/better conditioned problem. This parameter sets the number of consecutive retries before the minimizer gives up.

double
Solver::Options::
function_tolerance
¶ Default:
1e6
Solver terminates if
\[\frac{\Delta \text{cost}}{\text{cost}} <= \text{function_tolerance}\]where, \(\Delta \text{cost}\) is the change in objective function value (up or down) in the current iteration of LevenbergMarquardt.

double
Solver::Options::
gradient_tolerance
¶ Default:
1e10
Solver terminates if
\[\x  \Pi \boxplus(x, g(x))\_\infty <= \text{gradient_tolerance}\]where \(\\cdot\_\infty\) refers to the max norm, \(\Pi\) is projection onto the bounds constraints and \(\boxplus\) is Plus operation for the overall local parameterization associated with the parameter vector.

double
Solver::Options::
parameter_tolerance
¶ Default:
1e8
Solver terminates if
\[\\Delta x\ <= (\x\ + \text{parameter_tolerance}) * \text{parameter_tolerance}\]where \(\Delta x\) is the step computed by the linear solver in the current iteration.

LinearSolverType
Solver::Options::
linear_solver_type
¶ Default:
SPARSE_NORMAL_CHOLESKY
/DENSE_QR
Type of linear solver used to compute the solution to the linear least squares problem in each iteration of the LevenbergMarquardt algorithm. If Ceres is built with support for
SuiteSparse
orCXSparse
orEigen
‘s sparse Cholesky factorization, the default isSPARSE_NORMAL_CHOLESKY
, it isDENSE_QR
otherwise.

PreconditionerType
Solver::Options::
preconditioner_type
¶ Default:
JACOBI
The preconditioner used by the iterative linear solver. The default is the block Jacobi preconditioner. Valid values are (in increasing order of complexity)
IDENTITY
,JACOBI
,SCHUR_JACOBI
,CLUSTER_JACOBI
andCLUSTER_TRIDIAGONAL
. See Preconditioner for more details.

VisibilityClusteringType
Solver::Options::
visibility_clustering_type
¶ Default:
CANONICAL_VIEWS
Type of clustering algorithm to use when constructing a visibility based preconditioner. The original visibility based preconditioning paper and implementation only used the canonical views algorithm.
This algorithm gives high quality results but for large dense graphs can be particularly expensive. As its worst case complexity is cubic in size of the graph.
Another option is to use
SINGLE_LINKAGE
which is a simple thresholded single linkage clustering algorithm that only pays attention to tightly coupled blocks in the Schur complement. This is a fast algorithm that works well.The optimal choice of the clustering algorithm depends on the sparsity structure of the problem, but generally speaking we recommend that you try
CANONICAL_VIEWS
first and if it is too expensive trySINGLE_LINKAGE
.

DenseLinearAlgebraLibrary
Solver::Options::
dense_linear_algebra_library_type
¶ Default:
EIGEN
Ceres supports using multiple dense linear algebra libraries for dense matrix factorizations. Currently
EIGEN
andLAPACK
are the valid choices.EIGEN
is always available,LAPACK
refers to the systemBLAS + LAPACK
library which may or may not be available.This setting affects the
DENSE_QR
,DENSE_NORMAL_CHOLESKY
andDENSE_SCHUR
solvers. For small to moderate sized probemEIGEN
is a fine choice but for large problems, an optimizedLAPACK + BLAS
implementation can make a substantial difference in performance.

SparseLinearAlgebraLibrary
Solver::Options::
sparse_linear_algebra_library_type
¶ Default: The highest available according to:
SUITE_SPARSE
>CX_SPARSE
>EIGEN_SPARSE
>NO_SPARSE
Ceres supports the use of three sparse linear algebra libraries,
SuiteSparse
, which is enabled by setting this parameter toSUITE_SPARSE
,CXSparse
, which can be selected by setting this parameter toCX_SPARSE
andEigen
which is enabled by setting this parameter toEIGEN_SPARSE
. Lastly,NO_SPARSE
means that no sparse linear solver should be used; note that this is irrespective of whether Ceres was compiled with support for one.SuiteSparse
is a sophisticated and complex sparse linear algebra library and should be used in general.If your needs/platforms prevent you from using
SuiteSparse
, consider usingCXSparse
, which is a much smaller, easier to build library. As can be expected, its performance on large problems is not comparable to that ofSuiteSparse
.Last but not the least you can use the sparse linear algebra routines in
Eigen
. Currently the performance of this library is the poorest of the three. But this should change in the near future.Another thing to consider here is that the sparse Cholesky factorization libraries in Eigen are licensed under
LGPL
and building Ceres with support forEIGEN_SPARSE
will result in an LGPL licensed library (since the corresponding code from Eigen is compiled into the library).The upside is that you do not need to build and link to an external library to use
EIGEN_SPARSE
.

int
Solver::Options::
num_linear_solver_threads
¶ Default:
1
Number of threads used by the linear solver.

