Checks
If you have computed a gradient or differential and you are not sure whether it is correct.
Manopt.check_Hessian
— Functioncheck_Hessian(M, f, grad_f, Hess_f, p=rand(M), X=rand(M; vector_at=p), Y=rand(M, vector_at=p); kwargs...)
Check numerivcally whether the Hessian {operatorname{Hess} f(M,p, X)
of f(M,p)
is correct.
For this we require either a secondorder retraction or a critical point $p$ of f
.
given that we know that is whether
\[f(\operatorname{retr}_p(tX)) = f(p) + t⟨\operatorname{grad} f(p), X⟩ + \frac{t^2}{2}⟨\operatorname{Hess}f(p)[X], X⟩ + \mathcal O(t^3)\]
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 Section 6.8, Boumal, Cambridge Press, 2023.
Note that if the errors are below the given tolerance and the method is exact, no plot will be generated.
Keyword arguments
check_grad
– (true
) check whether $\operatorname{grad} f(p) \in T_p\mathcal M$.check_linearity
– (true
) check whether the Hessian is linear, seeis_Hessian_linear
usinga
,b
,X
, andY
check_symmetry
– (true
) check whether the Hessian is symmetric, seeis_Hessian_symmetric
check_vector
– (false
) check whether $\operatorname{Hess} f(p)[X] \in T_p\mathcal M$ usingis_vector
.mode
 (:Default
) specify the mode, by default we assume to have a second order retraction given byretraction_method=
you can also this method if you already have a cirtical pointp
. Set to:CritalPoint
to usegradient_descent
to find a critical point. Note: This requires (and evaluates) new tangent vectorsX
andY
atol
,rtol
– (same defaults asisapprox
) tolerances that are passed down to all checksa
,b
– two real values to check linearity of the Hessian (ifcheck_linearity=true
)N
 (101
) number of points to check within thelog_range
default range $[10^{8},10^{0}]$exactness_tol
 (1e12
) if all errors are below this tolerance, the check is considered to be exactio
– (nothing
) provide anIO
to print the check result togradient
 (grad_f(M, p)
) instead of the gradient function you can also provide the gradient atp
directlyHessian
 (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
 ((1e8,1)
) specify the limits in thelog_range
log_range
 (range(limits[1], limits[2]; length=N)
) specify the range of points (in log scale) to sample the Hessian lineN
 (101
) number of points to check within thelog_range
default range $[10^{8},10^{0}]$plot
 (false
) whether to plot the resulting check (ifPlots.jl
is loaded). The plot is in loglogscale. This is returned and can then also be saved.retraction_method
 (default_retraction_method(M, typeof(p))
) retraction method to use for the checkslope_tol
– (0.1
) tolerance for the slope (global) of the approximationthrow_error
 (false
) throw an error message if the Hessian is wrongwindow
– (nothing
) specify window sizes within thelog_range
that are used for the slope estimation. the default is, to use all window sizes2:N
.
The kwargs...
are also passed down to the check_vector
call, such that tolerances can easily be set.
Manopt.check_differential
— Functioncheck_differential(M, F, dF, p=rand(M), X=rand(M; vector_at=p); kwargs...)
Check numerivcally whether the differential dF(M,p,X)
of F(M,p)
is correct.
This implements the method described in Section 4.8, Boumal, Cambridge Press, 2023.
Note that if the errors are below the given tolerance and the method is exact, no plot will be generated,
Keyword arguments
exactness_tol
 (1e12
) if all errors are below this tolerance, the check is considered to be exactio
– (nothing
) provide anIO
to print the check result tolimits
((1e8,1)
) specify the limits in thelog_range
log_range
(range(limits[1], limits[2]; length=N)
)  specify the range of points (in log scale) to sample the differential lineN
(101
) – number of points to check within thelog_range
default range $[10^{8},10^{0}]$name
("differential"
) – name to display in the check (e.g. if checking differential)plot
 (false
) whether to plot the resulting check (ifPlots.jl
is loaded). The plot is in loglogscale. This is returned and can then also be saved.retraction_method
 (default_retraction_method(M, typeof(p))
) retraction method to use for the checkslope_tol
– (0.1
) tolerance for the slope (global) of the approximationthrow_error
 (false
) throw an error message if the differential is wrongwindow
– (nothing
) specify window sizes within thelog_range
that are used for the slope estimation. the default is, to use all window sizes2:N
.
