def _project(self, t):
'''
Compute the projection of a tensor on the functional bases.
The method takes an AlgebraicTensor t whose entries are the values of
the function on a product grid, and returns a FunctionalTensor obtained
by applying the projections obtained by the method
_projectionOperators.
Parameters
----------
t : tensap.Tensor
The tensor used for the projection.
Returns
-------
tensap.FunctionalTensor
The obtained projection.
'''
tensor = deepcopy(t)
P = self._projection_operators()
for nu in range(tensor.order):
alpha = tensor.tree.dim2ind[nu]
if tensor.is_active_node[alpha - 1]:
data = np.matmul(P[nu], tensor.tensors[alpha - 1].data)
tensor.tensors[alpha - 1] = tensap.FullTensor(
data, 2, data.shape)
else:
pa = tensor.tree.parent(alpha)
ch = tensor.tree.child_number(alpha)
tensor.tensors[pa-1] = \
tensor.tensors[pa-1].tensor_matrix_product(P[nu], ch-1)
return tensap.FunctionalTensor(tensor, self.bases)
def tensor_product_interpolation(self, fun, grid=None):
'''
Return the interpolation of function fun on a product grid.
Parameters
----------
fun : function or tensap.Function or tensap.Tensor
The function to interpolate, or a tensor of order d whose entries
are the evaluations of the function on a product grid.
grid : list, optional
The grid of points used for the interpolation. If one grid has more
points than the dimension of the corresponding basis, use
magicPoints for the selection of a subset of points adapted to the
basis. The default is None, indicating to use the method
self.bases.interpolation_points().
Raises
------
ValueError
If the argument fun is neither a tensap.Function, a function nor
a tensap.Tensor.
Returns
-------
tensap.FunctionalTensor
The interpolation of the function.
output : dict
A dictionnary of outputs of the method.
'''
if grid is None:
grid = self.bases.interpolation_points()
else:
grid = self.bases.interpolation_points(grid)
grid = tensap.FullTensorGrid(grid)
output = {}
if hasattr(fun, '__call__') or isinstance(fun, tensap.Function):
x_grid = grid.array()
y = fun(x_grid)
y = tensap.FullTensor(y, grid.dim, grid.shape)
output['number_of_evaluations'] = y.storage()
# TODO Create an empty class Tensor for when using isinstance?
elif isinstance(fun, (tensap.FullTensor, tensap.CanonicalTensor,
tensap.TreeBasedTensor), tensap.DiagonalTensor):
y = fun
else:
raise ValueError('The argument fun should be a Function, ' +
'function, or a Tensor.')
output['grid'] = grid
B = self.bases.eval(grid.grids)
B = [np.linalg.inv(x) for x in B]
y = y.tensor_matrix_product(B)
return tensap.FunctionalTensor(y, self.bases), output
def projection(self, fun, I):
'''
Compute the projection of the function fun on the basis functions of
self.
Parameters
----------
fun : tensap.Function
The function to project.
I : tensap.IntegrationRule
The integration rule used to compute the projection.
Raises
------
NotImplementedError
If the provided integration rule is not a
tensap.FullTensorProductIntegrationRule.
Returns
-------
tensap.FunctionalTensor
The projection of the function fun on the basis functions of self.
output : dict
Dictionnary containing the number of evaluations of the function in
the key 'number_of_evaluations'.
'''
if not isinstance(I, tensap.FullTensorProductIntegrationRule):
raise NotImplementedError('Method not implemented.')
u = fun.eval_on_tensor_grid(I.points)
output = {'number_of_evaluations': u.storage()}
Hx = self.bases.eval(np.hstack(I.points.grids))
Mx = [
np.matmul(np.transpose(x), np.matmul(np.diag(y), x))
for x, y in zip(Hx, I.weights)
]
Hx_w = [
np.linalg.solve(m, np.matmul(np.transpose(x), np.diag(y)))
for m, x, y in zip(Mx, Hx, I.weights)
]
f_dims = np.arange(len(Hx_w))
u_coeff = u.tensor_matrix_product(Hx_w, f_dims)
return tensap.FunctionalTensor(u_coeff, self.bases, f_dims), output
def _solve_adaptation(self):
'''
Solver for the learning problem with tensor formats using the adaptive
algorithm.
Returns
-------
f : tensap.FunctionalTensor
The learned approximation.
output : dict
The outputs of the solver.