shared_ptr<ParameterBlockOrdering>
Solver::Options::
linear_solver_ordering
¶ Default:
NULL
An instance of the ordering object informs the solver about the desired order in which parameter blocks should be eliminated by the linear solvers. See section~ref{sec:ordering`` for more details.
If
NULL
, the solver is free to choose an ordering that it thinks is best.See Ordering for more details.

bool
Solver::Options::
use_explicit_schur_complement
¶ Default:
false
Use an explicitly computed Schur complement matrix with
ITERATIVE_SCHUR
.By default this option is disabled and
ITERATIVE_SCHUR
evaluates evaluates matrixvector products between the Schur complement and a vector implicitly by exploiting the algebraic expression for the Schur complement.The cost of this evaluation scales with the number of nonzeros in the Jacobian.
For small to medium sized problems there is a sweet spot where computing the Schur complement is cheap enough that it is much more efficient to explicitly compute it and use it for evaluating the matrixvector products.
Enabling this option tells
ITERATIVE_SCHUR
to use an explicitly computed Schur complement. This can improve the performance of theITERATIVE_SCHUR
solver significantly.

bool
Solver::Options::
use_post_ordering
¶ Default:
false
Sparse Cholesky factorization algorithms use a fillreducing ordering to permute the columns of the Jacobian matrix. There are two ways of doing this.
 Compute the Jacobian matrix in some order and then have the factorization algorithm permute the columns of the Jacobian.
 Compute the Jacobian with its columns already permuted.
The first option incurs a significant memory penalty. The factorization algorithm has to make a copy of the permuted Jacobian matrix, thus Ceres prepermutes the columns of the Jacobian matrix and generally speaking, there is no performance penalty for doing so.
In some rare cases, it is worth using a more complicated reordering algorithm which has slightly better runtime performance at the expense of an extra copy of the Jacobian matrix. Setting
use_postordering
totrue
enables this tradeoff.

bool
Solver::Options::
dynamic_sparsity
¶ Some nonlinear least squares problems are symbolically dense but numerically sparse. i.e. at any given state only a small number of Jacobian entries are nonzero, but the position and number of nonzeros is different depending on the state. For these problems it can be useful to factorize the sparse jacobian at each solver iteration instead of including all of the zero entries in a single general factorization.
If your problem does not have this property (or you do not know), then it is probably best to keep this false, otherwise it will likely lead to worse performance.
This setting only affects the SPARSE_NORMAL_CHOLESKY solver.

int
Solver::Options::
min_linear_solver_iterations
¶ Default:
0
Minimum number of iterations used by the linear solver. This only makes sense when the linear solver is an iterative solver, e.g.,
ITERATIVE_SCHUR
orCGNR
.

int
Solver::Options::
max_linear_solver_iterations
¶ Default:
500
Minimum number of iterations used by the linear solver. This only makes sense when the linear solver is an iterative solver, e.g.,
ITERATIVE_SCHUR
orCGNR
.

double
Solver::Options::
eta
¶ Default:
1e1
Forcing sequence parameter. The truncated Newton solver uses this number to control the relative accuracy with which the Newton step is computed. This constant is passed to
ConjugateGradientsSolver
which uses it to terminate the iterations when\[\frac{Q_i  Q_{i1}}{Q_i} < \frac{\eta}{i}\]

bool
Solver::Options::
jacobi_scaling
¶ Default:
true
true
means that the Jacobian is scaled by the norm of its columns before being passed to the linear solver. This improves the numerical conditioning of the normal equations.

bool
Solver::Options::
use_inner_iterations
¶ Default:
false
Use a nonlinear version of a simplified variable projection algorithm. Essentially this amounts to doing a further optimization on each Newton/Trust region step using a coordinate descent algorithm. For more details, see Inner Iterations.

double
Solver::Options::
inner_iteration_tolerance
¶ Default:
1e3
Generally speaking, inner iterations make significant progress in the early stages of the solve and then their contribution drops down sharply, at which point the time spent doing inner iterations is not worth it.
Once the relative decrease in the objective function due to inner iterations drops below
inner_iteration_tolerance
, the use of inner iterations in subsequent trust region minimizer iterations is disabled.

shared_ptr<ParameterBlockOrdering>
Solver::Options::
inner_iteration_ordering
¶ Default:
NULL
If
Solver::Options::use_inner_iterations
true, then the user has two choices. Let the solver heuristically decide which parameter blocks to
optimize in each inner iteration. To do this, set
Solver::Options::inner_iteration_ordering
toNULL
.  Specify a collection of of ordered independent sets. The lower numbered groups are optimized before the higher number groups during the inner optimization phase. Each group must be an independent set. Not all parameter blocks need to be included in the ordering.
See Ordering for more details.
 Let the solver heuristically decide which parameter blocks to
optimize in each inner iteration. To do this, set

