tensap.approximation.tensor_approximation.principal_component_analysis package¶

Submodules¶

tensap.approximation.tensor_approximation.principal_component_analysis.functional_tensor_principal_component_analysis module¶

Module functional_tensor_principal_component_analysis.

class tensap.approximation.tensor_approximation.principal_component_analysis.functional_tensor_principal_component_analysis.FunctionalTensorPrincipalComponentAnalysis¶

Bases: object

Class FunctionalTensorPrincipalComponentAnalysis: principal component analysis of multivariate functions based on TensorPrincipalComponentAnalysis for algebraic tensors.

Attributes

tolint or float or list or numpy.ndarray

An array containing the prescribed relative precisions (the length depends on the format). If len(tol)==1, use the same value for all alpha. Set tol = inf to prescribe the rank.

max_rankint or list or numpy.ndarray

An array containing the maximum alpha-ranks (the length depends on the format). If len(max_rank)==1, use the same value for all alpha. Set max_rank = np.inf to prescribe the precision.

basestensap.FunctionalBases

The functional bases used for the projection of the function.

gridtensap.FullTensorGrid

FullTensorGrid for the projection of the function on the functional bases.

displaybool

Boolean specifying the verbosity of the methods.

pca_sampling_factorint

A factor to determine the number of samples N for the estimation of the principal components (1 by default):

if prescribed precision, N = pca_sampling_factor*N_alpha,
if prescribed rank, N = pca_sampling_factor*t.

pca_adaptive_samplingbool

Adaptive sampling to determine the principal components with prescribed precision.

projection_typestr

The type of projection. The default is ‘interpolation’.

Methods

`hopca`(fun)	Return the set of alpha-principal components of a tensor, for all alpha in {0,1,…,d-1}.
`tree_based_approximation`(fun, tree[, …])	Approximation of a function of d variables in tree based tensor format based on a Principal Component Analysis.
`tt_approximation`(fun)	Approximation of a function of d variables in tensor train format based on a Principal Component Analysis.
`tucker_approximation`(fun)	Approximation of a function of d variables in Tucker format based on a Principal Component Analysis.

hopca(fun)¶

Return the set of alpha-principal components of a tensor, for all alpha in {0,1,…,d-1}.

For prescribed precision, set FPCA.max_rank = np.inf and FPCA.tol to the desired precision (possibly an array of length d).

For prescribed rank, set FPCA.tol = np.inf and FPCA.max_rank to the desired rank (possibly an array of length d).

See also the documentation of the class FunctionalTensorPrincipalComponentAnalysis.

Parameters

funfunction: A function of d variables.

Returns

f_pclist: List of the alpha-principal components of the function.
outputslist: List containing the outputs of the method alpha_principal_components.

Raises

ValueError: If the provided tolerance and max ranks are not correct.

tree_based_approximation(fun, tree, is_active_node=None)¶

Approximation of a function of d variables in tree based tensor format based on a Principal Component Analysis.

For a prescribed precision, set TPCA.max_rank = np.inf and TPCA.tol to the desired precision (possibly an array of length d-1).

For a prescribed rank, set TPCA.tol = np.inf and TPCA.max_rank to the desired rank (possibly an array of length d-1).

See also the documentation of the class FunctionalTensorPrincipalComponentAnalysis.

Parameters

funfun or tensap.Function: Function of d variables i_1, …, i_d which returns the entries of the tensor.
shapelist or numpy.ndarray: The shape of the tensor.
treetensap.DimensionTree: The required dimension tree.
is_active_nodelist or numpy.ndarray, optional: An array of booleans indicating which nodes of the tree are active. The default is None, settings all the nodes active.

Returns

tensap.FunctionalTensor: A function in tree based format.
dict: Dictionnary containing the outputs of the method.

Raises

ValueError: If the provided tolerance and max ranks are not correct.

tt_approximation(fun)¶

Approximation of a function of d variables in tensor train format based on a Principal Component Analysis.

For a prescribed precision, set TPCA.max_rank = np.inf and TPCA.tol to the desired precision (possibly an array of length d).

For a prescribed rank, set TPCA.tol = np.inf and TPCA.max_rank to the desired rank (possibly an array of length d).

