Verifying gradients and Hessians

check_Hessian(M, f, grad_f, Hess_f, p=rand(M), X=rand(M; vector_at=p), Y=rand(M, vector_at=p); kwargs...)

Verify numerically whether the Hessian $\operatorname{Hess} f(M,p, X)$ of f(M,p) is correct.

For this either a second-order retraction or a critical point $p$ of f is required. The approximation is then

\[f(\operatorname{retr}_p(tX)) = f(p) + t⟨\operatorname{grad} f(p), X⟩ + \frac{t^2}{2}⟨\operatorname{Hess}f(p)[X], X⟩ + \mathcal O(t^3)\]

or in other words, that the error between the function $f$ and its second order Taylor behaves in error $\mathcal O(t^3)$, which indicates that the Hessian is correct, cf. also [Bou23, Section 6.8].

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

Keyword arguments

• check_grad: (true) verify that $\operatorname{grad} f(p) ∈ T_p\mathcal M$.

• check_linearity: (true) verify that the Hessian is linear, see is_Hessian_linear using a, b, X, and Y

• check_symmetry: (true) verify that the Hessian is symmetric, see is_Hessian_symmetric

• check_vector: (false) verify that $\operatorname{Hess} f(p)[X] ∈ T_p\mathcal M$ using is_vector.

• mode: (:Default) specify the mode for the verification; the default assumption is, that the retraction provided is of second order. Otherwise one can also verify the Hessian if the point p is a critical point. THen set the mode to :CritalPoint to use gradient_descent to find a critical point. Note: this requires (and evaluates) new tangent vectors X and Y

• atol, rtol: (same defaults as isapprox) tolerances that are passed down to all checks

• a, b two real values to verify linearity of the Hessian (if check_linearity=true)

• N: (101) number of points to verify within the log_range default range $[10^{-8},10^{0}]$

• exactness_tol: (1e-12) if all errors are below this tolerance, the verification is considered to be exact

• io: (nothing) provide an IO to print the result to

• gradient: (grad_f(M, p)) instead of the gradient function you can also provide the gradient at p directly

• Hessian: (Hess_f(M, p, X)) instead of the Hessian function you can provide the result of $\operatorname{Hess} f(p)[X]$ directly. Note that evaluations of the Hessian might still be necessary for checking linearity and symmetry and/or when using :CriticalPoint mode.

• limits: ((1e-8,1)) specify the limits in the log_range

• log_range: (range(limits[1], limits[2]; length=N)) specify the range of points (in log scale) to sample the Hessian line

• N: (101) number of points to use within the log_range default range $[10^{-8},10^{0}]$

• plot: (false) whether to plot the resulting verification (requires Plots.jl to be loaded). The plot is in log-log-scale. This is returned and can then also be saved.

• retraction_method: (default_retraction_method(M, typeof(p))) retraction method to use for

• slope_tol: (0.1) tolerance for the slope (global) of the approximation

• throw_error: (false) throw an error message if the Hessian is wrong

• window: (nothing) specify window sizes within the log_range that are used for the slope estimation. the default is, to use all window sizes 2:N.

The kwargs... are also passed down to the check_vector and the check_gradient call, such that tolerances can easily be set.

While check_vector is also passed to the inner call to check_gradient as well as the retraction_method, this inner check_gradient is meant to be just for inner verification, so it does not throw an error nor produce a plot itself.

check_differential(M, F, dF, p=rand(M), X=rand(M; vector_at=p); kwargs...)

Check numerically whether the differential dF(M,p,X) of F(M,p) is correct.

This implements the method described in [Bou23, Section 4.8].