LoggingType
Solver::Options::
logging_type
¶ Default:
PER_MINIMIZER_ITERATION

bool
Solver::Options::
minimizer_progress_to_stdout
¶ Default:
false
By default the
Minimizer
progress is logged toSTDERR
depending on thevlog
level. If this flag is set to true, andSolver::Options::logging_type
is notSILENT
, the logging output is sent toSTDOUT
.For
TRUST_REGION_MINIMIZER
the progress display looks likeiter cost cost_change gradient step tr_ratio tr_radius ls_iter iter_time total_time 0 4.185660e+06 0.00e+00 1.09e+08 0.00e+00 0.00e+00 1.00e+04 0 7.59e02 3.37e01 1 1.062590e+05 4.08e+06 8.99e+06 5.36e+02 9.82e01 3.00e+04 1 1.65e01 5.03e01 2 4.992817e+04 5.63e+04 8.32e+06 3.19e+02 6.52e01 3.09e+04 1 1.45e01 6.48e01
Here
cost
is the value of the objective function.cost_change
is the change in the value of the objective function if the step computed in this iteration is accepted.gradient
is the max norm of the gradient.step
is the change in the parameter vector.tr_ratio
is the ratio of the actual change in the objective function value to the change in the value of the trust region model.tr_radius
is the size of the trust region radius.ls_iter
is the number of linear solver iterations used to compute the trust region step. For direct/factorization based solvers it is always 1, for iterative solvers likeITERATIVE_SCHUR
it is the number of iterations of the Conjugate Gradients algorithm.iter_time
is the time take by the current iteration.total_time
is the total time taken by the minimizer.
For
LINE_SEARCH_MINIMIZER
the progress display looks like0: f: 2.317806e+05 d: 0.00e+00 g: 3.19e01 h: 0.00e+00 s: 0.00e+00 e: 0 it: 2.98e02 tt: 8.50e02 1: f: 2.312019e+05 d: 5.79e+02 g: 3.18e01 h: 2.41e+01 s: 1.00e+00 e: 1 it: 4.54e02 tt: 1.31e01 2: f: 2.300462e+05 d: 1.16e+03 g: 3.17e01 h: 4.90e+01 s: 2.54e03 e: 1 it: 4.96e02 tt: 1.81e01
Here
f
is the value of the objective function.d
is the change in the value of the objective function if the step computed in this iteration is accepted.g
is the max norm of the gradient.h
is the change in the parameter vector.s
is the optimal step length computed by the line search.it
is the time take by the current iteration.tt
is the total time taken by the minimizer.

vector<int>
Solver::Options::
trust_region_minimizer_iterations_to_dump
¶ Default:
empty
List of iterations at which the trust region minimizer should dump the trust region problem. Useful for testing and benchmarking. If
empty
, no problems are dumped.

string
Solver::Options::
trust_region_problem_dump_directory
¶ Default:
/tmp
Directory to which the problems should be written to. Should be nonempty ifSolver::Options::trust_region_minimizer_iterations_to_dump
is nonempty andSolver::Options::trust_region_problem_dump_format_type
is notCONSOLE
.

DumpFormatType
Solver::Options::
trust_region_problem_dump_format
¶ Default:
TEXTFILE
The format in which trust region problems should be logged when
Solver::Options::trust_region_minimizer_iterations_to_dump
is nonempty. There are three options:CONSOLE
prints the linear least squares problem in a humanreadable format to
stderr
. The Jacobian is printed as a dense matrix. The vectors \(D\), \(x\) and \(f\) are printed as dense vectors. This should only be used for small problems.
TEXTFILE
Write out the linear least squares problem to the directory pointed to bySolver::Options::trust_region_problem_dump_directory
as text files which can be read intoMATLAB/Octave
. The Jacobian is dumped as a text file containing \((i,j,s)\) triplets, the vectors \(D\), x and f are dumped as text files containing a list of their values.A
MATLAB/Octave
script calledceres_solver_iteration_???.m
is also output, which can be used to parse and load the problem into memory.

bool
Solver::Options::
check_gradients
¶ Default:
false
Check all Jacobians computed by each residual block with finite differences. This is expensive since it involves computing the derivative by normal means (e.g. user specified, autodiff, etc), then also computing it using finite differences. The results are compared, and if they differ substantially, the optimization fails and the details are stored in the solver summary.

double
Solver::Options::
gradient_check_relative_precision
¶ Default:
1e08
Precision to check for in the gradient checker. If the relative difference between an element in a Jacobian exceeds this number, then the Jacobian for that cost term is dumped.

double
Solver::Options::
gradient_check_numeric_derivative_relative_step_size
¶ Default:
1e6
Note
This option only applies to the numeric differentiation used for checking the user provided derivatives when when Solver::Options::check_gradients is true. If you are using
NumericDiffCostFunction
and are interested in changing the step size for numeric differentiation in your cost function, please have a look atNumericDiffOptions
.Relative shift used for taking numeric derivatives when Solver::Options::check_gradients is true.
For finite differencing, each dimension is evaluated at slightly shifted values, e.g., for forward differences, the numerical derivative is
\[\begin{split}\delta &= gradient\_check\_numeric\_derivative\_relative\_step\_size\\ \Delta f &= \frac{f((1 + \delta) x)  f(x)}{\delta x}\end{split}\]The finite differencing is done along each dimension. The reason to use a relative (rather than absolute) step size is that this way, numeric differentiation works for functions where the arguments are typically large (e.g. \(10^9\)) and when the values are small (e.g. \(10^{5}\)). It is possible to construct torture cases which break this finite difference heuristic, but they do not come up often in practice.

