TransportMaps.Maps.ParametricComponentwiseMapBase

Module Contents

Classes

ParametricComponentwiseMap

Map \(T[{\bf a}_{1:d_y}]({\bf x})= [T_1[{\bf a}_1]({\bf x}_{{\bf j}_{1}}),\ldots,T_{d_y}[{\bf a}_{d_y}]({\bf x}_{{\bf j}_{d_y}})]^\top\), where \(T_i[{\bf a}_i]({\bf x}_{{\bf j}_{i}}):\mathbb{R}^{n_i}\times\mathbb{R}^{\text{dim}({{\bf j}_{i}})}\rightarrow\mathbb{R}\).

class TransportMaps.Maps.ParametricComponentwiseMapBase.ParametricComponentwiseMap(**kwargs)

Bases: TransportMaps.Maps.ComponentwiseMapBase.ComponentwiseMap, TransportMaps.Maps.ParametricMapBase.ParametricMap

Map \(T[{\bf a}_{1:d_y}]({\bf x})= [T_1[{\bf a}_1]({\bf x}_{{\bf j}_{1}}),\ldots,T_{d_y}[{\bf a}_{d_y}]({\bf x}_{{\bf j}_{d_y}})]^\top\), where \(T_i[{\bf a}_i]({\bf x}_{{\bf j}_{i}}):\mathbb{R}^{n_i}\times\mathbb{R}^{\text{dim}({{\bf j}_{i}})}\rightarrow\mathbb{R}\).

property n_coeffs

Returns the total number of coefficients.

Returns:

total number \(N\) of coefficients characterizing the transport map.

property coeffs

Returns the actual value of the coefficients.

Returns:

(ndarray [\(N\)]) – coefficients.

abstract get_identity_coeffs()

[Abstract] Returns the coefficients corresponding to the identity map

Returns:

coefficients

Return type:

(ndarray [\(N\)])

Raises:

NotImplementedError – must be implemented in subclasses.

grad_a(x, precomp=None, idxs_slice=slice(None), cache=None)

Compute \(\nabla_{\bf a} T[{\bf a}]({\bf x})\)

By the definition of the transport map \(T[{\bf a}]({\bf x})\), the components \(T_1[{\bf a}^{(1)}] ({\bf x}_1)\), \(T_2[{\bf a}^{(2)}] ({\bf x}_{1:2})\), … are defined by different sets of parameters \({\bf a}^{(1)}\), \({\bf a}^{(2)}\), etc.

For this reason \(\nabla_{\bf a} T[{\bf a}]({\bf x})\) is block diagonal:

\nabla_a T[{\bf a}]({\bf x}) = \begin{bmatrix} \left[ \nabla_{{\bf a}^{(1)}} T_1[{\bf a}^{(1)}] ({\bf x}_1) \right]^T & {\bf 0} & \cdots \\ {\bf 0} & \left[ \nabla_{{\bf a}^{(2)}} T_2[{\bf a}^{(2)}] ({\bf x}_{1:2}) \right]^T & \\ \vdots & & \ddots \end{bmatrix}

Consequentely this function will return only the diagonal blocks of the gradient.

Parameters:

• x (ndarray [\(m,d\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

• cache (dics) – cache

Returns:

(list of ndarray [\(n_i\)]) –

list containing \(\nabla_{{\bf a}^{(1)}} T_1[{\bf a}^{(1)}] ({\bf x}_1)\), \(\nabla_{{\bf a}^{(2)}} T_2[{\bf a}^{(2)}] ({\bf x}_{1:2})\), etc.

Raises:

ValueError – if \(d\) does not match the dimension of the transport map.

abstract grad_a_inverse(x, precomp=None, idxs_slice=slice(None))

[Abstract] Compute \(\nabla_{\bf a} T^{-1}({\bf x},{\bf a})\)

Parameters:

• x (ndarray [\(m,d\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(ndarray [\(m,d,N\)]) –

\(\nabla_{\bf a} T^{-1}({\bf x},{\bf a})\)

Raises:

ValueError – if \(d\) does not match the dimension of the transport map.

hess_a(x, precomp=None, idxs_slice=slice(None), cache=None)

Compute \(\nabla^2_{\bf a} T[{\bf a}]({\bf x})\).

As in the case of grad_a(), the \(d \times N \times N\) Hessian of T[{bf a}]({bf x}) is (hyper) block diagonal.

Parameters:

• x (ndarray [\(m,d\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(list of ndarray [\(n_i,n_i\)]) –

list containing \(\nabla^2_{{\bf a}^{(1)}} T_1[{\bf a}^{(1)}] ({\bf x}_1)\), \(\nabla^2_{{\bf a}^{(2)}} T_2[{\bf a}^{(2)}] ({\bf x}_{1:2})\), etc.

Raises:

ValueError – if \(d\) does not match the dimension of the transport map.

action_hess_a(x, da, precomp=None, idxs_slice=slice(None), cache=None)

Compute \(\langle\nabla^2_{\bf a} T[{\bf a}]({\bf x}), \delta{\bf a}\rangle\).