'''
flag = 0
output = {
'enriched_nodes_iterations':
np.empty(self.rank_adaptation_options['max_iterations'],
dtype=object)
}
tree_adapt = False
f = None
errors = np.zeros(self.rank_adaptation_options['max_iterations'])
test_errors = np.zeros(self.rank_adaptation_options['max_iterations'])
iterates = np.empty(self.rank_adaptation_options['max_iterations'],
dtype=object)
# new_rank = s_local.rank
new_rank = self.rank
s_local = self.local_solver()
s_local.model_selection = False
enriched_nodes = np.array([])
for iteration in range(self.rank_adaptation_options['max_iterations']):
s_local.bases = self.bases
s_local.bases_eval = self.bases_eval
s_local.bases_eval_test = self.bases_eval_test
s_local.training_data = self.training_data
s_local.test_data = self.test_data
s_local.rank = new_rank
f_old = deepcopy(f)
f, output_local = s_local.solve()
s_local = self.local_solver()
if 'error' in output_local:
errors[iteration] = output_local['error']
if np.isinf(errors[iteration]):
print('Infinite error, returning to the previous iterate.')
f = f_old
iteration -= 1
flag = -2
break
if self.test_error:
f_eval_test = tensap.FunctionalTensor(f, self.bases_eval_test)
test_errors[iteration] = self.loss_function.test_error(
f_eval_test, self.test_data)
if self.store_iterates:
if isinstance(self.bases, tensap.FunctionalBases):
iterates[iteration] = tensap.FunctionalTensor(
f.tensor, self.bases)
else:
iterates[iteration] = f
if self.display:
if self.alternating_minimization_parameters['display']:
print('')
print('\nRank adaptation, iteration %i:' % (iteration))
self.adaptation_display(f, enriched_nodes)
print('\tStorage complexity = %i' % f.tensor.storage(),
flush=True)
if errors[iteration] != 0:
print('\tError = %2.5e' % errors[iteration])
if test_errors[iteration] != 0:
print('\tTest error = %2.5e' % test_errors[iteration])
if self.alternating_minimization_parameters['display']:
print('')
if iteration == self.rank_adaptation_options['max_iterations'] - 1:
break
if (self.test_error and test_errors[iteration] <
self.tolerance['on_error']) or \
('error' in output_local and errors[iteration] <
self.tolerance['on_error']):
flag = 1
break
fac = self.rank_adaptation_options['early_stopping_factor']
cond = iteration > 0 and (
self.test_error and
(np.isnan(test_errors[iteration]) or fac *
np.min(test_errors[:iteration]) < test_errors[iteration]) or
('error' in output_local and
(np.isnan(errors[iteration])
or fac * np.min(errors[:iteration]) < errors[iteration])))
if self.rank_adaptation_options['early_stopping'] and cond:
print('Early stopping', end='')
if 'error' in output_local:
print(', error = %2.5e' % errors[iteration], end='')
if self.test_error:
print(', test error = %2.5e' % test_errors[iteration],
end='')
print('\n')
iteration -= 1
f = f_old
flag = -1
break
adapted_tree = False
if s_local.tree_adaptation and iteration > 0 and \
(not self.tree_adaptation_options['force_rank_adaptation']
or not tree_adapt):
C_old = f.tensor.storage()
self, f, output = self.adapt_tree(f, errors[iteration], None,
output, iteration)
adapted_tree = output['adapted_tree']
if adapted_tree:
if self.display:
print('\t\tStorage complexity before permutation ' +
'= %i' % C_old)
print('\t\tStorage complexity after permutation ' +
'= %i' % f.tensor.storage())
if self.test_error:
f_eval_test = tensap.FunctionalTensor(
f, self.bases_eval_test)
test_errors[iteration] = self.loss_function.test_error(
f_eval_test, self.test_data)
if self.display:
print('\t\tTest error after permutation ' +
'= %2.5e' % test_errors[iteration])
if self.alternating_minimization_parameters['display']:
print('')
if not self.tree_adaptation or not adapted_tree:
if iteration > 0 and not tree_adapt:
stagnation = self.stagnation_criterion(
tensap.FunctionalTensor(f.tensor, self.bases_eval),
tensap.FunctionalTensor(f_old.tensor, self.bases_eval))
if stagnation < self.tolerance['on_stagnation'] or \
np.isnan(stagnation):
break
tree_adapt = False
new_rank, enriched_nodes, tensor_for_initialization = \
self.new_rank_selection(f)
output['enriched_nodes_iterations'][iteration] = enriched_nodes
s_local = self.initial_guess_new_rank(
s_local, tensor_for_initialization, new_rank)
else:
tree_adapt = True
enriched_nodes = []
new_rank = f.tensor.ranks
s_local.initialization_type = 'initial_guess'
s_local.initial_guess = f.tensor
if isinstance(self.bases, tensap.FunctionalBases):
f = tensap.FunctionalTensor(f.tensor, self.bases)
if self.store_iterates:
output['iterates'] = iterates[:iteration + 1]
output['flag'] = flag
output['enriched_nodes_iterations'] = \
output['enriched_nodes_iterations'][:iteration+1]
if 'error' in output_local:
output['error_iterations'] = errors[:iteration + 1]
output['error'] = errors[iteration]
if self.test_error:
output['test_error_iterations'] = test_errors[:iteration + 1]
output['test_error'] = test_errors[iteration]
if 'adapted_tree' in output:
del output['adapted_tree']
return f, output
def _solve_standard(self):
'''
Solver for the learning problem with tensor formats using the standard
algorithm (without adaptation).