vector<IterationCallback>
Solver::Options::
callbacks
¶ Callbacks that are executed at the end of each iteration of the
Minimizer
. They are executed in the order that they are specified in this vector. By default, parameter blocks are updated only at the end of the optimization, i.e., when theMinimizer
terminates. This behavior is controlled bySolver::Options::update_state_every_variable
. If the user wishes to have access to the update parameter blocks when his/her callbacks are executed, then setSolver::Options::update_state_every_iteration
to true.The solver does NOT take ownership of these pointers.

bool
Solver::Options::
update_state_every_iteration
¶ Default:
false
Normally the parameter blocks are only updated when the solver terminates. Setting this to true update them in every iteration. This setting is useful when building an interactive application using Ceres and using an
IterationCallback
.
ParameterBlockOrdering
¶

class
ParameterBlockOrdering
¶ ParameterBlockOrdering
is a class for storing and manipulating an ordered collection of groups/sets with the following semantics:Group IDs are nonnegative integer values. Elements are any type that can serve as a key in a map or an element of a set.
An element can only belong to one group at a time. A group may contain an arbitrary number of elements.
Groups are ordered by their group id.

bool
ParameterBlockOrdering::
AddElementToGroup
(const double *element, const int group)¶ Add an element to a group. If a group with this id does not exist, one is created. This method can be called any number of times for the same element. Group ids should be nonnegative numbers. Return value indicates if adding the element was a success.

void
ParameterBlockOrdering::
Clear
()¶ Clear the ordering.

bool
ParameterBlockOrdering::
Remove
(const double *element)¶ Remove the element, no matter what group it is in. If the element is not a member of any group, calling this method will result in a crash. Return value indicates if the element was actually removed.

void
ParameterBlockOrdering::
Reverse
()¶ Reverse the order of the groups in place.

int
ParameterBlockOrdering::
GroupId
(const double *element) const¶ Return the group id for the element. If the element is not a member of any group, return 1.

bool
ParameterBlockOrdering::
IsMember
(const double *element) const¶ True if there is a group containing the parameter block.

int
ParameterBlockOrdering::
GroupSize
(const int group) const¶ This function always succeeds, i.e., implicitly there exists a group for every integer.

int
ParameterBlockOrdering::
NumElements
() const¶ Number of elements in the ordering.

int
ParameterBlockOrdering::
NumGroups
() const¶ Number of groups with one or more elements.
IterationCallback
¶

class
IterationSummary
¶ IterationSummary
describes the state of the minimizer at the end of each iteration.

int32
IterationSummary::
iteration
¶ Current iteration number.

bool
IterationSummary::
step_is_valid
¶ Step was numerically valid, i.e., all values are finite and the step reduces the value of the linearized model.
Note:IterationSummary::step_is_valid
is false whenIterationSummary::iteration
= 0.

bool
IterationSummary::
step_is_nonmonotonic
¶ Step did not reduce the value of the objective function sufficiently, but it was accepted because of the relaxed acceptance criterion used by the nonmonotonic trust region algorithm.
Note:
IterationSummary::step_is_nonmonotonic
is false when whenIterationSummary::iteration
= 0.

bool
IterationSummary::
step_is_successful
¶ Whether or not the minimizer accepted this step or not.
If the ordinary trust region algorithm is used, this means that the relative reduction in the objective function value was greater than
Solver::Options::min_relative_decrease
. However, if the nonmonotonic trust region algorithm is used (Solver::Options::use_nonmonotonic_steps
= true), then even if the relative decrease is not sufficient, the algorithm may accept the step and the step is declared successful.Note:
IterationSummary::step_is_successful
is false when whenIterationSummary::iteration
= 0.

double
IterationSummary::
cost
¶ Value of the objective function.

double
IterationSummary::
cost_change
¶ Change in the value of the objective function in this iteration. This can be positive or negative.

double
IterationSummary::
gradient_max_norm
¶ Infinity norm of the gradient vector.

double
IterationSummary::
gradient_norm
¶ 2norm of the gradient vector.

double
IterationSummary::
step_norm
¶ 2norm of the size of the step computed in this iteration.

double
IterationSummary::
relative_decrease
¶ For trust region algorithms, the ratio of the actual change in cost and the change in the cost of the linearized approximation.
This field is not used when a linear search minimizer is used.

double
IterationSummary::
trust_region_radius
¶ Size of the trust region at the end of the current iteration. For the LevenbergMarquardt algorithm, the regularization parameter is 1.0 / member::IterationSummary::trust_region_radius.

double
IterationSummary::
eta
¶ For the inexact step LevenbergMarquardt algorithm, this is the relative accuracy with which the step is solved. This number is only applicable to the iterative solvers capable of solving linear systems inexactly. Factorizationbased exact solvers always have an eta of 0.0.

double
IterationSummary::
step_size
¶ Step sized computed by the line search algorithm.
This field is not used when a trust region minimizer is used.

int
IterationSummary::
line_search_function_evaluations
¶ Number of function evaluations used by the line search algorithm.
This field is not used when a trust region minimizer is used.

