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tfp.substrates.jax.bijectors.ScaleMatvecLU

Matrix-vector multiply using LU decomposition.

Inherits From: Bijector

tfp.substrates.jax.bijectors.ScaleMatvecLU(
    lower_upper, permutation, validate_args=False, name=None
)

This bijector is identical to the 'Convolution1x1' used in Glow [(Kingma and Dhariwal, 2018)[1].

Examples

Here's an example of initialization via random weights matrix:

def trainable_lu_factorization(
    event_size, batch_shape=(), seed=None, dtype=tf.float32, name=None):
  with tf.name_scope(name or 'trainable_lu_factorization'):
    event_size = tf.convert_to_tensor(
        event_size, dtype_hint=tf.int32, name='event_size')
    batch_shape = tf.convert_to_tensor(
        batch_shape, dtype_hint=event_size.dtype, name='batch_shape')
    random_matrix = tf.random.stateless_uniform(
        shape=tf.concat([batch_shape, [event_size, event_size]], axis=0),
        dtype=dtype,
        seed=seed)
    random_orthonormal = tf.linalg.qr(random_matrix)[0]
    lower_upper, permutation = tf.linalg.lu(random_orthonormal)
    lower_upper = tf.Variable(
        initial_value=lower_upper,
        trainable=True,
        name='lower_upper')
    # Initialize a non-trainable variable for the permutation indices so
    # that its value isn't re-sampled from run-to-run.
    permutation = tf.Variable(
        initial_value=permutation,
        trainable=False,
        name='permutation')
    return lower_upper, permutation

channels = 3
conv1x1 = tfb.ScaleMatvecLU(*trainable_lu_factorization(channels),
                            validate_args=True)
x = tf.random.stateless_uniform(shape=[2, 28, 28, channels])
fwd = conv1x1.forward(x)
rev_fwd = conv1x1.inverse(fwd)
# ==> x

To initialize this variable outside of TensorFlow, one can also use SciPy, e.g.,

def lu_factorized_random_orthonormal_matrix(channels, dtype=np.float32):
  random_matrix = np.random.rand(channels, channels).astype(dtype)
  lower_upper = scipy.linalg.qr(random_matrix)[0]
  permutation = scipy.linalg.lu(lower_upper, overwrite_a=True)[0]
  permutation = np.argmax(permutation, axis=-2)
  return lower_upper, permutation

References

[1]: Diederik P. Kingma, Prafulla Dhariwal. Glow: Generative Flow with Invertible 1x1 Convolutions. arXiv preprint arXiv:1807.03039, 2018. https://arxiv.org/abs/1807.03039

lower_upper The LU factorization as returned by tf.linalg.lu.
permutation The LU factorization permutation as returned by tf.linalg.lu.
validate_args Python bool indicating whether arguments should be checked for correctness. Default value: False.
name Python str name given to ops managed by this object. Default value: None (i.e., 'ScaleMatvecLU').

ValueError If both/neither channels and lower_upper/permutation are specified.

dtype

forward_min_event_ndims Returns the minimal number of dimensions bijector.forward operates on.

Multipart bijectors return structured ndims, which indicates the expected structure of their inputs. Some multipart bijectors, notably Composites, may return structures of None.

graph_parents Returns this Bijector's graph_parents as a Python list.
has_static_min_event_ndims Returns True if the bijector has statically-known min_event_ndims.
inverse_min_event_ndims Returns the minimal number of dimensions bijector.inverse operates on.

Multipart bijectors return structured event_ndims, which indicates the expected structure of their outputs. Some multipart bijectors, notably Composites, may return structures of None.

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

name Returns the string name of this Bijector.
parameters Dictionary of parameters used to instantiate this Bijector.
permutation

trainable_variables

validate_args Returns True if Tensor arguments will be validated.
variables

Methods

forward

View source

forward(
    x, name='forward', **kwargs
)

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

Args
x Tensor (structure). The input to the 'forward' evaluation.
name The name to give this op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
Tensor (structure).