Raises
------
ValueError
If the number of LinearModelLearning objects is not equal to
_numberOfParameters.
Returns
-------
f : tensap.FunctionalTensor
The learned approximation.
output : dict
The outputs of the solver.
'''
output = {'flag': 0}
self, f = self.initialize()
f = tensap.FunctionalTensor(f, self.bases_eval)
# Replication of the LinearModelLearning objects
if self.linear_model_learning_parameters[
'identical_for_all_parameters'] and \
not isinstance(self.linear_model_learning, (list, np.ndarray)):
self.linear_model_learning = list(
map(deepcopy,
[self.linear_model_learning] * self._number_of_parameters))
elif isinstance(self.linear_model_learning, (list, np.ndarray)) and \
len(self.linear_model_learning) != self._number_of_parameters:
raise ValueError('Must provide self._numberOfParameters ' +
'LinearModelLearning objects.')
if self.error_estimation:
for x in self.linear_model_learning:
setattr(x, 'error_estimation', True)
# Working set paths
if isinstance(self.linear_model_learning, (list, np.ndarray)) and \
np.any([x.basis_adaptation for
x in self.linear_model_learning]) and \
self.bases_adaptation_path is None:
self.bases_adaptation_path = self.bases.adaptation_path()
if self.alternating_minimization_parameters['max_iterations'] == 0:
return f, output
output['stagnation_criterion'] = []
output['iterates'] = []
output['error_iterations'] = []
output['test_error_iterations'] = []
# Alternating minimization loop
for iteration in range(
self.alternating_minimization_parameters['max_iterations']):
self, f = self.pre_processing(f)
f0 = deepcopy(f)
if self.alternating_minimization_parameters['random']:
# Randomize the exploration strategy
alpha_list = self.randomize_exploration_strategy()
else:
alpha_list = self._exploration_strategy
for alpha in alpha_list:
self, A, b, f = \
self.prepare_alternating_minimization_system(f, alpha)
self.linear_model_learning[alpha - 1].training_data = [None, b]
self.linear_model_learning[alpha - 1].basis = None
self.linear_model_learning[alpha - 1].basis_eval = A
coef, output_tmp = self.linear_model_learning[alpha -
1].solve()
if coef is None or np.count_nonzero(coef) == 0 or \
not np.all(np.isfinite(coef)):
print('Empty, zero or NaN solution, returning to the ' +
'previous iteration.')
output['flag'] = -2
output['error'] = np.inf
break
f = self.set_parameter(f, alpha, coef)
stagnation = self.stagnation_criterion(f, f0)
output['stagnation_criterion'].append(stagnation)
if self.store_iterates:
if isinstance(self.bases, tensap.FunctionalBases):
output['iterates'].append(
tensap.FunctionalTensor(f.tensor, self.bases))
else:
output['iterates'].append(f)
if 'error' in output_tmp:
output['error'] = output_tmp['error']
output['error_iterations'].append(output['error'])
if self.test_error:
f_eval_test = tensap.FunctionalTensor(f, self.bases_eval_test)
output['test_error'] = self.loss_function.test_error(
f_eval_test, self.test_data)
output['test_error_iterations'].append(output['test_error'])
if self.alternating_minimization_parameters['display']:
print('\tAlt. min. iteration %s: stagnation = %2.5e' %
(str(iteration).zfill(
len(
str(self.alternating_minimization_parameters[
'max_iterations'] - 1))), stagnation),
end='')
if 'error' in output:
print(', error = %2.5e' % output['error'], end='')
if self.test_error:
if not np.isscalar(output['test_error']):
output['test_error'] = output['test_error'].numpy()
print(', test error = %2.5e' % output['test_error'],
end='')
print('')
if iteration > 0 and stagnation < \
self.alternating_minimization_parameters['stagnation']:
output['flag'] = 1
break
if self.test_error and \
output['test_error'] < self.tolerance['on_error']:
output['flag'] = 1
break
if isinstance(self.bases, tensap.FunctionalBases):
f = tensap.FunctionalTensor(f.tensor, self.bases)
output['iter'] = iteration
if self.display and not self.model_selection:
if self.alternating_minimization_parameters['display']:
print('')
self.final_display(f)
if 'error' in output:
print(', error = %2.5e' % output['error'], end='')
if 'test_error' in output:
print(', test error = %2.5e' % output['test_error'], end='')
print('')
return f, output
def _solve_greedy(self):
'''
Greedy solver in canonical tensor format.