int
IterationSummary::
linear_solver_iterations
¶ Number of iterations taken by the linear solver to solve for the trust region step.
Currently this field is not used when a line search minimizer is used.

double
IterationSummary::
iteration_time_in_seconds
¶ Time (in seconds) spent inside the minimizer loop in the current iteration.

double
IterationSummary::
step_solver_time_in_seconds
¶ Time (in seconds) spent inside the trust region step solver.

double
IterationSummary::
cumulative_time_in_seconds
¶ Time (in seconds) since the user called Solve().

class
IterationCallback
¶ Interface for specifying callbacks that are executed at the end of each iteration of the minimizer.
class IterationCallback { public: virtual ~IterationCallback() {} virtual CallbackReturnType operator()(const IterationSummary& summary) = 0; };
The solver uses the return value of
operator()
to decide whether to continue solving or to terminate. The user can return three values.SOLVER_ABORT
indicates that the callback detected an abnormal situation. The solver returns without updating the parameter blocks (unlessSolver::Options::update_state_every_iteration
is set true). Solver returns withSolver::Summary::termination_type
set toUSER_FAILURE
.SOLVER_TERMINATE_SUCCESSFULLY
indicates that there is no need to optimize anymore (some user specified termination criterion has been met). Solver returns withSolver::Summary::termination_type`
set toUSER_SUCCESS
.SOLVER_CONTINUE
indicates that the solver should continue optimizing.
For example, the following
IterationCallback
is used internally by Ceres to log the progress of the optimization.class LoggingCallback : public IterationCallback { public: explicit LoggingCallback(bool log_to_stdout) : log_to_stdout_(log_to_stdout) {} ~LoggingCallback() {} CallbackReturnType operator()(const IterationSummary& summary) { const char* kReportRowFormat = "% 4d: f:% 8e d:% 3.2e g:% 3.2e h:% 3.2e " "rho:% 3.2e mu:% 3.2e eta:% 3.2e li:% 3d"; string output = StringPrintf(kReportRowFormat, summary.iteration, summary.cost, summary.cost_change, summary.gradient_max_norm, summary.step_norm, summary.relative_decrease, summary.trust_region_radius, summary.eta, summary.linear_solver_iterations); if (log_to_stdout_) { cout << output << endl; } else { VLOG(1) << output; } return SOLVER_CONTINUE; } private: const bool log_to_stdout_; };
CRSMatrix
¶

class
CRSMatrix
¶ A compressed row sparse matrix used primarily for communicating the Jacobian matrix to the user.

int
CRSMatrix::
num_rows
¶ Number of rows.

int
CRSMatrix::
num_cols
¶ Number of columns.

vector<int>
CRSMatrix::
rows
¶ CRSMatrix::rows
is aCRSMatrix::num_rows
+ 1 sized array that points into theCRSMatrix::cols
andCRSMatrix::values
array.

vector<int>
CRSMatrix::
cols
¶ CRSMatrix::cols
contain as many entries as there are nonzeros in the matrix.For each row
i
,cols[rows[i]]
...cols[rows[i + 1]  1]
are the indices of the nonzero columns of rowi
.

vector<int>
CRSMatrix::
values
¶ CRSMatrix::values
contain as many entries as there are nonzeros in the matrix.For each row
i
,values[rows[i]]
...values[rows[i + 1]  1]
are the values of the nonzero columns of rowi
.
e.g., consider the 3x4 sparse matrix
0 10 0 4
0 2 3 2
1 2 0 0
The three arrays will be:
row0 row1 row2
rows = [ 0, 2, 5, 7]
cols = [ 1, 3, 1, 2, 3, 0, 1]
values = [10, 4, 2, 3, 2, 1, 2]
Solver::Summary
¶

class
Solver::
Summary
¶ Summary of the various stages of the solver after termination.

string
Solver::Summary::
BriefReport
() const¶ A brief one line description of the state of the solver after termination.

string
Solver::Summary::
FullReport
() const¶ A full multiline description of the state of the solver after termination.

bool
Solver::Summary::
IsSolutionUsable
() const¶ Whether the solution returned by the optimization algorithm can be relied on to be numerically sane. This will be the case if Solver::Summary:termination_type is set to CONVERGENCE, USER_SUCCESS or NO_CONVERGENCE, i.e., either the solver converged by meeting one of the convergence tolerances or because the user indicated that it had converged or it ran to the maximum number of iterations or time.

MinimizerType
Solver::Summary::
minimizer_type
¶ Type of minimization algorithm used.

