# Solving¶

## 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 following optimization problem [1] .

(1)$\begin{split}\arg \min_x \frac{1}{2}\|F(x)\|^2\ . \\ L \le x \le U\end{split}$

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 non-linear 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 non-linear 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:

(2)$\min_{\Delta x} \frac{1}{2}\|J(x)\Delta x + F(x)\|^2$

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, non-linear optimization algorithms can be divided into two major categories [NocedalWright].

1. 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.
2. 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 Quasi-Newton 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.

1. Given an initial point $$x$$ and a trust region radius $$\mu$$.

2. Solve

$\begin{split}\arg \min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 \\ \text{such that} &\|D(x)\Delta x\|^2 \le \mu\\ &L \le x + \Delta x \le U.\end{split}$
3. $$\rho = \frac{\displaystyle \|F(x + \Delta x)\|^2 - \|F(x)\|^2}{\displaystyle \|J(x)\Delta x + F(x)\|^2 - \|F(x)\|^2}$$

4. if $$\rho > \epsilon$$ then $$x = x + \Delta x$$.

5. if $$\rho > \eta_1$$ then $$\rho = 2 \rho$$

6. else if $$\rho < \eta_2$$ then $$\rho = 0.5 * \rho$$

7. 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 non-linear 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 non-linear objective, which in turn is reflected in the value of $$\rho$$.

The key computational step in a trust-region algorithm is the solution of the constrained optimization problem

(3)$\begin{split}\arg \min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 \\ \text{such that} &\|D(x)\Delta x\|^2 \le \mu\\ &L \le x + \Delta x \le U.\end{split}$

There are a number of different ways of solving this problem, each giving rise to a different concrete trust-region algorithm. Currently Ceres, implements two trust-region algorithms - Levenberg-Marquardt 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 non-linear 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 non-linear least squares problem.

### Levenberg-Marquardt¶

The Levenberg-Marquardt algorithm [Levenberg] [Marquardt] is the most popular algorithm for solving non-linear 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 Levenberg-Marquardt algorithm [WrightHolt] [NashSofer].

It can be shown, that the solution to (3) can be obtained by solving an unconstrained optimization of the form

$\begin{split}\arg\min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 +\lambda \|D(x)\Delta x\|^2\end{split}$

Where, $$\lambda$$ is a Lagrange multiplier that is inverse related to $$\mu$$. In Ceres, we solve for

(4)$\begin{split}\arg\min_{\Delta x}& \frac{1}{2}\|J(x)\Delta x + F(x)\|^2 + \frac{1}{\mu} \|D(x)\Delta x\|^2\end{split}$

The matrix $$D(x)$$ is a non-negative 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.

(5)$\min_{\Delta x} \frac{1}{2} \|J(x)\Delta x + f(x)\|^2 .$

For all but the smallest problems the solution of (5) in each iteration of the Levenberg-Marquardt 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 Levenberg-Marquardt 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 non-linear 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

(6)$\|H(x) \Delta x + g(x)\| \leq \eta_k \|g(x)\|.$

Here, $$k$$ indicates the Levenberg-Marquardt iteration number and $$0 < \eta_k <1$$ is known as the forcing sequence. [WrightHolt] prove that a truncated Levenberg-Marquardt 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 Levenberg-Marquardt algorithm is used. When the user chooses an iterative linear solver, the inexact step Levenberg-Marquardt 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

$\begin{split}\Delta x^{\text{Gauss-Newton}} &= \arg \min_{\Delta x}\frac{1}{2} \|J(x)\Delta x + f(x)\|^2.\\ \Delta x^{\text{Cauchy}} &= -\frac{\|g(x)\|^2}{\|J(x)g(x)\|^2}g(x).\end{split}$

Note that the vector $$\Delta x^{\text{Gauss-Newton}}$$ 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 $$\Delta x^{\text{Gauss-Newton}}$$ 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 Gauss-Newton and Cauchy vectors and finds the point farthest along this line shaped like a dogleg (hence the name) that is contained in the trust-region. 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 Levenberg Marquardt 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, Levenberg-Marquardt 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 Gauss-Newton 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 non-linear 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. e.g., consider the following regression problem

$y = a_1 e^{b_1 x} + a_2 e^{b_3 x^2 + c_1}$

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 non-linear 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.,

$y = f_1(a_1, e^{b_1 x}) + f_2(a_2, e^{b_3 x^2 + c_1})$

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 non-linear 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 co-occur 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 non-overlapping independent sets and optimizing each of them.