See also the documentation of the class FunctionalTensorPrincipalComponentAnalysis.

Parameters

funfunction: A function of d variables.

Returns

tensap.FunctionalTensor: A function in tree based format with a linear tree.
dict: Dictionnary containing the outputs of the method.

Raises

ValueError: If the provided tolerance and max ranks are not correct.

tucker_approximation(fun)¶

Approximation of a function of d variables in Tucker format based on a Principal Component Analysis.

For a prescribed precision, set TPCA.max_rank = np.inf and TPCA.tol to the desired precision (possibly an array of length d).

For a prescribed rank, set TPCA.tol = np.inf and TPCA.max_rank to the desired rank (possibly an array of length d).

See also the documentation of the class FunctionalTensorPrincipalComponentAnalysis.

Parameters

funfunction: A function of d variables.

Returns

tensap.FunctionalTensor: A function in tree based format with a trivial tree.
dict: Dictionnary containing the outputs of the method.

Raises

ValueError: If the provided tolerance and max ranks are not correct.

tensap.approximation.tensor_approximation.principal_component_analysis.tensor_principal_component_analysis module¶

Module tensor_principal_component_analysis.

class tensap.approximation.tensor_approximation.principal_component_analysis.tensor_principal_component_analysis.TensorPrincipalComponentAnalysis¶

Bases: object

Class TensorPrincipalComponentAnalysis: principal component analysis of an algebraic tensor.

Attributes

displaybool

Boolean specifying the verbosity of the methods.

pca_sampling_factorint

A factor to determine the number of samples N for the estimation of the principal components (1 by default):

if prescribed precision, N = pca_sampling_factor*N_alpha,
if prescribed rank, N = pca_sampling_factor*t.

pca_adaptive_samplingbool

Adaptive sampling to determine the principal components with prescribed precision.

tolint or float or list or numpy.ndarray

An array containing the prescribed relative precisions (the length depends on the format). If len(tol)==1, use the same value for all alpha. Set tol = inf to prescribe the rank.

max_rankint or list or numpy.ndarray

An array containing the maximum alpha-ranks (the length depends on the format). If len(max_rank)==1, use the same value for all alpha. Set max_rank = np.inf to prescribe the precision.

Methods

`alpha_principal_components`(fun, shape, …)	Evaluate the alpha-principal components of a tensor f.
`hopca`(fun, shape)	Return the set of alpha-principal components of an algebraic tensor, for all alpha in {0,1,…,d-1}.
`tree_based_approximation`(fun, shape, tree[, …])	Approximation of a tensor of order d in tree based tensor format based on a Principal Component Analysis.
`tt_approximation`(fun, shape)	Approximation of a tensor of order d in tensor train format based on a Principal Component Analysis.
`tucker_approximation`(fun, shape)	Approximation of a tensor of order d in Tucker format based on a Principal Component Analysis.

alpha_principal_components(fun, shape, alpha, tol, B_alpha, I_alpha)¶

Evaluate the alpha-principal components of a tensor f.

For alpha in {0,…,d-1}, it evaluates the alpha-principal components of a tensor f, meaning the principal components of the matricisations f_alpha(i_alpha,i_notalpha), where i_alpha and i_notalpha are groups of indices.

It evaluates f_alpha on the product of a set of indices in dimension alpha (of size Nalpha) and a set of random indices (N samples) in the complementary dimensions. Then, it computes approximations of the alpha-principal components in a given basis phi_1(i_alpha) … phi_Nalpha(i_alpha).

If t is an integer, t is the rank (number of principal components). If t<1, the rank (number of principal components) is determined such that the relative error after truncation is t.

Parameters

funfun or tensap.Function: Function of d variables i_1, …, i_d which returns the entries of the tensor.
shapelist or numpy.ndarray: The shape of the tensor.
alphaint: An array containing a tuple in {0,…,d-1}.
tolint or float: The number of principal components or a positive number smaller than 1 (tolerance).
B_alphanumpy.ndarray: Array of shape (N_lpha, N_lpha) whose i-th column is the evaluation of phi_i at the set of indices i_alpha in Ialpha.
I_alphanumpy.ndarray: Array of shape (N_alpha, #alpha) containing N_alpha tuples i_alpha.