TerminationType
Solver::Summary::
termination_type
¶ The cause of the minimizer terminating.

string
Solver::Summary::
message
¶ Reason why the solver terminated.

double
Solver::Summary::
initial_cost
¶ Cost of the problem (value of the objective function) before the optimization.

double
Solver::Summary::
final_cost
¶ Cost of the problem (value of the objective function) after the optimization.

double
Solver::Summary::
fixed_cost
¶ The part of the total cost that comes from residual blocks that were held fixed by the preprocessor because all the parameter blocks that they depend on were fixed.

vector<IterationSummary>
Solver::Summary::
iterations
¶ IterationSummary
for each minimizer iteration in order.

int
Solver::Summary::
num_successful_steps
¶ Number of minimizer iterations in which the step was accepted. Unless
Solver::Options::use_non_monotonic_steps
is true this is also the number of steps in which the objective function value/cost went down.

int
Solver::Summary::
num_unsuccessful_steps
¶ Number of minimizer iterations in which the step was rejected either because it did not reduce the cost enough or the step was not numerically valid.

int
Solver::Summary::
num_inner_iteration_steps
¶  Number of times inner iterations were performed.

int
Solver::Summary::
num_line_search_steps
¶ Total number of iterations inside the line search algorithm across all invocations. We call these iterations “steps” to distinguish them from the outer iterations of the line search and trust region minimizer algorithms which call the line search algorithm as a subroutine.

int

double
Solver::Summary::
preprocessor_time_in_seconds
¶ Time (in seconds) spent in the preprocessor.

double
Solver::Summary::
minimizer_time_in_seconds
¶ Time (in seconds) spent in the Minimizer.

double
Solver::Summary::
postprocessor_time_in_seconds
¶ Time (in seconds) spent in the post processor.

double
Solver::Summary::
total_time_in_seconds
¶ Time (in seconds) spent in the solver.

double
Solver::Summary::
linear_solver_time_in_seconds
¶ Time (in seconds) spent in the linear solver computing the trust region step.

double
Solver::Summary::
residual_evaluation_time_in_seconds
¶ Time (in seconds) spent evaluating the residual vector.

double
Solver::Summary::
jacobian_evaluation_time_in_seconds
¶ Time (in seconds) spent evaluating the Jacobian matrix.

double
Solver::Summary::
inner_iteration_time_in_seconds
¶ Time (in seconds) spent doing inner iterations.

int
Solver::Summary::
num_parameter_blocks
¶ Number of parameter blocks in the problem.

int
Solver::Summary::
num_parameters
¶ Number of parameters in the problem.

int
Solver::Summary::
num_effective_parameters
¶ Dimension of the tangent space of the problem (or the number of columns in the Jacobian for the problem). This is different from
Solver::Summary::num_parameters
if a parameter block is associated with aLocalParameterization
.

int
Solver::Summary::
num_residual_blocks
¶ Number of residual blocks in the problem.

int
Solver::Summary::
num_residuals
¶ Number of residuals in the problem.

int
Solver::Summary::
num_parameter_blocks_reduced
¶ Number of parameter blocks in the problem after the inactive and constant parameter blocks have been removed. A parameter block is inactive if no residual block refers to it.

int
Solver::Summary::
num_parameters_reduced
¶ Number of parameters in the reduced problem.

int
Solver::Summary::
num_effective_parameters_reduced
¶ Dimension of the tangent space of the reduced problem (or the number of columns in the Jacobian for the reduced problem). This is different from
Solver::Summary::num_parameters_reduced
if a parameter block in the reduced problem is associated with aLocalParameterization
.

int
Solver::Summary::
num_residual_blocks_reduced
¶ Number of residual blocks in the reduced problem.

int
Solver::Summary::
num_residuals_reduced
¶ Number of residuals in the reduced problem.

int
Solver::Summary::
num_threads_given
¶ Number of threads specified by the user for Jacobian and residual evaluation.

int
Solver::Summary::
num_threads_used
¶ Number of threads actually used by the solver for Jacobian and residual evaluation. This number is not equal to
Solver::Summary::num_threads_given
if OpenMP is not available.

int
Solver::Summary::
num_linear_solver_threads_given
¶ Number of threads specified by the user for solving the trust region problem.

int
Solver::Summary::
num_linear_solver_threads_used
¶ Number of threads actually used by the solver for solving the trust region problem. This number is not equal to
Solver::Summary::num_linear_solver_threads_given
if OpenMP is not available.

LinearSolverType
Solver::Summary::
linear_solver_type_given
¶ Type of the linear solver requested by the user.

LinearSolverType
Solver::Summary::
linear_solver_type_used
¶ Type of the linear solver actually used. This may be different from
Solver::Summary::linear_solver_type_given
if Ceres determines that the problem structure is not compatible with the linear solver requested or if the linear solver requested by the user is not available, e.g. The user requested SPARSE_NORMAL_CHOLESKY but no sparse linear algebra library was available.

vector<int>
Solver::Summary::
linear_solver_ordering_given
¶ Size of the elimination groups given by the user as hints to the linear solver.

vector<int>
Solver::Summary::
linear_solver_ordering_used
¶ Size of the parameter groups used by the solver when ordering the columns of the Jacobian. This maybe different from
Solver::Summary::linear_solver_ordering_given
if the user leftSolver::Summary::linear_solver_ordering_given
blank and asked for an automatic ordering, or if the problem contains some constant or inactive parameter blocks.

std::string
Solver::Summary::
schur_structure_given
¶ For Schur type linear solvers, this string describes the template specialization which was detected in the problem and should be used.