Setting Solver::Options::use_inner_iterations to true enables the use of this non-linear 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.

### Non-monotonic Steps¶

Note that the basic trust-region 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 non-decreasing 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 non-monotonic 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 non-monotonic 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

1. Given an initial point $$x$$
2. $$\Delta x = -H^{-1}(x) g(x)$$
3. $$\arg \min_\mu \frac{1}{2} \| F(x + \mu \Delta x) \|^2$$
4. $$x = x + \mu \Delta x$$
5. 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.

1. 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.
2. NONLINEAR_CONJUGATE_GRADIENT A generalization of the Conjugate Gradient method to non-linear functions. The generalization can be performed in a number of different ways, resulting in a variety of search directions. Ceres Solver currently supports FLETCHER_REEVES, POLAK_RIBIRERE and HESTENES_STIEFEL directions.
3. 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 quasi-Newton step [NocedalWright]. BFGS is currently the best known general quasi-Newton algorithm.
4. LBFGS A limited memory approximation to the full BFGS method in which the last M iterations are used to approximate the inverse Hessian used to compute a quasi-Newton 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 trust-region methods described above, the key computational cost is the solution of a linear least squares problem of the form

(7)$\min_{\Delta x} \frac{1}{2} \|J(x)\Delta x + f(x)\|^2 .$

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.

(8)$H \Delta x = g$

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 QR-decomposition 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

$\Delta x^* = -R^{-1}Q^\top f$

Ceres uses Eigen ‘s dense QR factorization routines.

### DENSE_NORMAL_CHOLESKY & SPARSE_NORMAL_CHOLESKY¶

Large non-linear 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

$\Delta x^* = R^{-1} R^{-\top} g.$

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].

### 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

(9)$\begin{split}H = \left[ \begin{matrix} B & E\\ E^\top & C \end{matrix} \right]\ ,\end{split}$

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

(10)$\begin{split}\left[ \begin{matrix} B & E\\ E^\top & C \end{matrix} \right]\left[ \begin{matrix} \Delta y \\ \Delta z \end{matrix} \right] = \left[ \begin{matrix} v\\ w \end{matrix} \right]\ ,\end{split}$

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

(11)$\left[B - EC^{-1}E^\top\right] \Delta y = v - EC^{-1}w\ .$

The matrix

$S = B - EC^{-1}E^\top$

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 non-zero if and only if the two images observe at least one common point.

Now, eq-linear2 can be solved by first forming $$S$$, solving for $$\Delta y$$, and then back-substituting $$\Delta y$$ 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 matrix-matrix and matrix-vector 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 re-ordering algorithms to maximize the sparsity of the Cholesky decomposition, and focus their compute effort on the non-zero 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.

### 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

$H x = J^\top J x = J^\top(J x)$

When the user chooses ITERATIVE_SCHUR as the linear solver, Ceres automatically switches from the exact step algorithm to an inexact step algorithm.

### ITERATIVE_SCHUR¶

Another option for bundle adjustment problems is to apply PCG 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 $$\kappa(S)\leq \kappa(H)$$. Cseres implements PCG 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 cost of forming and storing the Schur complement $$S$$ can be prohibitive for large problems. Indeed, for an inexact Newton solver that computes $$S$$ and runs PCG on it, almost all of its time is spent in constructing $$S$$; the time spent inside the PCG algorithm is negligible in comparison. Because PCG only needs access to $$S$$ via its product with a vector, one way to evaluate $$Sx$$ is to observe that

$\begin{split}x_1 &= E^\top x\end{split}$
$\begin{split}x_2 &= C^{-1} x_1\end{split}$
$\begin{split}x_3 &= Ex_2\\\end{split}$
$\begin{split}x_4 &= Bx\\\end{split}$
(12)$\begin{split}Sx &= x_4 - x_3\end{split}$

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 sub-category of techniques known as Iterative Sub-structuring [Saad] [Mathew].