Manopt.check_gradient
— Functioncheck_gradient(M, F, gradF, p=rand(M), X=rand(M; vector_at=p); kwargs...)
Check numerivcally 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 Section 4.8, Boumal, Cambridge Press, 2023.
Note that if the errors are below the given tolerance and the method is exact, no plot will be generated.
Keyword arguments
check_vector
– (true
) check whether $\operatorname{grad} f(p) \in T_p\mathcal M$ usingis_vector
.exactness_tol
 (1e12
) if all errors are below this tolerance, the check is considered to be exactio
– (nothing
) provide anIO
to print the check result togradient
 (grad_f(M, p)
) instead of the gradient function you can also provide the gradient atp
directlylimits
 ((1e8,1)
) specify the limits in thelog_range
log_range
 (range(limits[1], limits[2]; length=N)
)  specify the range of points (in log scale) to sample the gradient lineN
 (101
) – number of points to check within thelog_range
default range $[10^{8},10^{0}]$plot
 (false
) whether to plot the resulting check (ifPlots.jl
is loaded). The plot is in loglogscale. This is returned and can then also be saved.retraction_method
 (default_retraction_method(M, typeof(p))
) retraction method to use for the checkslope_tol
– (0.1
) tolerance for the slope (global) of the approximationatol
,rtol
– (same defaults asisapprox
) tolerances that are passed down tois_vector
ifcheck_vector
is set totrue
throw_error
 (false
) throw an error message if the gradient is wrongwindow
– (nothing
) specify window sizes within thelog_range
that are used for the slope estimation. the default is, to use all window sizes2:N
.
The kwargs...
are also passed down to the check_vector
call, such that tolerances can easily be set.
Manopt.find_best_slope_window
— Function(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 are checked. For all these (shifted) windows the regression line is computed (i.e. 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 linea + t*b
i
,j
determining the window, i.e the regression line stems from dataX[i], ..., X[j]
Manopt.is_Hessian_linear
— Functionis_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...
)
Check whether the Hessian function Hess_f
fulfills linearity, i.e. that
\[\operatorname{Hess} f(p)[aX + bY] = b\operatorname{Hess} f(p)[X] + b\operatorname{Hess} f(p)[Y]\]
which is checked using isapprox
and the kwargs...
are passed to this function.
Optional Arguments
throw_error
 (false
) throw an error message if the Hessian is wrong
Manopt.is_Hessian_symmetric
— Functionis_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
)
Check whether the Hessian function Hess_f
fulfills symmetry, i.e. 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 usualisapprox
throw_error
 (false
) throw an error message if the Hessian is wrong
Manopt.plot_slope
— Methodplot_slope(x, y; slope=2, line_base=0, a=0, b=2.0, i=1,j=length(x))
Plot the result from the error check functions, e.g. check_gradient
, check_differential
, check_Hessian
on data x,y
with two comparison lines
line_base
+ tslope
as the global slope the plot should havea
+b*t
on the interval [x[i]
,x[j]
] for some (best fitting) comparison slope
Manopt.prepare_check_result
— Methodprepare_check_result(log_range, errors, slope)
Given a range of values log_range
, where we computed errors
, check whether this yields a slope of slope
in logscale
Note that if the errors are below the given tolerance and the method is exact, no plot will be generated,
Keyword arguments
exactness_tol
 (1e3*eps(eltype(errors))
) is all errors are below this tolerance, the check is considered to be exactio
– (nothing
) provide anIO
to print the check result toname
("differntial"
) – name to display in the check (e.g. if checking gradient)plot
 (false
) whether to plot the resulting check (ifPlots.jl
is loaded). The plot is in loglogscale. This is returned and can then also be saved.slope_tol
– (0.1
) tolerance for the slope (global) of the approximationthrow_error
 (false
) throw an error message if the gradient or Hessian is wrong
Literature
 [Bou23]

N. Boumal. An Introduction to Optimization on Smooth Manifolds. Cambridge University Press (2023).