std::string
Solver::Summary::
schur_structure_used
¶ For Schur type linear solvers, this string describes the template specialization that was actually instantiated and used. The reason this will be different from
Solver::Summary::schur_structure_given
is because the corresponding template specialization does not exist.Template specializations can be added to ceres by editing
internal/ceres/generate_template_specializations.py

bool
Solver::Summary::
inner_iterations_given
¶ True if the user asked for inner iterations to be used as part of the optimization.

bool
Solver::Summary::
inner_iterations_used
¶ True if the user asked for inner iterations to be used as part of the optimization and the problem structure was such that they were actually performed. For example, in a problem with just one parameter block, inner iterations are not performed.

vector<int>
inner_iteration_ordering_given
¶ Size of the parameter groups given by the user for performing inner iterations.

vector<int>
inner_iteration_ordering_used
¶ Size of the parameter groups given used by the solver for performing inner iterations. This maybe different from
Solver::Summary::inner_iteration_ordering_given
if the user leftSolver::Summary::inner_iteration_ordering_given
blank and asked for an automatic ordering, or if the problem contains some constant or inactive parameter blocks.

PreconditionerType
Solver::Summary::
preconditioner_type_given
¶ Type of the preconditioner requested by the user.

PreconditionerType
Solver::Summary::
preconditioner_type_used
¶ Type of the preconditioner actually used. This may be different from
Solver::Summary::linear_solver_type_given
if Ceres determines that the problem structure is not compatible with the linear solver requested or if the linear solver requested by the user is not available.

VisibilityClusteringType
Solver::Summary::
visibility_clustering_type
¶ Type of clustering algorithm used for visibility based preconditioning. Only meaningful when the
Solver::Summary::preconditioner_type
isCLUSTER_JACOBI
orCLUSTER_TRIDIAGONAL
.

TrustRegionStrategyType
Solver::Summary::
trust_region_strategy_type
¶ Type of trust region strategy.

DoglegType
Solver::Summary::
dogleg_type
¶ Type of dogleg strategy used for solving the trust region problem.

DenseLinearAlgebraLibraryType
Solver::Summary::
dense_linear_algebra_library_type
¶ Type of the dense linear algebra library used.

SparseLinearAlgebraLibraryType
Solver::Summary::
sparse_linear_algebra_library_type
¶ Type of the sparse linear algebra library used.

LineSearchDirectionType
Solver::Summary::
line_search_direction_type
¶ Type of line search direction used.

LineSearchType
Solver::Summary::
line_search_type
¶ Type of the line search algorithm used.

LineSearchInterpolationType
Solver::Summary::
line_search_interpolation_type
¶ When performing line search, the degree of the polynomial used to approximate the objective function.

NonlinearConjugateGradientType
Solver::Summary::
nonlinear_conjugate_gradient_type
¶ If the line search direction is NONLINEAR_CONJUGATE_GRADIENT, then this indicates the particular variant of nonlinear conjugate gradient used.

int
Solver::Summary::
max_lbfgs_rank
¶ If the type of the line search direction is LBFGS, then this indicates the rank of the Hessian approximation.
Covariance Estimation¶
Background¶
One way to assess the quality of the solution returned by a nonlinear least squares solve is to analyze the covariance of the solution.
Let us consider the nonlinear regression problem
i.e., the observation \(y\) is a random nonlinear function of the independent variable \(x\) with mean \(f(x)\) and identity covariance. Then the maximum likelihood estimate of \(x\) given observations \(y\) is the solution to the nonlinear least squares problem:
And the covariance of \(x^*\) is given by
Here \(J(x^*)\) is the Jacobian of \(f\) at \(x^*\). The above formula assumes that \(J(x^*)\) has full column rank.
If \(J(x^*)\) is rank deficient, then the covariance matrix \(C(x^*)\) is also rank deficient and is given by the MoorePenrose pseudo inverse.
Note that in the above, we assumed that the covariance matrix for \(y\) was identity. This is an important assumption. If this is not the case and we have
Where \(S\) is a positive semidefinite matrix denoting the covariance of \(y\), then the maximum likelihood problem to be solved is
and the corresponding covariance estimate of \(x^*\) is given by
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 \(S^{1/2} f(x)\) instead of just \(f(x)\), where \(S^{1/2}\) is the inverse square root of the covariance matrix \(S\).
Gauge Invariance¶
In structure from motion (3D reconstruction) problems, the reconstruction is ambiguous upto a similarity transform. This is known as a Gauge Ambiguity. Handling Gauges correctly requires the use of SVD or custom inversion algorithms. For small problems the user can use the dense algorithm. For more details see the work of Kanatani & Morris [KanataniMorris].
Covariance
¶
Covariance
allows the user to evaluate the covariance for a
nonlinear 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. 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 \(J'J\) is not defined and instead a pseudo inverse needs to be computed.
The rank deficiency in \(J\) 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.
 The rank deficiency arises from overparameterization. e.g., a four dimensional quaternion used to parameterize \(SO(3)\), which is a three dimensional manifold. In cases like this, the user should use an appropriate
LocalParameterization
. Not only will this lead to better numerical behaviour of the Solver, it will also expose the rank deficiency to theCovariance
object so that it can handle it correctly. More general numerical rank deficiency in the Jacobian requires the computation of the so called Singular Value Decomposition (SVD) of \(J'J\). 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
¶

int
Covariance::Options::
num_threads
¶ Default:
1
Number of threads to be used for evaluating the Jacobian and estimation of covariance.