### 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.

For ITERATIVE_SCHUR there are two obvious choices for block diagonal preconditioners for $$S$$. The block diagonal of the matrix $$B$$ [Mandel] and the block diagonal $$S$$, i.e, the block Jacobi preconditioner for $$S$$. Ceres’s implements both of these preconditioners and refers to them as JACOBI and SCHUR_JACOBI respectively.

For bundle adjustment problems arising in reconstruction from community photo collections, more effective preconditioners can be constructed by analyzing and exploiting the camera-point 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. e.g. Consider the linear system

$\begin{split}x + y &= 3\\ 2x + 3y &= 7\end{split}$

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.

1. $$\{0: x\}, \{1: y\}$$ : Eliminate $$x$$ first.
2. $$\{0: y\}, \{1: x\}$$ : Eliminate $$y$$ first.
3. $$\{0: x, y\}$$ : 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 co-occur 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.

MinimizerType Solver::Options::minimizer_type

Default: TRUST_REGION

Choose between LINE_SEARCH and TRUST_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 and LBFGS.

LineSearchType Solver::Options::line_search_type

Default: WOLFE

Choices are ARMIJO and WOLFE (strong Wolfe conditions). Note that in order for the assumptions underlying the BFGS and LBFGS line search direction algorithms to be guaranteed to be satisifed, the WOLFE line search should be used.

Default: FLETCHER_REEVES

Choices are FLETCHER_REEVES, POLAK_RIBIRERE and HESTENES_STIEFEL.

int Solver::Options::max_lbfs_rank

Default: 20

The L-BFGS 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.

1. The method only uses secant information and not actual derivatives.
2. 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 right-multiply 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. Setting use_approximate_eigenvalue_bfgs_scaling to true enables this scaling in BFGS (before first iteration) and LBFGS (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 low-sensitivity 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 and CUBIC.

double Solver::Options::min_line_search_step_size

The line search terminates if:

$\begin{split}\|\Delta x_k\|_\infty < \text{min_line_search_step_size}\end{split}$

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: 1e-4

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: 1e-3

In each iteration of the line search,

$\begin{split}\text{new_step_size} >= \text{max_line_search_step_contraction} * \text{step_size}\end{split}$

Note that by definition, for contraction:

$\begin{split}0 < \text{max_step_contraction} < \text{min_step_contraction} < 1\end{split}$
double Solver::Options::min_line_search_step_contraction

Default: 0.6

In each iteration of the line search,

$\begin{split}\text{new_step_size} <= \text{min_line_search_step_contraction} * \text{step_size}\end{split}$

Note that by definition, for contraction:

$\begin{split}0 < \text{max_step_contraction} < \text{min_step_contraction} < 1\end{split}$
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 sub-optimal 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.

$\begin{split}\|f'(\text{step_size})\| <= \text{sufficient_curvature_decrease} * \|f'(0)\|\end{split}$

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 containinqg a point satisfying the conditions is found. Precisely, at each iteration of the expansion:

$\begin{split}\text{new_step_size} <= \text{max_step_expansion} * \text{step_size}\end{split}$

By definition for expansion

$\begin{split}\text{max_step_expansion} > 1.0\end{split}$
TrustRegionStrategyType Solver::Options::trust_region_strategy_type

Default: LEVENBERG_MARQUARDT

The trust region step computation algorithm used by Ceres. Currently LEVENBERG_MARQUARDT and DOGLEG are the two valid choices. See Levenberg-Marquardt and Dogleg for more details.

DoglegType Solver::Options::dogleg_type

Ceres supports two different dogleg strategies. TRADITIONAL_DOGLEG method by Powell and the SUBSPACE_DOGLEG method described by [ByrdSchnabel] . See Dogleg for more details.

bool Solver::Options::use_nonmonotonic_steps

Default: false

Relax the requirement that the trust-region algorithm take strictly decreasing steps. See Non-monotonic Steps for more details.

int Solver::Options::max_consecutive_nonmonotonic_steps

Default: 5

The window size used by the step selection algorithm to accept non-monotonic 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.

