Radial Flow¶
The radial flow introduced in [REZENDE2015] with the parametrization used in [TRIPPE2018]. The flow was originally designed for variational inference and sampling. Therefore it doesn’t easily fit our use-case of density estimation. Since we especially need the inverse \(f^{-1}(x)\) of the flow to be easily computable, we invert it’s direction, defining it as a mapping from the transformed distribution \(p_1(x)\) to the base distribution \(p_0(x)\). Hence the flow is called InvertedRadialFlow in our implementation and the forward method is not implemented.
To ensure \(f(x)\) exists we have to constrain the parameters of the flow:
\(\alpha \geq 0\) needs to hold. Therefore we apply a softplus transformation to \(\alpha\)
\(\beta \geq -1\) needs to hold. We apply \(f(x) = \exp(x) - 1\) to \(\beta\) before assignment
Jacobian determinant:
-
class
cde.density_estimator.normalizing_flows.
InvertedRadialFlow
(params, n_dims, validate_args=False, name='InvertedRadialFlow')[source]¶ Implements a bijector x = y + (alpha * beta * (y - y_0)) / (alpha + abs(y - y_0)).
- Parameters
params – Tensor shape (?, n_dims+2). This will be split into the parameters alpha (?, 1), beta (?, 1), gamma (?, n_dims). Furthermore alpha will be constrained to assure the invertability of the flow
n_dims – The dimension of the distribution that will be transformed
name – The name to give this particular flow
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event_ndims
¶ Returns then number of event dimensions this bijector operates on.
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forward
(x)[source]¶ We don’t require sampling and it would be slow, therefore it is not implemented
- Raises
NotImplementedError –
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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_shape – TensorShape 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
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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_shape – Tensor, 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
x – Tensor. 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.
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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
-
graph_parents
¶ Returns this Bijector’s graph_parents as a Python list.
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inverse
(y, name='inverse')¶ Returns the inverse Bijector evaluation, i.e., X = g^{-1}(Y).
- Parameters
y – Tensor. 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.
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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_shape – TensorShape 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_shape – Tensor, 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
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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
y – Tensor. 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
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name
¶ Returns the string name of this Bijector.
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validate_args
¶ Returns True if Tensor arguments will be validated.
- REZENDE2015
Rezende, Mohamed (2015). Variational Inference with Normalizing Flows (http://arxiv.org/abs/1505.05770)
- TRIPPE2018
Trippe, Turner (2018). Conditional Density Estimation with Bayesian Normalising Flows (http://arxiv.org/abs/1802.04908)