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

Keyword arguments

• exactness_tol: (1e-12) if all errors are below this tolerance, the differential is considered to be exact
• io: (nothing) provide an IO to print the result to
• limits: ((1e-8,1)) specify the limits in the log_range
• log_range: (range(limits[1], limits[2]; length=N)) specify the range of points (in log scale) to sample the differential line
• N: (101) number of points to verify within the log_range default range $[10^{-8},10^{0}]$
• name: ("differential") name to display in the plot
• plot: (false) 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.
• retraction_method: (default_retraction_method(M, typeof(p))) retraction method to use
• slope_tol: (0.1) tolerance for the slope (global) of the approximation
• throw_error: (false) throw an error message if the differential is wrong
• window: (nothing) specify window sizes within the log_range that are used for the slope estimation. the default is, to use all window sizes 2:N.

check_gradient(M, F, gradF, p=rand(M), X=rand(M; vector_at=p); kwargs...)

Verify numerically whether the gradient gradF(M,p) of F(M,p) is correct, that is whether

\[f(\operatorname{retr}_p(tX)) = f(p) + t⟨\operatorname{grad} f(p), X⟩ + \mathcal O(t^2)\]

or in other words, that the error between the function $f$ and its first order Taylor behaves in error $\mathcal O(t^2)$, which indicates that the gradient is correct, cf. also [Bou23, Section 4.8].

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

Keyword arguments

• check_vector: (true) verify that $\operatorname{grad} f(p) ∈ T_p\mathcal M$ using is_vector.
• exactness_tol: (1e-12) if all errors are below this tolerance, the gradient is considered to be exact
• io: (nothing) provide an IO to print the result to
• gradient: (grad_f(M, p)) instead of the gradient function you can also provide the gradient at p directly
• limits: ((1e-8,1)) specify the limits in the log_range
• log_range: (range(limits[1], limits[2]; length=N)) - specify the range of points (in log scale) to sample the gradient line
• N: (101) number of points to verify within the log_range default range $[10^{-8},10^{0}]$
• plot: (false) 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.
• retraction_method: (default_retraction_method(M, typeof(p))) retraction method to use
• slope_tol: (0.1) tolerance for the slope (global) of the approximation
• atol, rtol: (same defaults as isapprox) tolerances that are passed down to is_vector if check_vector is set to true
• throw_error: (false) throw an error message if the gradient is wrong
• window: (nothing) specify window sizes within the log_range that are used for the slope estimation. the default is, to use all window sizes 2:N.

The remaining keyword arguments are also passed down to the check_vector call, such that tolerances can easily be set.

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

is_Hessian_linear(M, Hess_f, p,
    X=rand(M; vector_at=p), Y=rand(M; vector_at=p), a=randn(), b=randn();
    throw_error=false, io=nothing, kwargs...
)

Verify whether the Hessian function Hess_f fulfills linearity,

\[\operatorname{Hess} f(p)[aX + bY] = b\operatorname{Hess} f(p)[X] + b\operatorname{Hess} f(p)[Y]\]

which is checked using isapprox and the keyword arguments are passed to this function.

Optional arguments

• throw_error: (false) throw an error message if the Hessian is wrong

is_Hessian_symmetric(M, Hess_f, p=rand(M), X=rand(M; vector_at=p), Y=rand(M; vector_at=p);
throw_error=false, io=nothing, atol::Real=0, rtol::Real=atol>0 ? 0 : √eps

)

Verify whether the Hessian function Hess_f fulfills symmetry, which means that

\[⟨\operatorname{Hess} f(p)[X], Y⟩ = ⟨X, \operatorname{Hess} f(p)[Y]⟩\]

which is checked using isapprox and the kwargs... are passed to this function.

Optional arguments

• atol, rtol with the same defaults as the usual isapprox
• throw_error: (false) throw an error message if the Hessian is wrong

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 check_gradient, check_differential, check_Hessian on data x,y with two comparison lines

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

prepare_check_result(log_range, errors, slope)

Given a range of values log_range, with computed errors, verify whether this yields a slope of slope in log-scale

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

Keyword arguments

• exactness_tol: (1e3*eps(eltype(errors))) is all errors are below this tolerance, the verification is considered to be exact
• io: (nothing) provide an IO to print the result to
• name: ("differential") name to display in the plot title
• plot: (false) 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.
• slope_tol: (0.1) tolerance for the slope (global) of the approximation
• throw_error: (false) throw an error message if the gradient or Hessian is wrong