Default: 1

Number of threads used by Ceres to evaluate the Jacobian.

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.

Default: 1e16

The trust region radius is not allowed to grow beyond this value.

Default: 1e-32

The solver terminates, when the trust region becomes smaller than this value.

double Solver::Options::min_relative_decrease

Default: 1e-3

Lower threshold for relative decrease before a trust-region step is accepted.

double Solver::Options::min_lm_diagonal

Default: 1e6

The LEVENBERG_MARQUARDT strategy, uses a diagonal matrix to regularize the 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 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: 1e-6

Solver terminates if

$\begin{split}\frac{|\Delta \text{cost}|}{\text{cost} < \text{function_tolerance}}\end{split}$

where, $$\Delta \text{cost}$$ is the change in objective function value (up or down) in the current iteration of Levenberg-Marquardt.

Default: 1e-10

Solver terminates if

$\begin{split}\|x - \Pi \boxplus(x, -g(x))\|_\infty < \text{gradient_tolerance}\end{split}$

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: 1e-8

Solver terminates if

$\begin{split}\|\Delta x\| < (\|x\| + \text{parameter_tolerance}) * \text{parameter_tolerance}\end{split}$

where $$\Delta x$$ is the step computed by the linear solver in the current iteration of Levenberg-Marquardt.

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 Levenberg-Marquardt algorithm. If Ceres is build with SuiteSparse linked in then the default is SPARSE_NORMAL_CHOLESKY, it is DENSE_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 and CLUSTER_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 try SINGLE_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 and LAPACK are the valid choices. EIGEN is always available, LAPACK refers to the system BLAS + LAPACK library which may or may not be available.

This setting affects the DENSE_QR, DENSE_NORMAL_CHOLESKY and DENSE_SCHUR solvers. For small to moderate sized probem EIGEN is a fine choice but for large problems, an optimized LAPACK + BLAS implementation can make a substantial difference in performance.

SparseLinearAlgebraLibrary Solver::Options::sparse_linear_algebra_library_type

Default:SUITE_SPARSE

Ceres supports the use of two sparse linear algebra libraries, SuiteSparse, which is enabled by setting this parameter to SUITE_SPARSE and CXSparse, which can be selected by setting this parameter to CX_SPARSE. 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 using CXSparse, which is a much smaller, easier to build library. As can be expected, its performance on large problems is not comparable to that of SuiteSparse.

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_post_ordering

Default: false

Sparse Cholesky factorization algorithms use a fill-reducing ordering to permute the columns of the Jacobian matrix. There are two ways of doing this.

1. Compute the Jacobian matrix in some order and then have the factorization algorithm permute the columns of the Jacobian.
2. 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 pre-permutes 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 to true enables this tradeoff.

bool Solver::Options::dynamic_sparsity

Some non-linear least squares problems are symbolically dense but numerically sparse. i.e. at any given state only a small number of Jacobian entries are non-zero, but the position and number of non-zeros 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 settings affects the SPARSE_NORMAL_CHOLESKY solver.

int Solver::Options::min_linear_solver_iterations

Default: 1

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 or CGNR.

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 or CGNR.

double Solver::Options::eta

Default: 1e-1

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

$\begin{split}\frac{Q_i - Q_{i-1}}{Q_i} < \frac{\eta}{i}\end{split}$
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 non-linear 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_itearation_tolerance

Default: 1e-3

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.

1. Let the solver heuristically decide which parameter blocks to optimize in each inner iteration. To do this, set Solver::Options::inner_iteration_ordering to NULL.
2. 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.

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 to STDERR depending on the vlog level. If this flag is set to true, and Solver::Options::logging_type is not SILENT, the logging output is sent to STDOUT.

