# Affine Flow¶

A scale-and-shift bijector

\begin{align}\begin{aligned}a,b \in \mathbb{R}^d\\f(x) = \exp(a^T) \cdot x + b\end{aligned}\end{align}

Dimension of parameter space: $$d+d$$

Determinant of the Jabobian:

$\det(\mathbf{J}) = \prod_{j=1}^{d}\exp(\mathbf{a}_j) = \exp(\sum_{j=1}^{d}\mathbf{a}_j)$

Hence the Inverse Log Det Jacobian is:

$\log(\det(\mathbf{J}^{-1})) = -(\sum_{j=1}^{d}\mathbf{a}_j)$
class cde.density_estimator.normalizing_flows.AffineFlow(params, n_dims, name='AffineFlow')[source]

Implements a bijector y = a*x + b

Parameters
• params – tensor of shape (?, 2*n_dims). This will be split into the parameters a, b

• n_dims – The dimension of the distribution that is being transformed

• name – The name to give this flow

dtype

dtype of Tensors transformable by this distribution.

event_ndims

Returns then number of event dimensions this bijector operates on.

forward(x, name='forward')

Returns the forward Bijector evaluation, i.e., X = g(Y).

Parameters
• xTensor. The input to the “forward” evaluation.

• name – The name to give this op.

Returns

Tensor.

Raises
• TypeError – if self.dtype is specified and x.dtype is not self.dtype.

• NotImplementedError – if _forward is not implemented.

forward_event_shape(input_shape)

Shape of a single sample from a single batch as a TensorShape.

Same meaning as forward_event_shape_tensor. May be only partially defined.

Parameters

input_shapeTensorShape indicating event-portion shape passed into forward function.

Returns

TensorShape indicating event-portion shape

after applying forward. Possibly unknown.

Return type

forward_event_shape_tensor

forward_event_shape_tensor(input_shape, name='forward_event_shape_tensor')

Shape of a single sample from a single batch as an int32 1D Tensor.

Parameters
• input_shapeTensor, int32 vector indicating event-portion shape passed into forward function.

• name – name to give to the op

Returns

Tensor, int32 vector indicating

event-portion shape after applying forward.

Return type

forward_event_shape_tensor

forward_log_det_jacobian(x, name='forward_log_det_jacobian')

Returns both the forward_log_det_jacobian.

Parameters
• xTensor. The input to the “forward” Jacobian evaluation.

• name – The name to give this op.

Returns

Tensor, if this bijector is injective.

If not injective this is not implemented.

Raises
• TypeError – if self.dtype is specified and y.dtype is not self.dtype.

• NotImplementedError – if neither _forward_log_det_jacobian nor {_inverse, _inverse_log_det_jacobian} are implemented, or this is a non-injective bijector.

static get_param_size(n_dims)[source]
Parameters

n_dims – The dimension of the distribution to be transformed by the flow.

Returns

(int) The dimension of the parameter space for the flow. Here it’s n_dims + n_dims

graph_parents

Returns this Bijector’s graph_parents as a Python list.

inverse(y, name='inverse')

Returns the inverse Bijector evaluation, i.e., X = g^{-1}(Y).

Parameters
• yTensor. The input to the “inverse” evaluation.

• name – The name to give this op.

Returns

Tensor, if this bijector is injective.

If not injective, returns the k-tuple containing the unique k points (x1, …, xk) such that g(xi) = y.

Raises
• TypeError – if self.dtype is specified and y.dtype is not self.dtype.

• NotImplementedError – if _inverse is not implemented.

inverse_event_shape(output_shape)

Shape of a single sample from a single batch as a TensorShape.

Same meaning as inverse_event_shape_tensor. May be only partially defined.

Parameters

output_shapeTensorShape indicating event-portion shape passed into inverse function.

Returns

TensorShape indicating event-portion shape

after applying inverse. Possibly unknown.

Return type

inverse_event_shape_tensor

inverse_event_shape_tensor(output_shape, name='inverse_event_shape_tensor')

Shape of a single sample from a single batch as an int32 1D Tensor.

Parameters
• output_shapeTensor, int32 vector indicating event-portion shape passed into inverse function.

• name – name to give to the op

Returns

Tensor, int32 vector indicating

event-portion shape after applying inverse.

Return type

inverse_event_shape_tensor

inverse_log_det_jacobian(y, name='inverse_log_det_jacobian')

Returns the (log o det o Jacobian o inverse)(y).

Mathematically, returns: log(det(dX/dY))(Y). (Recall that: X=g^{-1}(Y).)

Note that forward_log_det_jacobian is the negative of this function, evaluated at g^{-1}(y).

Parameters
• yTensor. The input to the “inverse” Jacobian evaluation.

• name – The name to give this op.

Returns

Tensor, if this bijector is injective.

If not injective, returns the tuple of local log det Jacobians, log(det(Dg_i^{-1}(y))), where g_i is the restriction of g to the ith partition Di.

Raises
• TypeError – if self.dtype is specified and y.dtype is not self.dtype.

• NotImplementedError – if _inverse_log_det_jacobian is not implemented.

is_constant_jacobian

Returns true iff the Jacobian is not a function of x.

Note: Jacobian is either constant for both forward and inverse or neither.

Returns

Python bool.

Return type

is_constant_jacobian

name

Returns the string name of this Bijector.

validate_args

Returns True if Tensor arguments will be validated.