TransportMaps.Maps.DiagonalIsotropicTransportMapBase

Module Contents

Classes

DiagonalIsotropicTransportMap

Diagonal transport map \(T({\bf x})=[T_1,T_2,\ldots,T_{d_x}]^\top\) where \(T_i(x_{i})=F(x_i):\mathbb{R}\rightarrow\mathbb{R}\).

class TransportMaps.Maps.DiagonalIsotropicTransportMapBase.DiagonalIsotropicTransportMap(**kwargs)

Bases: TransportMaps.Maps.DiagonalComponentwiseTransportMapBase.DiagonalComponentwiseTransportMap

Diagonal transport map \(T({\bf x})=[T_1,T_2,\ldots,T_{d_x}]^\top\) where \(T_i(x_{i})=F(x_i):\mathbb{R}\rightarrow\mathbb{R}\).

evaluate(x, *args, **kwargs)

Evaluate the transport map at the points \({\bf x} \in \mathbb{R}^{m \times d}\).

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 (dict) – cache

Returns:

(ndarray [\(m,d\)]) – transformed points

Raises:

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

grad_x(x, *args, **kwargs)

Compute \(\nabla_{\bf x} T({\bf x})\).

This is

\nabla_{\bf x} T({\bf x},{\bf a}) = \begin{bmatrix} \nabla_{\bf x} T_1({\bf x}) \\ \nabla_{\bf x} T_2({\bf x}) \\ \vdots \\ \nabla_{\bf x} T_d({\bf x}) \end{bmatrix}

for every evaluation point.

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 [\(m,d,d\)]) – gradient matrices for every evaluation point.

Raises:

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

hess_x(x, *args, **kwargs)

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

This is the tensor

\[\left[\nabla^2_{\bf x} T({\bf x})\right]_{i,k,:,:} = \nabla^2_{\bf x} T_k({\bf x}^{(i)})\]

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 [\(m,d,d,d\)]) – Hessian matrices for every evaluation point and every dimension.

Raises:

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

log_det_grad_x(x, *args, **kwargs)

Compute: \(\log \det \nabla_{\bf x} T({\bf x})\).

Since the map is lower triangular,

\[\log \det \nabla_{\bf x} T({\bf x}) = \sum_{k=1}^d \log \partial_{{\bf x}_k} T_k({\bf x}_{1:k})\]

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 (dict) – cache

Returns:

(ndarray [\(m\)]) – \(\log \det \nabla_{\bf x} T({\bf x})\) at every evaluation point

Raises:

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

grad_x_log_det_grad_x(x, *args, **kwargs)

Compute: \(\nabla_{\bf x} \log \det \nabla_{\bf x} T({\bf x})\)

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 [\(m,d\)]) – \(\nabla_{\bf x} \log \det \nabla_{\bf x} T({\bf x})\) at every evaluation point

Raises:

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

See also

log_det_grad_x().

hess_x_log_det_grad_x(x, *args, **kwargs)

Compute: \(\nabla^2_{\bf x} \log \det \nabla_{\bf x} T({\bf x})\)

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 [\(m,d,d\)]) – \(\nabla^2_{\bf x} \log \det \nabla_{\bf x} T({\bf x})\) at every evaluation point

Raises:

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

See also

log_det_grad_x() and grad_x_log_det_grad_x().