As in the case of grad_a(), the :math:`d times N ` actions of the Hessian of T[{bf a}]({bf x}) is (hyper) block diagonal.

Parameters:

• x (ndarray [\(m,d\)]) – evaluation points

• da (ndarray [\(N\)]) – direction on which to evaluate the Hessian

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

• cache (dict) – cache

Returns:

(list of ndarray [\(n_i,n_i\)]) –

list containing \(\langle\nabla^2_{{\bf a}^{(1)}} T_1[{\bf a}^{(1)}] ({\bf x}_1),\delta{\bf a}^{(1)}\rangle\), \(\langle\nabla^2_{{\bf a}^{(2)}} T_2[{\bf a}^{(2)}] ({\bf x}_{1:2}),\delta{\bf a}^{(2)}\rangle\), etc.

Raises:

ValueError – if \(d\) does not match the dimension of the transport map.

grad_a_grad_x(x, precomp=None, idxs_slice=slice(None), *args, **kwargs)

Compute \(\nabla_{\bf a} \nabla_{\bf x} T[{\bf a}]({\bf x})\)

By the definition of the transport map \(T[{\bf a}]({\bf x})\), the components \(T_1[{\bf a}^{(1)}] ({\bf x}_1)\), \(T_2[{\bf a}^{(2)}] ({\bf x}_{1:2})\), … are defined by different sets of parameters \({\bf a}^{(1)}\), \({\bf a}^{(2)}\), etc.

For this reason \(\nabla_{\bf a} \nabla_{\bf x} T[{\bf a}]({\bf x})\) is block diagonal:

\nabla_a \nabla_{\bf x} T[{\bf a}]({\bf x}) = \begin{bmatrix} \left[ \nabla_{{\bf a}^{(1)}} \nabla_{\bf x}_1 T_1[{\bf a}^{(1)}] ({\bf x}_1) \right]^T & {\bf 0} & \cdots \\ {\bf 0} & \left[ \nabla_{{\bf a}^{(2)}} \nabla_{\bf x}_{1:2} T_2[{\bf a}^{(2)}] ({\bf x}_{1:2}) \right]^T & \\ \vdots & & \ddots \end{bmatrix}

Consequentely this function will return only the diagonal blocks of the gradient.

Parameters:

• x (ndarray [\(m,d\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(list of ndarray [\(n_i\)]) –

list containing \(\nabla_{{\bf a}^{(1)}} \nabla_{\bf x}_1 T_1[{\bf a}^{(1)}] ({\bf x}_1)\), \(\nabla_{{\bf a}^{(2)}} \nabla_{\bf x}_{1:2} T_2[{\bf a}^{(2)}] ({\bf x}_{1:2})\), etc.

Raises:

ValueError – if \(d\) does not match the dimension of the transport map.

grad_a_hess_x(x, precomp=None, idxs_slice=slice(None), *args, **kwargs)

Compute \(\nabla_{\bf a} \nabla^2_{\bf x} T[{\bf a}]({\bf x})\).

By the definition of the transport map \(T[{\bf a}]({\bf x})\), the components \(T_1[{\bf a}^{(1)}] ({\bf x}_1)\), \(T_2[{\bf a}^{(2)}] ({\bf x}_{1:2})\), … are defined by different sets of parameters \({\bf a}^{(1)}\), \({\bf a}^{(2)}\), etc.

For this reason \(\nabla_{\bf a} \nabla^2_{\bf x} T[{\bf a}]({\bf x})\) is block diagonal:

\nabla_a \nabla^2_{\bf x} T[{\bf a}]({\bf x}) = \begin{bmatrix} \left[ \nabla_{{\bf a}^{(1)}} \nabla^2_{\bf x}_1 T_1[{\bf a}^{(1)}] ({\bf x}_1) \right]^T & {\bf 0} & \cdots \\ {\bf 0} & \left[ \nabla_{{\bf a}^{(2)}} \nabla^2_{\bf x}_{1:2} T_2[{\bf a}^{(2)}] ({\bf x}_{1:2}) \right]^T & \\ \vdots & & \ddots \end{bmatrix}

Consequentely this function will return only the diagonal blocks of the hessian.

Parameters:

• x (ndarray [\(m,d\)]) – evaluation points

• precomp (dict) – dictionary of precomputed values

• idxs_slice (slice) – if precomputed values are present, this parameter indicates at which of the points to evaluate. The number of indices represented by idxs_slice must match x.shape[0].

Returns:

(list of ndarray [\(n_i\)]) –

list containing \(\nabla_{{\bf a}^{(1)}} \nabla^2_{\bf x}_1 T_1[{\bf a}^{(1)}] ({\bf x}_1)\), \(\nabla_{{\bf a}^{(2)}} \nabla^2_{\bf x}_{1:2} T_2[{\bf a}^{(2)}] ({\bf x}_{1:2})\), etc.

Raises:

ValueError – if \(d\) does not match the dimension of the transport map.