Feature Map¶

Module for the feature map class

References

Francesco Romor, Marco Tezzele, Andrea Lario, Gianluigi Rozza. Kernel-based active subspaces with application to computational fluid dynamics parametric problems using discontinuous Galerkin method. International Journal for Numerical Methods in Engineering, 123(23):6000–6027, 2022. doi:10.1002/nme.7099

`FeatureMap._compute_pr_matrix`	Sample the projection matrixx from the spectral distribution.
`FeatureMap.compute_fmap`	Evaluate the feature map at inputs.
`FeatureMap.compute_fmap_jac`	Evaluate the Jacobian matrix of feature map at inputs.
`FeatureMap.pr_matrix`	Get the projection matrix.
`FeatureMap.tune_pr_matrix`	Tune the parameters of the spectral distribution.
`rff_map`	Random Fourier Features map.
`rff_jac`	Random Fourier Features map’s Jacobian.

class FeatureMap(distr, bias, input_dim, n_features, params, sigma_f)[source]

Bases: object

Feature map class.

Parameters

distr (str) – name of the spectral distribution to sample from the matrix.
bias (numpy.ndarray) – n_features dimensional bias. It is sampled from a Unifrom distribution on the interval [0, 2*PI].
input_dim (int) – dimension of the input space.
n_features (int) – dimension of the Reproducing Kernel Hilbert Space.
params (list) – number of hyperparameters of the spectral distribution.
sigma_f (int) – multiplicative constant of the feature map. Usually it is set as the empirical variance of the outputs.

Variables

fmap (callable) – feature map used to project the input samples to the RKHS. Default value is rff_map.
fjac (callable) – Jacobian matrix of fmap. Default value is rff_jac, the analytical Jacobian of fmap.
_pr_matrix (numpy.ndarray) – n_features-by-input_dim projection matrix, whose rows are sampled from the spectral distribution distr.

:raises TypeError

_compute_pr_matrix()[source]

Sample the projection matrixx from the spectral distribution.

Returns: the projection matrix.
Return type: numpy.ndarray

compute_fmap(inputs)[source]

Evaluate the feature map at inputs.

Parameters: inputs (numpy.ndarray) – the inputs to project on the RKHS.
Returns: the n_features dimensional projection of the inputs.
Return type: numpy.ndarray

compute_fmap_jac(inputs)[source]

Evaluate the Jacobian matrix of feature map at inputs.

Parameters: inputs (numpy.ndarray) – the inputs at which compute the Jacobian matrix of the feature map.
Returns: the n_features-by-input_dim dimensional Jacobian of the feature map at the inputs.
Return type: numpy.ndarray

property pr_matrix

Get the projection matrix.

Returns: the projection matrix.
Return type: numpy.ndarray

tune_pr_matrix(func, bounds, method=None, maxiter=50, save_file=False, fn_args=None)[source]

Tune the parameters of the spectral distribution. Three methods are available: log-grid-search (brute), annealing (dual_annealing) and Bayesian stochastic optimization (bso) from GpyOpt. The default object function to optimize is athena.utils.average_rrmse, which uses a cross-validation procedure from athena.utils, see Example and tutorial 06_kernel-based_AS.

Example

>>> from athena.kas import KernelActiveSubspaces
>>> from athena.feature_map import FeatureMap
>>> from athena.projection_factory import ProjectionFactory
>>> from athena.utils import CrossValidation, average_rrmse
>>> input_dim, output_dim, n_samples = 2, 1, 30
>>> n_features, n_params = 10, 1
>>> xx = np.ones((n_samples, input_dim))
>>> f = np.ones((n_samples, output_dim))
>>> df = np.ones((n_samples, output_dim, input_dim))
>>> fm = FeatureMap(distr='laplace',
                    bias=np.random.uniform(0, 2 * np.pi, n_features),
                    input_dim=input_dim,
                    n_features=n_features,
                    params=np.zeros(n_params),
                    sigma_f=f.var())
>>> kss = KernelActiveSubspaces(feature_map=fm, dim=1, n_features=n_features)
>>> csv = CrossValidation(inputs=xx,
                        outputs=f.reshape(-1, 1),
                        gradients=df.reshape(n_samples, output_dim, input_dim),
                        folds=3,
                        subspace=kss)
>>> best = fm.tune_pr_matrix(func=average_rrmse,
                            bounds=[slice(-2, 1, 0.2) for i in range(n_params)],
                            args=(csv, ),
                            method='bso',
                            maxiter=20,
                            save_file=False)

Parameters

func (callable) – the objective function to be minimized. Must be in the form f(x, *args), where x is the argument in the form of a 1-D array and args is a tuple of any additional fixed parameters needed to completely specify the function.
bounds (tuple) – each component of the bounds tuple must be a slice tuple of the form (low, high, step). It defines bounds for the objective function parameter in a logarithmic scale. Step will be ignored for ‘dual_annealing’ method.
args (tuple) – any additional fixed parameters needed to completely specify the objective function.
method (str) – method used to optimize the objective function. Possible values are ‘brute’, or ‘dual_annealing’. Default value is None, and the choice is made automatically wrt the dimension of self.params. If the dimension is less than 4 brute force is used, otherwise a traditional Generalized Simulated Annealing will be performed with no local search strategy applied.
maxiter (int) – the maximum number of global search iterations. Default value is 50.
save_file (bool) – True to save the optimal projection matrix
fn_args (dict) – dictionary of arguments passed to func.

Raises

ValueError

Returns

list that records the best score and the best projection matrix. The initial values are 0.8 and a n_features-by-input_dim numpy.ndarray of zeros.

Return type

list

feature_map.rff_map(pr_matrix, bias, n_features, sigma_f)¶

Random Fourier Features map. It can be vectorized for inputs of shape n_samples-by-input_dim.

Parameters

inputs (numpy.ndarray) – input_dim dimensional inputs to project to the RKHS.
pr_matrix (numpy.ndarray) – n_features-by-input_dim projection matrix, whose rows are sampled from the spectral distribution.
bias (numpy.ndarray) – n_features dimensional bias. It is sampled from a Unifrom distribution on the interval [0, 2*PI].
n_features (int) – dimension of the RKHS
sigma_f (int) – multiplicative term representing the empirical variance the outptus.

Returns

n_features dimensional projection of the inputs to the RKHS

Return type

numpy.ndarray

feature_map.rff_jac(pr_matrix, bias, n_features, sigma_f)¶

Random Fourier Features map’s Jacobian. It can be vectorized for inputs of shape n_samples-by-input_dim.

Parameters

inputs (numpy.ndarray) – input_dim dimensional inputs to project to the RKHS.
pr_matrix (numpy.ndarray) – n_features-by-input_dim projection matrix, whose rows are sampled from the spectral distribution.
bias (numpy.ndarray) – n_features dimensional bias. It is sampled from a Unifrom distribution on the interval [0, 2*PI].
n_features (int) – dimension of the RKHS
sigma_f (int) – multiplicative term representing the empirical variance the outptus.

Returns

n_features-by-input_dim dimensional projection of the inputs to the RKHS

Return type

numpy.ndarray