tf.contrib.distributions.bijectors.BatchNormalization

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Compute `Y = g(X) s.t.

Inherits From: Bijector

tf.contrib.distributions.bijectors.BatchNormalization(
    batchnorm_layer=None, training=True, validate_args=False,
    name='batch_normalization'
)

X = g^-1(Y) = (Y - mean(Y)) / std(Y)`.

Applies Batch Normalization [(Ioffe and Szegedy, 2015)][1] to samples from a data distribution. This can be used to stabilize training of normalizing flows ([Papamakarios et al., 2016][3]; [Dinh et al., 2017][2])

When training Deep Neural Networks (DNNs), it is common practice to normalize or whiten features by shifting them to have zero mean and scaling them to have unit variance.

The inverse() method of the BatchNormalization bijector, which is used in the log-likelihood computation of data samples, implements the normalization procedure (shift-and-scale) using the mean and standard deviation of the current minibatch.

Conversely, the forward() method of the bijector de-normalizes samples (e.g. X*std(Y) + mean(Y) with the running-average mean and standard deviation computed at training-time. De-normalization is useful for sampling.


dist = tfd.TransformedDistribution(
    distribution=tfd.Normal()),
    bijector=tfb.BatchNorm())

y = tfd.MultivariateNormalDiag(loc=1., scale=2.).sample(100)  # ~ N(1, 2)
x = dist.bijector.inverse(y)  # ~ N(0, 1)
y = dist.sample()  # ~ N(1, 2)

During training time, BatchNorm.inverse and BatchNorm.forward are not guaranteed to be inverses of each other because inverse(y) uses statistics of the current minibatch, while forward(x) uses running-average statistics accumulated from training. In other words, BatchNorm.inverse(BatchNorm.forward(...)) and BatchNorm.forward(BatchNorm.inverse(...)) will be identical when training=False but may be different when training=True.

References

[1]: Sergey Ioffe and Christian Szegedy. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. In International Conference on Machine Learning, 2015. https://arxiv.org/abs/1502.03167

[2]: Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density Estimation using Real NVP. In International Conference on Learning Representations, 2017. https://arxiv.org/abs/1605.08803

[3]: George Papamakarios, Theo Pavlakou, and Iain Murray. Masked Autoregressive Flow for Density Estimation. In Neural Information Processing Systems, 2017. https://arxiv.org/abs/1705.07057

batchnorm_layer tf.compat.v1.layers.BatchNormalization layer object. If None, defaults to `tf.compat.v1.layers.BatchNormalization(gamma_constraint=nn_ops.relu(x)

  • 1e-6). This ensures positivity of the scale variable. </td> </tr><tr> <td>training</td> <td> If True, updates running-average statistics during call toinverse(). </td> </tr><tr> <td>validate_args</td> <td> Pythonboolindicating whether arguments should be checked for correctness. </td> </tr><tr> <td>name</td> <td> Pythonstr` name given to ops managed by this object.

ValueError If bn_layer is not an instance of tf.compat.v1.layers.BatchNormalization, or if it is specified with renorm=True or a virtual batch size.

dtype dtype of Tensors transformable by this distribution.
forward_min_event_ndims Returns the minimal number of dimensions bijector.forward operates on.
graph_parents Returns this Bijector's graph_parents as a Python list.
inverse_min_event_ndims Returns the minimal number of dimensions bijector.inverse operates on.
is_constant_jacobian Returns true iff the Jacobian matrix is not a function of x.
name Returns the string name of this Bijector.
validate_args Returns True if Tensor arguments will be validated.

Methods

forward

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forward(
    x, name='forward'
)

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

Args
x Tensor. 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

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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.

Args
input_shape TensorShape indicating event-portion shape passed into forward function.

Returns
forward_event_shape_tensor TensorShape indicating event-portion shape after applying forward. Possibly unknown.

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.

Args
input_shape Tensor, int32 vector indicating event-portion shape passed into forward function.
name name to give to the op

Returns
forward_event_shape_tensor Tensor, int32 vector indicating event-portion shape after applying forward.

forward_log_det_jacobian

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forward_log_det_jacobian(
    x, event_ndims, name='forward_log_det_jacobian'
)

Returns both the forward_log_det_jacobian.

Args
x Tensor. The input to the "forward" Jacobian determinant evaluation.
event_ndims Number of dimensions in the probabilistic events being transformed. Must be greater than or equal to self.forward_min_event_ndims. The result is summed over the final dimensions to produce a scalar Jacobian determinant for each event, i.e. it has shape x.shape.ndims - event_ndims dimensions.
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.

inverse

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inverse(
    y, name='inverse'
)

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

Args
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.

inverse_event_shape

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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.

Args
output_shape TensorShape indicating event-portion shape passed into inverse function.

Returns
inverse_event_shape_tensor TensorShape indicating event-portion shape after applying inverse. Possibly unknown.

inverse_event_shape_tensor

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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.

Args
output_shape Tensor, int32 vector indicating event-portion shape passed into inverse function.
name name to give to the op

Returns
inverse_event_shape_tensor Tensor, int32 vector indicating event-portion shape after applying inverse.

inverse_log_det_jacobian

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inverse_log_det_jacobian(
    y, event_ndims, 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).

Args
y Tensor. The input to the "inverse" Jacobian determinant evaluation.
event_ndims Number of dimensions in the probabilistic events being transformed. Must be greater than or equal to self.inverse_min_event_ndims. The result is summed over the final dimensions to produce a scalar Jacobian determinant for each event, i.e. it has shape y.shape.ndims - event_ndims dimensions.
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.