Numerical Verification

check_geodesic(
    M::AbstractManifold,
    p=rand(M),
    X=rand(M; vector_at=p);
    #
    tol::Real = 1e-12,
    io::Union{IO,Nothing} = nothing,
    N::Int = 101,
    name::String = "geodesic",
    plot::Bool = false,
    error::Symbol = :none,
    inverse_retraction_method::AbstractInverseRetractionMethod = LogarithmicInverseRetraction(),
    vector_transport_method::AbstractVectorTransportMethod = ParallelTransport(),
)

Numerically check whether the geodesic implementation is correct. This check requires both the log and parallel_transport_to functions to be implemented in order to an exact numerical verification. We further require the norm function to be implemented for the AbstractManifold M.

You can provide an inverse_retraction_method and vector_transport_method to be used, but the verification will then be less accurate.

The tests performed are the following based on sampling the geodesic $\gamma(t) =$geodesic(M, p, X) at N equidistant points $t_i = i/(N-1)$, i=0,...,N-1 we denote by $p_i = \gamma(t_i)$

1. Compute the norms of the tangent vectors $\dot\gamma(t_i) ≈ X_i = \log_{p_i}(p_{i+1})$, $i=0,...,N-2$ and check how far these norms deviate from their mean value.
2. Check how far this mean deviates from being $\frac{||X||}{N-1}$
3. Check that the parallel transport of each $X_i$ from $p_i$ to $p_{i+1}$ is close to $X_{i+1}$.

Arguments

• M: the manifold to check
• p: point on the manifold to start the geodesic, a random point is used by default
• X: tangent vector at p to start the geodesic, a random tangent vector is used by default

Keyword arguments

• tol: if all errors are below this tolerance, the geodesic is considered to be exact
• io: provide an IO to print the result to
• N: number of points to sample the geodesic on $[0,1]$ (default: 101)
• plot: whether to plot the result (if Plots.jl is loaded).
• error: specify how to report errors: :none, :info, :warn, or :error are available
• inverse_retraction_method: method to use for the inverse retraction, it is recommended to use LogarithmicInverseRetraction
• vector_transport_method: method to use for the vector transport, it is recommended to use ParallelTransport

Note that since the plot yields more information than throwing an error, when both are specified, the plot is generated first and returned (to be shown/displayed), such that no error is thrown. You can switch to e.g. :warn to get a warning together with the plot.

check_inverse_retraction(
    M::AbstractManifold,
    inverse_rectraction_method::AbstractInverseRetractionMethod,
    p=rand(M),
    X=rand(M; vector_at=p);
    #
    exactness_tol::Real = 1e-12,
    io::Union{IO,Nothing} = nothing,
    limits::Tuple = (-8.0, 0.0),
    log_range::AbstractVector = range(limits[1], limits[2]; length=N),
    N::Int = 101,
    name::String = "inverse retraction",
    plot::Bool = false,
    second_order::Bool = true
    slope_tol::Real = 0.1,
    error::Symbol = :none,
    window = nothing,
)

Check numerically wether the inverse retraction inverse_retraction_method is correct. This requires the exp and norm functions to be implemented for the AbstractManifold M.

This implements a method similar to [Bou23, Section 4.8 or Section 6.8].

Note that if the errors are below the given tolerance and the method is exact, no plot is generated,

Keyword arguments

• exactness_tol: if all errors are below this tolerance, the inverse retraction is considered to be exact
• io: provide an IO to print the result to
• limits: specify the limits in the log_range, that is the exponent for the range
• log_range: specify the range of points (in log scale) to sample the length of the tangent vector X
• N: number of points to verify within the log_range default range $[10^{-8},10^{0}]$
• name: name to display in the plot
• plot: whether to plot the result (see plot_slope) The plot is in log-log-scale. This is returned and can then also be saved.
• second_order: check whether the retraction is of second order. if set to false, first order is checked.
• slope_tol: tolerance for the slope (global) of the approximation
• error: specify how to report errors: :none, :info, :warn, or :error are available
• window: specify window sizes within the log_range that are used for the slope estimation. the default is, to use all window sizes 2:N.

Note that since the plot yields more information than throwing an error, when both are specified, the plot is generated first and returned (to be shown/displayed), such that no error is thrown. You can switch to e.g. :warn to get a warning together with the plot.

check_retraction(
    M::AbstractManifold,
    rectraction_method::AbstractRetractionMethod,
    p=rand(M),
    X=rand(M; vector_at=p);
    #
    exactness_tol::Real = 1e-12,
    io::Union{IO,Nothing} = nothing,
    limits::Tuple = (-8.0, 0.0),
    log_range::AbstractVector = range(limits[1], limits[2]; length=N),
    N::Int = 101,
    name::String = "retraction",
    plot::Bool = false,
    second_order::Bool = true
    slope_tol::Real = 0.1,
    error::Symbol = :none,
    window = nothing,
)

Check numerically whether the retraction vector_transport_to is correct, by selecting a set of points $q_i = \exp_p (t_i X)$ where $t$ takes all values from log_range, to then compare parallel_transport_to to the vector_transport_method applied to the vector Y.

