# Covariance Estimation

## Introduction

One way to assess the quality of the solution returned by a non-linear least squares solver 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:

\[x^* = \arg \min_x \|f(x) - y\|^2\]

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

o.. math:: 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 up to 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 `Manifold`. 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

int Covariance::Options::num_threads

  Default: `1`

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

SparseLinearAlgebraLibraryType Covariance::Options::sparse_linear_algebra_library_type

  Default: `SUITE_SPARSE` Ceres Solver is built with support for SuiteSparse and `EIGEN_SPARSE` otherwise. Note that `EIGEN_SPARSE` is always available.

CovarianceAlgorithmType Covariance::Options::algorithm_type

  Default: `SPARSE_QR`

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

  1. `SPARSE_QR` uses the sparse QR factorization algorithm to compute the decomposition

     > \[\begin{split}QR &= J\\ \left(J^\top J\right)^{-1} &= \left(R^\top R\right)^{-1}\end{split}\]

     The speed of this algorithm depends on the sparse linear algebra library being used. `Eigen`’s sparse QR factorization is a moderately fast algorithm suitable for small to medium sized matrices. For best performance we recommend using `SuiteSparseQR` which is enabled by setting `Covariance::Options::sparse_linear_algebra_library_type` to `SUITE_SPARSE`.

     `SPARSE_QR` cannot compute the covariance if the Jacobian is rank deficient.

  2. `DENSE_SVD` uses `Eigen`’s `JacobiSVD` to perform the computations. It computes the singular value decomposition

     \[U D V^\top = J\]

     and then uses it to compute the pseudo inverse of J’J as

     \[(J'J)^{\dagger} = V D^{2\dagger} V^\top\]

     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.

double Covariance::Options::column_pivot_threshold

  Default: \(-1\)

  > During QR factorization, if a column with Euclidean norm less than `column_pivot_threshold` is encountered it is treated as zero.

  > If `column_pivot_threshold < 0`, then an automatic default value of 20*(m+n)*eps*sqrt(max(diag(J’*J))) is used. Here m and n are the number of rows and columns of the Jacobian (J) respectively.

  > This is an advanced option meant for users who know enough about their Jacobian matrices that they can determine a value better than the default.

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`

     \[\frac{\sigma_{\text{min}}}{\sigma_{\text{max}}} < \sqrt{\text{min_reciprocal_condition_number}}\]

     where \(\sigma_{\text{min}}\) and \(\sigma_{\text{max}}\) are the minimum and maximum singular values of \(J\) respectively.

  2. `SPARSE_QR`

     > \[\operatorname{rank}(J) < \operatorname{num\_col}(J)\]

     > Here \(\operatorname{rank}(J)\) 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 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

  \[\frac{\lambda_i}{\lambda_{\textrm{max}}} < \textrm{min_reciprocal_condition_number}\]

  This option has no effect on `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 for `Covariance::Options` before using `Covariance`.

bool Covariance::Compute(const std::vector<std::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 cross-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.

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

  Return the block of the cross-covariance matrix corresponding to `parameter_block1` and `parameter_block2`. Returns cross-covariance in the tangent space if a local parameterization is associated with either parameter block; else returns cross-covariance 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 when `Compute` was called. Otherwise `GetCovarianceBlock` will return false.

  `covariance_block` must point to a memory location that can store a `parameter_block1_local_size x parameter_block2_local_size` matrix. The returned covariance will be a row-major 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);

    std::vector<std::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)