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function checkmanifold(M)
% Run a collection of tests on a manifold obtained from a manopt factory
%
% function checkmanifold(M)
%
% M should be a manifold structure obtained from a Manopt factory. This
% tool runs a collection of tests on M to verify (to some extent) that M is
% indeed a valid description of a Riemannian manifold.
%
% This tool is work in progress: your suggestions for additional tests are
% welcome on our forum or as pull requests on GitHub.
%
% See also: checkretraction
% This file is part of Manopt: www.manopt.org.
% Original author: Nicolas Boumal, Aug. 31, 2018.
% Contributors:
% Change log:
% April 12, 2020 (NB):
% Now checking M.dist(x, M.exp(x, v, t)) for several values of t
% because this test is only valid for norm(x, tv) <= inj(x).
% May 19, 2020 (NB):
% Now checking M.dim().
% Jan 8, 2021 (NB):
% Added partial checks of M.inner, M.tangent2ambient, M.proj, ...
assert(isstruct(M), 'M must be a structure.');
%% List required fields that must be function handles here
list_of_functions = {'name', 'dim', 'inner', 'norm', 'typicaldist', ...
'proj', 'tangent', 'egrad2rgrad', 'retr', ...
'rand', 'randvec', 'zerovec', 'lincomb'};
for k = 1 : numel(list_of_functions)
field = list_of_functions{k};
if ~(isfield(M, field) && isa(M.(field), 'function_handle'))
fprintf('M.%s must be a function handle.\n', field);
end
end
%% List recommended fields that should be function handles here
list_of_functions = {'dist', 'ehess2rhess', 'exp', 'log', 'hash', ...
'transp', 'pairmean', 'vec', 'mat', ...
'vecmatareisometries'};
for k = 1 : numel(list_of_functions)
field = list_of_functions{k};
if ~(isfield(M, field) && isa(M.(field), 'function_handle'))
fprintf(['M.%s should ideally (but does not have to) be ' ...
'a function handle.\n'], field);
end
end
%% Check random generators
try
x = M.rand();
v = M.randvec(x);
fprintf('Random tangent vector norm: %g (should be 1).\n', ...
M.norm(x, v));
z = M.lincomb(x, 1, v, -1, v);
fprintf('norm(v - v)_x = %g (should be 0).\n', M.norm(x, z));
catch up %#ok<NASGU>
fprintf('Couldn''t check rand, randvec, lincomb.\n');
end
%% Check inner product
try
x = M.rand();
% Check symmetry
u = M.randvec(x);
v = M.randvec(x);
uv = M.inner(x, u, v);
vu = M.inner(x, v, u);
fprintf('<u, v>_x = %g, <v, u>_x = %g, difference = %g (should be 0).\n', uv, vu, uv-vu);
% Check linearity (and owing to symmetry: bilinearity)
a = randn();
b = randn();
w = M.lincomb(x, a, u, b, v);
z = M.randvec(x);
wz = M.inner(x, w, z);
wzbis = a*M.inner(x, u, z) + b*M.inner(x, v, z);
fprintf('<au+bv, z>_x = %g, a<u, z>_x + b<v, z>_x = %g, difference = %g (should be 0).\n', wz, wzbis, wz-wzbis);
% Should check positive definiteness too: it's somehow part of the
% check for M.dim() below.
catch up %#ok<NASGU>
fprintf('Couldn''t check inner.\n');
end
%% Check tangent2ambient, proj, norm
try
x = M.rand();
v = M.randvec(x);
va = M.tangent2ambient(x, v);
vp = M.proj(x, va);
v_min_vp = M.lincomb(x, 1, v, -1, vp);
df = M.norm(x, v_min_vp);
fprintf('Norm of tangent vector minus its projection to tangent space: %g (should be zero).\n', df);
% Should check that proj is linear, self-adjoint, idempotent.
% The issue for generic code is that manifolds do not provide means
% to generate random vectors in the ambient space.
catch up %#ok<NASGU>
fprintf('Couldn''t check tangent2ambient, proj, norm\n');
end
%% Checking exp and dist
try
x = M.rand();
v = M.randvec(x);
for t = logspace(-8, 1, 10)
y = M.exp(x, v, t);
d = M.dist(x, y);
err = d - abs(t)*M.norm(x, v);
fprintf(['dist(x, M.exp(x, v, t)) - abs(t)*M.norm(x, v) = ' ...
'%g (t = %.1e; should be zero for small enough t).\n'], ...
err, t);
end
catch up %#ok<NASGU>
fprintf('Couldn''t check exp and dist.\n');
% Perhaps we want to rethrow(up) ?
% Alternatively, we could check if exp and dist are available and
% silently pass this test if not, but this way is more informative.
end
%% Checking mat, vec, vecmatareisometries
try
x = M.rand();
u = M.randvec(x);
v = M.randvec(x);
U = M.vec(x, u);
V = M.vec(x, v);
if ~iscolumn(U) || ~iscolumn(V)
fprintf('M.vec should return column vectors: they are not.\n');
end
if ~isreal(U) || ~isreal(V)
fprintf('M.vec should return real vectors: they are not real.\n');
end
fprintf(['Unless otherwise stated, M.vec seems to return real ' ...
'column vectors, as intended.\n']);
ru = M.norm(x, M.lincomb(x, 1, M.mat(x, U), -1, u));
rv = M.norm(x, M.lincomb(x, 1, M.mat(x, V), -1, v));
fprintf(['Checking mat/vec are inverse pairs: ' ...
'%g, %g (should be two zeros).\n'], ru, rv);
a = randn(1);
b = randn(1);
fprintf('Checking if vec is linear: %g (should be zero).\n', ...
norm(M.vec(x, M.lincomb(x, a, u, b, v)) - (a*U + b*V)));
if M.vecmatareisometries()
fprintf('M.vecmatareisometries says true.\n');
else
fprintf('M.vecmatareisometries says false.\n');
end
fprintf('If true, this should be zero: %g.\n', ...
U(:).'*V(:) - M.inner(x, u, v));
catch up %#ok<NASGU>
fprintf('Couldn''t check mat, vec, vecmatareisometries.\n');
end
%% Checking dim
dim_threshold = 200;
if M.dim() <= dim_threshold
x = M.rand();
n = M.dim() + 1;
B = cell(n, 1);
for k = 1 : n
B{k} = M.randvec(x);
end
G = grammatrix(M, x, B);
eigG = sort(real(eig(G)), 'descend');
fprintf('Testing M.dim() (works best when dimension is small):\n');
fprintf('\tIf this number is machine-precision zero, then M.dim() may be too large: %g\n', eigG(n-1));
fprintf('\tIf this number is not machine-precision zero, then M.dim() may be too small: %g\n', eigG(n));
else
fprintf('M.dim() not tested because it is > %d.\n', dim_threshold);
end
%% Recommend calling checkretraction
fprintf('It is recommended also to call checkretraction.\n');
end