For TRUST_REGION_MINIMIZER the progress display looks like

0: f: 1.250000e+01 d: 0.00e+00 g: 5.00e+00 h: 0.00e+00 rho: 0.00e+00 mu: 1.00e+04 li:  0 it: 6.91e-06 tt: 1.91e-03
1: f: 1.249750e-07 d: 1.25e+01 g: 5.00e-04 h: 5.00e+00 rho: 1.00e+00 mu: 3.00e+04 li:  1 it: 2.81e-05 tt: 1.99e-03
2: f: 1.388518e-16 d: 1.25e-07 g: 1.67e-08 h: 5.00e-04 rho: 1.00e+00 mu: 9.00e+04 li:  1 it: 1.00e-05 tt: 2.01e-03


Here

1. f is the value of the objective function.
2. d is the change in the value of the objective function if the step computed in this iteration is accepted.
3. g is the max norm of the gradient.
4. h is the change in the parameter vector.
5. rho is the ratio of the actual change in the objective function value to the change in the the value of the trust region model.
6. mu is the size of the trust region radius.
7. li 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 like ITERATIVE_SCHUR it is the number of iterations of the Conjugate Gradients algorithm.
8. it is the time take by the current iteration.
9. tt is the the total time taken by the minimizer.

For LINE_SEARCH_MINIMIZER the progress display looks like

0: f: 2.317806e+05 d: 0.00e+00 g: 3.19e-01 h: 0.00e+00 s: 0.00e+00 e:  0 it: 2.98e-02 tt: 8.50e-02
1: f: 2.312019e+05 d: 5.79e+02 g: 3.18e-01 h: 2.41e+01 s: 1.00e+00 e:  1 it: 4.54e-02 tt: 1.31e-01
2: f: 2.300462e+05 d: 1.16e+03 g: 3.17e-01 h: 4.90e+01 s: 2.54e-03 e:  1 it: 4.96e-02 tt: 1.81e-01


Here

1. f is the value of the objective function.
2. d is the change in the value of the objective function if the step computed in this iteration is accepted.
3. g is the max norm of the gradient.
4. h is the change in the parameter vector.
5. s is the optimal step length computed by the line search.
6. it is the time take by the current iteration.
7. tt is the 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 non-empty if Solver::Options::trust_region_minimizer_iterations_to_dump is non-empty and Solver::Options::trust_region_problem_dump_format_type is not CONSOLE.
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 non-empty. There are three options:

• CONSOLE prints the linear least squares problem in a human

readable 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 by Solver::Options::trust_region_problem_dump_directory as text files which can be read into MATLAB/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 called ceres_solver_iteration_???.m is also output, which can be used to parse and load the problem into memory.

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, details are printed to the log.

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::numeric_derivative_relative_step_size

Default: 1e-6

Relative shift used for taking numeric derivatives. For finite differencing, each dimension is evaluated at slightly shifted values, e.g., for forward differences, the numerical derivative is

$\begin{split}\delta &= numeric\_derivative\_relative\_step\_size\\ \Delta f &= \frac{f((1 + \delta) x) - f(x)}{\delta x}\end{split}$

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 the Minimizer terminates. This behavior is controlled by Solver::Options::update_state_every_variable. If the user wishes to have access to the update parameter blocks when his/her callbacks are executed, then set Solver::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 non-negative 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 non-negative 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.

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 non-monotonic trust region algorithm.

Note: IterationSummary::step_is_nonmonotonic is false when when IterationSummary::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 non-monotonic 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 when IterationSummary::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.

Infinity norm of the gradient vector.

2-norm of the gradient vector.

double IterationSummary::step_norm

2-norm 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.

Size of the trust region at the end of the current iteration. For the Levenberg-Marquardt algorithm, the regularization parameter is 1.0 / member::IterationSummary::trust_region_radius.

double IterationSummary::eta

For the inexact step Levenberg-Marquardt 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. Factorization-based 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.

1. SOLVER_ABORT indicates that the callback detected an abnormal situation. The solver returns without updating the parameter blocks (unless Solver::Options::update_state_every_iteration is set true). Solver returns with Solver::Summary::termination_type set to USER_FAILURE.
2. SOLVER_TERMINATE_SUCCESSFULLY indicates that there is no need to optimize anymore (some user specified termination criterion has been met). Solver returns with Solver::Summary::termination_type set to USER_SUCCESS.
3. 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.step_norm,
summary.relative_decrease,
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 a CRSMatrix::num_rows + 1 sized array that points into the CRSMatrix::cols and CRSMatrix::values array.

vector<int> CRSMatrix::cols

CRSMatrix::cols contain as many entries as there are non-zeros in the matrix.