Returns

pcnumpy.ndarray: The principal components of the tensor.
outputdict: A dictionnary of outputs, containing the singular values corresponding to the principal components, as well as the set of indices at which the tensor has been evaluated.

hopca(fun, shape)¶

Return the set of alpha-principal components of an algebraic tensor, for all alpha in {0,1,…,d-1}.

For prescribed precision, set TPCA.max_rank = np.inf and TPCA.tol to the desired precision (possibly an array of length d).

For prescribed rank, set TPCA.tol = np.inf and TPCA.max_rank to the desired rank (possibly an array of length d).

See also the documentation of the class TensorPrincipalComponentAnalysis.

Parameters

funfun or tensap.Function: Function of d variables i_1, …, i_d which returns the entries of the tensor.
shapelist or numpy.ndarray: The shape of the tensor.

Returns

f_pclist: List of the alpha-principal components of the tensor.
outputlist: List containing the outputs of the method alpha_principal_components.

Raises

ValueError: If the provided tolerance and max ranks are not correct.

tree_based_approximation(fun, shape, tree, is_active_node=None)¶

Approximation of a tensor of order d in tree based tensor format based on a Principal Component Analysis.

For a prescribed precision, set TPCA.max_rank = np.inf and TPCA.tol to the desired precision (possibly an array of length d-1).

For a prescribed rank, set TPCA.tol = np.inf and TPCA.max_rank to the desired rank (possibly an array of length d-1).

See also the documentation of the class TensorPrincipalComponentAnalysis.

Parameters

funfun or tensap.Function: Function of d variables i_1, …, i_d which returns the entries of the tensor.
shapelist or numpy.ndarray: The shape of the tensor.
treetensap.DimensionTree: The required dimension tree.
is_active_nodelist or numpy.ndarray, optional: An array of booleans indicating which nodes of the tree are active. The default is None, settings all the nodes active.

Returns

tensap.TreeBasedTensor: A tensor in tree based format.
dict: Dictionnary containing the outputs of the method.

Raises

ValueError: If the provided tolerance and max ranks are not correct.

tt_approximation(fun, shape)¶

Approximation of a tensor of order d in tensor train format based on a Principal Component Analysis.

For a prescribed precision, set TPCA.max_rank = np.inf and TPCA.tol to the desired precision (possibly an array of length d-1).

For a prescribed rank, set TPCA.tol = np.inf and TPCA.max_rank to the desired rank (possibly an array of length d-1).

See also the documentation of the class TensorPrincipalComponentAnalysis.

Parameters

funfun or tensap.Function: Function of d variables i_1, …, i_d which returns the entries of the tensor.
shapelist or numpy.ndarray: The shape of the tensor.

Returns

tensap.TreeBasedTensor: A tensor in tree based format with a linear tree.
dict: Dictionnary containing the outputs of the method.

Raises

ValueError: If the provided tolerance and max ranks are not correct.

tucker_approximation(fun, shape)¶

Approximation of a tensor of order d in Tucker format based on a Principal Component Analysis.

For a prescribed precision, set TPCA.max_rank = np.inf and TPCA.tol to the desired precision (possibly an array of length d).

For a prescribed rank, set TPCA.tol = np.inf and TPCA.max_rank to the desired rank (possibly an array of length d).

See also the documentation of the class TensorPrincipalComponentAnalysis.

Parameters

funfun or tensap.Function: Function of d variables i_1, …, i_d which returns the entries of the tensor.
shapelist or numpy.ndarray: The shape of the tensor.

Returns

tensap.TreeBasedTensor: A tensor in tree based format with a trivial tree.
dict: Dictionnary containing the outputs of the method.

Raises

ValueError: If the provided tolerance and max ranks are not correct.

tensap.approximation.tensor_approximation.principal_component_analysis package¶

Submodules¶

tensap.approximation.tensor_approximation.principal_component_analysis.functional_tensor_principal_component_analysis module¶

tensap.approximation.tensor_approximation.principal_component_analysis.tensor_principal_component_analysis module¶

Module contents¶

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