Raises
TypeError if self.dtype is specified and x.dtype is not self.dtype.
NotImplementedError if _forward is not implemented.

forward_dtype

View source

forward_dtype(
    dtype=UNSPECIFIED, name='forward_dtype', **kwargs
)

Returns the dtype returned by forward for the provided input.

forward_event_ndims

View source

forward_event_ndims(
    event_ndims, **kwargs
)

Returns the number of event dimensions produced by forward.

forward_event_shape

View source

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 (structure) indicating event-portion shape passed into forward function.

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

forward_event_shape_tensor

View source

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 (structure) indicating event-portion shape passed into forward function.
name name to give to the op

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

forward_log_det_jacobian

View source

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

Returns both the forward_log_det_jacobian.

Args
x Tensor (structure). 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 rank(x) - event_ndims dimensions. Multipart bijectors require structured event_ndims, such that rank(y[i]) - rank(event_ndims[i]) is the same for all elements i of the structured input. Furthermore, the first event_ndims[i] of each x[i].shape must be the same for all i (broadcasting is not allowed).
name The name to give this op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
Tensor (structure), if this bijector is injective. If not injective this is not implemented.

Raises
TypeError if y's dtype is incompatible with the expected output 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', **kwargs
)

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

Args
y Tensor (structure). The input to the 'inverse' evaluation.
name The name to give this op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
Tensor (structure), 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 y's structured dtype is incompatible with the expected output dtype.
NotImplementedError if _inverse is not implemented.

inverse_dtype

View source

inverse_dtype(
    dtype=UNSPECIFIED, name='inverse_dtype', **kwargs
)

Returns the dtype returned by inverse for the provided input.

inverse_event_ndims

View source

inverse_event_ndims(
    event_ndims, **kwargs
)

Returns the number of event dimensions produced by inverse.

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 (structure) indicating event-portion shape passed into inverse function.

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

inverse_event_shape_tensor

View source

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 (structure) indicating event-portion shape passed into inverse function.
name name to give to the op

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

inverse_log_det_jacobian

View source

inverse_log_det_jacobian(
    y, event_ndims, name='inverse_log_det_jacobian', **kwargs
)

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 (structure). 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 rank(y) - event_ndims dimensions. Multipart bijectors require structured event_ndims, such that rank(y[i]) - rank(event_ndims[i]) is the same for all elements i of the structured input. Furthermore, the first event_ndims[i] of each x[i].shape must be the same for all i (broadcasting is not allowed).
name The name to give this op.
**kwargs Named arguments forwarded to subclass implementation.

Returns
ildj 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 x's dtype is incompatible with the expected inverse-dtype.
NotImplementedError if _inverse_log_det_jacobian is not implemented.

__call__

View source

__call__(
    value, name=None, **kwargs
)

Applies or composes the Bijector, depending on input type.

This is a convenience function which applies the Bijector instance in three different ways, depending on the input:

  1. If the input is a tfd.Distribution instance, return tfd.TransformedDistribution(distribution=input, bijector=self).
  2. If the input is a tfb.Bijector instance, return tfb.Chain([self, input]).
  3. Otherwise, return self.forward(input)

Args
value A tfd.Distribution, tfb.Bijector, or a (structure of) Tensor.
name Python str name given to ops created by this function.
**kwargs Additional keyword arguments passed into the created tfd.TransformedDistribution, tfb.Bijector, or self.forward.

Returns
composition A tfd.TransformedDistribution if the input was a tfd.Distribution, a tfb.Chain if the input was a tfb.Bijector, or a (structure of) Tensor computed by self.forward.

Examples

sigmoid = tfb.Reciprocal()(
    tfb.Shift(shift=1.)(
      tfb.Exp()(
        tfb.Scale(scale=-1.))))
# ==> `tfb.Chain([
#         tfb.Reciprocal(),
#         tfb.Shift(shift=1.),
#         tfb.Exp(),
#         tfb.Scale(scale=-1.),
#      ])`  # ie, `tfb.Sigmoid()`

log_normal = tfb.Exp()(tfd.Normal(0, 1))
# ==> `tfd.TransformedDistribution(tfd.Normal(0, 1), tfb.Exp())`

tfb.Exp()([-1., 0., 1.])
# ==> tf.exp([-1., 0., 1.])

__eq__

View source

__eq__(
    other
)

Return self==value.