For each row i, cols[rows[i]] ... cols[rows[i + 1] - 1] are the indices of the non-zero columns of row i.

vector<int> CRSMatrix::values

CRSMatrix::values contain as many entries as there are non-zeros in the matrix.

For each row i, values[rows[i]] ... values[rows[i + 1] - 1] are the values of the non-zero columns of row i.

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.

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 a LocalParameterization.

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 a LocalParameterization.

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.

Number of threads specified by the user for Jacobian and residual evaluation.

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.

Number of threads specified by the user for solving the trust region problem.

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 left Solver::Summary::linear_solver_ordering_given blank and asked for an automatic ordering, or if the problem contains some constant or inactive parameter blocks.

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. e.g., 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 left Solver::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

Type of preconditioner used for solving the trust region step. Only meaningful when an iterative linear solver is used.

VisibilityClusteringType Solver::Summary::visibility_clustering_type

Type of clustering algorithm used for visibility based preconditioning. Only meaningful when the Solver::Summary::preconditioner_type is CLUSTER_JACOBI or CLUSTER_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.

If the line search direction is NONLINEAR_CONJUGATE_GRADIENT, then this indicates the particular variant of non-linear 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 non-linear least squares solve is to analyze the covariance of the solution.

Let us consider the non-linear regression problem

$y = f(x) + N(0, I)$

i.e., the observation $$y$$ is a random non-linear 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 non-linear least squares problem:

$x^* = \arg \min_x \|f(x)\|^2$

And the covariance of $$x^*$$ is given by

$C(x^*) = \left(J'(x^*)J(x^*)\right)^{-1}$

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 Moore-Penrose pseudo inverse.

$C(x^*) = \left(J'(x^*)J(x^*)\right)^{\dagger}$

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

$y = f(x) + N(0, S)$

Where $$S$$ is a positive semi-definite matrix denoting the covariance of $$y$$, then the maximum likelihood problem to be solved is

$x^* = \arg \min_x f'(x) S^{-1} f(x)$

and the corresponding covariance estimate of $$x^*$$ is given by

$C(x^*) = \left(J'(x^*) S^{-1} J(x^*)\right)^{-1}$

So, if it is the case that the observations being fitted to have a covariance matrix not equal to identity, then it is the user’s responsibility that the corresponding cost functions are correctly scaled, e.g. in the above case the cost function for this problem should evaluate $$S^{-1/2} f(x)$$ 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 non-linear least squares problem and provides random access to its blocks. The computation assumes that the cost functions compute residuals such that their covariance is identity.

Since the computation of the covariance matrix requires computing the inverse of a potentially large matrix, this can involve a rather large amount of time and memory. However, it is usually the case that the user is only interested in a small part of the covariance matrix. Quite often just the block diagonal. 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.

1. 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 the Covariance object so that it can handle it correctly.
2. 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.

Covariance::Options

class Covariance::Options

Default: 1

Number of threads to be used for evaluating the Jacobian and estimation of covariance.

CovarianceAlgorithmType Covariance::Options::algorithm_type

Default: SPARSE_QR or DENSE_SVD

Ceres supports three different algorithms for covariance estimation, which represent different tradeoffs in speed, accuracy and reliability.

1. DENSE_SVD uses Eigen‘s JacobiSVD 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 full-rank as well as rank deficient Jacobians.

2. SPARSE_CHOLESKY uses the CHOLMOD sparse Cholesky factorization library to compute the decomposition :

$R^\top R = J^\top J$

and then

$\left(J^\top J\right)^{-1} = \left(R^\top R\right)^{-1}$

It a fast algorithm for sparse matrices that should be used when the Jacobian matrix J is well conditioned. For ill-conditioned matrices, this algorithm can fail unpredictabily. This is because Cholesky factorization is not a rank-revealing factorization, i.e., it cannot reliably detect when the matrix being factorized is not of full rank. SuiteSparse/CHOLMOD supplies a heuristic for checking if the matrix is rank deficient (cholmod_rcond), but it is only a heuristic and can have both false positive and false negatives.