CovarianceAlgorithmType
Covariance::Options::
algorithm_type
¶ Default:
SUITE_SPARSE_QR
ifSuiteSparseQR
is installed andEIGEN_SPARSE_QR
otherwise.Ceres supports three different algorithms for covariance estimation, which represent different tradeoffs in speed, accuracy and reliability.
DENSE_SVD
usesEigen
‘sJacobiSVD
to perform the computations. It computes the singular value decomposition\[U S V^\top = J\]and then uses it to compute the pseudo inverse of J’J as
\[(J'J)^{\dagger} = V S^{\dagger} V^\top\]It is an accurate but slow method and should only be used for small to moderate sized problems. It can handle fullrank as well as rank deficient Jacobians.
EIGEN_SPARSE_QR
uses the sparse QR factorization algorithm inEigen
to compute the decomposition\[\begin{split}QR &= J\\ \left(J^\top J\right)^{1} &= \left(R^\top R\right)^{1}\end{split}\]It is a moderately fast algorithm for sparse matrices.
SUITE_SPARSE_QR
uses the sparse QR factorization algorithm inSuiteSparse
. It uses dense linear algebra and is multi threaded, so for large sparse sparse matrices it is significantly faster thanEIGEN_SPARSE_QR
.
Neither
EIGEN_SPARSE_QR
norSUITE_SPARSE_QR
are capable of computing the covariance if the Jacobian is rank deficient.

int
Covariance::Options::
min_reciprocal_condition_number
¶ Default: \(10^{14}\)
If the Jacobian matrix is near singular, then inverting \(J'J\) will result in unreliable results, e.g, if
\[\begin{split}J = \begin{bmatrix} 1.0& 1.0 \\ 1.0& 1.0000001 \end{bmatrix}\end{split}\]which is essentially a rank deficient matrix, we have
\[\begin{split}(J'J)^{1} = \begin{bmatrix} 2.0471e+14& 2.0471e+14 \\ 2.0471e+14 2.0471e+14 \end{bmatrix}\end{split}\]This is not a useful result. Therefore, by default
Covariance::Compute()
will returnfalse
if a rank deficient Jacobian is encountered. How rank deficiency is detected depends on the algorithm being used.DENSE_SVD
\[\frac{\sigma_{\text{min}}}{\sigma_{\text{max}}} < \sqrt{\text{min_reciprocal_condition_number}}\]where \(\sigma_{\text{min}}\) and \(\sigma_{\text{max}}\) are the minimum and maxiumum singular values of \(J\) respectively.
EIGEN_SPARSE_QR
andSUITE_SPARSE_QR
\[\operatorname{rank}(J) < \operatorname{num\_col}(J)\]Here :math:operatorname{rank}(J) is the estimate of the rank of J returned by the sparse QR factorization algorithm. It is a fairly reliable indication of rank deficiency.

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 \(J'J\), the MoorePenrose pseudoinverse of \(J'J\) should be computed.
If \(J'J\) has the eigen decomposition \((\lambda_i, e_i)\), where \(lambda_i\) is the \(i^\textrm{th}\) eigenvalue and \(e_i\) is the corresponding eigenvector, then the inverse of \(J'J\) is
\[(J'J)^{1} = \sum_i \frac{1}{\lambda_i} e_i e_i'\]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 nonnegative, then the smallest null_space_rank eigenvalue/eigenvectors are dropped irrespective of the magnitude of \(\lambda_i\). If the ratio of the smallest nonzero 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
\[\frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}\]This option has no effect on
EIGEN_SPARSE_QR
andSUITE_SPARSE_QR
.

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
¶ 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 forCovariance::Options
before usingCovariance
.

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 blockwise 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>
, thenGetCovarianceBlock
can be called withblock1
,block2
as well asblock2
,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
Covariance::Options
for more on the conditions under which this function returnsfalse
.

bool
GetCovarianceBlock
(const double *parameter_block1, const double *parameter_block2, double *covariance_block) const¶ Return the block of the crosscovariance matrix corresponding to
parameter_block1
andparameter_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 whenCompute
was called. OtherwiseGetCovarianceBlock
will return false.covariance_block
must point to a memory location that can store aparameter_block1_size x parameter_block2_size
matrix. The returned covariance will be a rowmajor matrix.

bool
GetCovarianceBlockInTangentSpace
(const double *parameter_block1, const double *parameter_block2, double *covariance_block) const¶ Return the block of the crosscovariance matrix corresponding to
parameter_block1
andparameter_block2
. Returns crosscovariance in the tangent space if a local parameterization is associated with either parameter block; else returns crosscovariance 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 whenCompute
was called. OtherwiseGetCovarianceBlock
will return false.covariance_block
must point to a memory location that can store aparameter_block1_local_size x parameter_block2_local_size
matrix. The returned covariance will be a rowmajor matrix.
Example Usage¶
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)