Raises
------
ValueError
If the number of LinearModelLearning objects is not equal to
_numberOfParameters.
Returns
-------
f : tensap.FunctionalTensor
The learned approximation.
output : dict
The outputs of the solver.
'''
assert isinstance(self.loss_function, tensap.SquareLossFunction), \
'Method not implemented for this loss function.'
bases_eval = self.bases_eval
output = {}
output['sequence'] = np.empty(self.rank, dtype=object)
# Replication of the LinearModelLearning objects
if self.linear_model_learning_parameters[
'identical_for_all_parameters'] and \
not isinstance(self.linear_model_learning, (list, np.ndarray)):
self.linear_model_learning = list(
map(deepcopy,
[self.linear_model_learning] * self._number_of_parameters))
elif isinstance(self.linear_model_learning, (list, np.ndarray)) and \
len(self.linear_model_learning) != self._number_of_parameters:
raise ValueError('Must provide self._numberOfParameters ' +
'LinearModelLearning objects.')
# Working set paths
if isinstance(self.linear_model_learning, (list, np.ndarray)) and \
np.any([x.basis_adaptation for
x in self.linear_model_learning]) and \
self.bases_adaptation_path is None:
self.bases_adaptation_path = self.bases.adaptation_path()
y = self.training_data[1]
s_local = deepcopy(self)
s_local.algorithm = 'standard'
s_local.rank = 1
s_local.display = False
s_local.test_error = False
s_local.model_selection = False
f = tensap.CanonicalTensor.zeros(0, [x.shape[1] for x in bases_eval])
f_0 = deepcopy(f)
stagnation = np.zeros((1, self.rank))
ls_local = deepcopy(
s_local.linear_model_learning[self._number_of_parameters - 1])
is_error = False
error = np.ones((1, self.rank))
pure_greedy = False
if not pure_greedy:
for linear_solver in self.linear_model_learning:
setattr(linear_solver, 'error_estimation', False)
for k in np.arange(1, self.rank + 1):
s_local.training_data[1] = \
y - f.tensor_matrix_product(bases_eval).eval_diag().data
f_add, output_greedy = s_local.solve()
if isinstance(f_add, tensap.FunctionalTensor):
f += f_add.tensor
else:
f += f_add
if not pure_greedy:
f_H = f.tensor_matrix_product(bases_eval)
A = np.ones((bases_eval[0].shape[0], len(f_H.core.data)))
for space in f_H.space:
A *= space
ls_local.basis_adaptation = False
ls_local.basis = None
ls_local.basis_eval = A
ls_local.training_data = [None, y]
coef, output_greedy = ls_local.solve()
f.core.data = np.ravel(coef)
stagnation[k - 1] = 2 * (f - f_0).norm() / (f.norm() + f_0.norm())
current_rank = len(f.core.data)
output['sequence'][k - 1] = f
if 'error' in output_greedy:
error[k - 1] = output_greedy['error']
is_error = True
if self.alternating_minimization_parameters['display']:
print(
'Alternating minimization (greedy): rank = %i, ' +
'error = %2.5e, stagnation = %2.5e', current_rank,
error[k - 1], stagnation[k - 1])
else:
if self.alternating_minimization_parameters['display']:
print(
'Alternating minimization (greedy): rank = %i, ' +
'stagnation = %2.5e', current_rank, stagnation[k - 1])
if error[k-1] < self.tolerance['on_error'] or \
stagnation[k-1] < self.tolerance['on_stagnation'] or \
(k > 2 and error[k-1] > error[k-2] and
error[k-2] > error[k-3]):
break
f_0 = deepcopy(f)
if self.test_error:
f_eval_test = tensap.FunctionalTensor(f, self.bases_eval_test)
output['test_error'] = self.loss_function.test_error(
f_eval_test, self.test_data)
output['test_error_iterations'].append(output['test_error'])
if self.display:
print('Greedy: iteration %i, test error = %2.5e', k,
output['test_error'])
output['stagnation'] = stagnation[:k - 1]
if is_error:
output['errors'] = error[:k - 1]
K = np.argmin(output['errors'])
f = output['sequence'][K]
output['selected_iterate'] = K
output['error'] = output['errors'][K]
if isinstance(self.bases, tensap.FunctionalBases):
f = tensap.FunctionalTensor(f, self.bases)
return f, output