Recent versions of SuiteSparse (>= 4.2.0) provide a much more efficient method for solving for rows of the covariance matrix. Therefore, if you are doing SPARSE_CHOLESKY, we strongly recommend using a recent version of SuiteSparse.

3. SPARSE_QR uses the SuiteSparseQR sparse QR factorization library 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, which at the price of more time and memory than the SPARSE_CHOLESKY algorithm is numerically better behaved and is rank revealing, i.e., it can reliably detect when the Jacobian matrix is rank deficient.

Neither SPARSE_CHOLESKY or 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 return false if a rank deficient Jacobian is encountered. How rank deficiency is detected depends on the algorithm being used.

1. DENSE_SVD

$\begin{split}\frac{\sigma_{\text{min}}}{\sigma_{\text{max}}} < \sqrt{\text{min_reciprocal_condition_number}}\end{split}$

where $$\sigma_{\text{min}}$$ and $$\sigma_{\text{max}}$$ are the minimum and maxiumum singular values of $$J$$ respectively.

1. SPARSE_CHOLESKY

$\begin{split}\text{cholmod_rcond} < \text{min_reciprocal_conditioner_number}\end{split}$
Here cholmod_rcond is a crude estimate of the reciprocal condition number of $$J^\top J$$ by using the maximum and minimum diagonal entries of the Cholesky factor $$R$$. There are no theoretical guarantees associated with this test. It can give false positives and negatives. Use at your own risk. The default value of min_reciprocal_condition_number has been set to a conservative value, and sometimes the Covariance::Compute() may return false even if it is possible to estimate the covariance reliably. In such cases, the user should exercise their judgement before lowering the value of min_reciprocal_condition_number.
1. SPARSE_QR

$\begin{split}\operatorname{rank}(J) < \operatorname{num\_col}(J)\end{split}$

Here :math:operatorname{rank}(J) is the estimate of the rank of J returned by the SuiteSparseQR 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 Moore-Penrose 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 non-negative, then the smallest null_space_rank eigenvalue/eigenvectors are dropped irrespective of the magnitude of $$\lambda_i$$. If the ratio of the smallest non-zero eigenvalue to the largest eigenvalue in the truncated matrix is still below min_reciprocal_condition_number, then the Covariance::Compute() will fail and return false.

Setting null_space_rank = -1 drops all terms for which

$\begin{split}\frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}\end{split}$
This option has no effect on SPARSE_QR and SPARSE_CHOLESKY
algorithms.
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 for Covariance::Options before using Covariance.

bool Covariance::Compute(const vector<pair<const double*, const double*>>& covariance_blocks, Problem* problem)

Compute a part of the covariance matrix.

The vector covariance_blocks, indexes into the covariance matrix block-wise using pairs of parameter blocks. This allows the covariance estimation algorithm to only compute and store these blocks.

Since the covariance matrix is symmetric, if the user passes <block1, block2>, then GetCovarianceBlock can be called with block1, block2 as well as block2, block1.

covariance_blocks cannot contain duplicates. Bad things will happen if they do.

Note that the list of covariance_blocks is only used to determine what parts of the covariance matrix are computed. The full Jacobian is used to do the computation, i.e. they do not have an impact on what part of the Jacobian is used for computation.

The return value indicates the success or failure of the covariance computation. Please see the documentation for Covariance::Options for more on the conditions under which this function returns false.

bool GetCovarianceBlock(const double* parameter_block1, const double* parameter_block2, double* covariance_block) const

Return the block of the covariance matrix corresponding to parameter_block1 and parameter_block2.

Compute must be called before the first call to GetCovarianceBlock and the pair <parameter_block1, parameter_block2> OR the pair <parameter_block2, parameter_block1> must have been present in the vector covariance_blocks when Compute was called. Otherwise GetCovarianceBlock will return false.

covariance_block must point to a memory location that can store a parameter_block1_size x parameter_block2_size matrix. The returned covariance will be a row-major matrix.

### Example Usage¶

double x[3];
double y[2];

Problem problem;
<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)