This requires the exp, parallel_transport_to and norm function to be implemented for the AbstractManifold M.

This implements a method similar to [Bou23, Section 4.8 or Section 6.8].

Note that if the errors are below the given tolerance and the method is exact, no plot is generated,

Keyword arguments

• exactness_tol: if all errors are below this tolerance, the retraction is considered to be exact
• io: provide an IO to print the result to
• limits: specify the limits in the log_range, that is the exponent for the range
• log_range: specify the range of points (in log scale) to sample the length of the tangent vector X
• N: number of points to verify within the log_range default range $[10^{-8},10^{0}]$
• name: name to display in the plot
• plot: whether to plot the result (if Plots.jl is loaded). The plot is in log-log-scale. This is returned and can then also be saved.
• second_order: check whether the retraction is of second order. if set to false, first order is checked.
• slope_tol: tolerance for the slope (global) of the approximation
• error: specify how to report errors: :none, :info, :warn, or :error are available
• window: specify window sizes within the log_range that are used for the slope estimation. the default is, to use all window sizes 2:N.

Note that since the plot yields more information than throwing an error, when both are specified, the plot is generated first and returned (to be shown/displayed), such that no error is thrown. You can switch to e.g. :warn to get a warning together with the plot.

check_vector_transport(
    M::AbstractManifold,
    vector_transport_method::AbstractVectorTransportMethod,
    p=rand(M),
    X=rand(M; vector_at=p),
    Y=rand(M; vector_at=p);
    #
    exactness_tol::Real = 1e-12,
    io::Union{IO,Nothing} = nothing,
    limits::Tuple = (-8.0, 0.0),
    log_range::AbstractVector = range(limits[1], limits[2]; length=N),
    N::Int = 101,
    name::String = "inverse retraction",
    plot::Bool = false,
    second_order::Bool = true
    slope_tol::Real = 0.1,
    error::Symbol = :none,
    window = nothing,
)

Check numerically wether the retraction vector_transport_to is correct, by selecting a set of points $q_i = \exp_p (t_i X)$ where $t$ takes all values from log_range, to then compare parallel_transport_to to the vector_transport_method applied to the vector Y.

This requires the exp, parallel_transport_to and norm function to be implemented for the AbstractManifold M.

This implements a method similar to [Bou23, Section 4.8 or Section 6.8].

Note that if the errors are below the given tolerance and the method is exact, no plot is generated,

Keyword arguments

• exactness_tol: if all errors are below this tolerance, the differential is considered to be exact
• io: provide an IO to print the result to
• limits: specify the limits in the log_range, that is the exponent for the range
• log_range: specify the range of points (in log scale) to sample the differential line
• N: number of points to verify within the log_range default range $[10^{-8},10^{0}]$
• name: name to display in the plot
• plot: whether to plot the result (if Plots.jl is loaded). The plot is in log-log-scale. This is returned and can then also be saved.
• second_order: check whether the retraction is of second order. if set to false, first order is checked.
• slope_tol: tolerance for the slope (global) of the approximation
• error: specify how to report errors: :none, :info, :warn, or :error are available
• window: specify window sizes within the log_range that are used for the slope estimation. the default is, to use all window sizes 2:N.

Note that since the plot yields more information than throwing an error, when both are specified, the plot is generated first and returned (to be shown/displayed), such that no error is thrown. You can switch to e.g. :warn to get a warning together with the plot.

(a, b, i, j) = find_best_slope_window(X, Y, window=nothing; slope::Real=2.0, slope_tol::Real=0.1)

Check data X,Y for the largest contiguous interval (window) with a regression line fitting “best”. Among all intervals with a slope within slope_tol to slope the longest one is taken. If no such interval exists, the one with the slope closest to slope is taken.

If the window is set to nothing (default), all window sizes 2,...,length(X) are checked. You can also specify a window size or an array of window sizes.

For each window size, all its translates in the data is checked. For all these (shifted) windows the regression line is computed (with a,b in a + t*b) and the best line is computed.

From the best line the following data is returned

• a, b specifying the regression line a + t*b
• i, j determining the window, i.e the regression line stems from data X[i], ..., X[j]

plot_slope(x, y;
    slope=2,
    line_base=0,
    a=0,
    b=2.0,
    i=1,
    j=length(x)
)

Plot the result from the verification functions on data x,y with two comparison lines

1. line_base + tslope as the global slope(s) the plot could have
2. a + b*t on the interval [x[i], x[j]] for some (best fitting) comparison slope