Given x, compute a differentiable approximation to tf.math.floor(x).
It is parameterized by a temperature parameter t to control the closeness
of the approximation at the cost of numerical stability of the inverse.
This Bijector has the following properties:
This Bijector is a map between R to R.
For t close to 0, this bijector mimics the identity function.
For t approaching infinity, this bijector converges pointwise
to tf.math.floor (except at integer points).
Note that for lower temperatures t, this bijector becomes more numerically
unstable. In particular, the inverse for this bijector is not numerically
stable at lower temperatures, because flooring is not a bijective function (
and hence any pointwise limit towards the floor function will start to have a
non-numerically stable inverse).
Mathematical details
Let x be in [0.5, 1.5]. We would like to simulate the floor function on
this interval. We will do this via a shifted and rescaled sigmoid.
floor(x) = 0 for x < 1 and floor(x) = 1 for x >= 1.
If we take f(x) = sigmoid((x - 1.) / t), where t > 0, we can see that
when t goes to zero, we get that when x > 1, the f(x) tends towards 1
while f(x) tends to 0 when x < 1, thus giving us a function that looks
like the floor function. If we shift f(x) by -sigmoid(-0.5 / t) and
rescale by 1 / (sigmoid(0.5 / t) - sigmoid(-0.5 / t)), we preserve the
pointwise limit, but also fix f(0.5) = 0. and f(1.5) = 1..
Thus we can define softfloor(x, t) = a * sigmoid((x - 1.) / t) + b
The implementation of the Softfloor bijector follows this, with the caveat
that we extend the function to all of the real line, by appropriately shifting
this function for each integer.
Examples
Example use:
# High temperature.
soft_floor = Softfloor(temperature=100.)
x = [2.1, 3.2, 5.5]
soft_floor.forward(x)
# Low temperature. This acts like a floor.
soft_floor = Softfloor(temperature=0.01)
soft_floor.forward(x) # Should be close to [2., 3., 5.]
# Ceiling is just a shifted floor at non-integer points.
soft_ceiling = tfb.Chain(
[tfb.AffineScalar(1.),
tfb.Softfloor(temperature=1.)])
soft_ceiling.forward(x) # Should be close to [3., 5., 6.]
Args
graph_parents
Python list of graph prerequisites of this Bijector.
is_constant_jacobian
Python bool indicating that the Jacobian matrix is
not a function of the input.
validate_args
Python bool, default False. Whether to validate input
with asserts. If validate_args is False, and the inputs are invalid,
correct behavior is not guaranteed.
dtype
tf.dtype supported by this Bijector. None means dtype is not
enforced.
forward_min_event_ndims
Python integer indicating the minimum number of
dimensions forward operates on.
inverse_min_event_ndims
Python integer indicating the minimum number of
dimensions inverse operates on. Will be set to
forward_min_event_ndims by default, if no value is provided.
parameters
Python dict of parameters used to instantiate this
Bijector.
name
The name to give Ops created by the initializer.
Raises
ValueError
If neither forward_min_event_ndims and
inverse_min_event_ndims are specified, or if either of them is
negative.
ValueError
If a member of graph_parents is not a Tensor.
Attributes
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.
parameters
Dictionary of parameters used to instantiate this Bijector.
temperature
trainable_variables
validate_args
Returns True if Tensor arguments will be validated.
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 rank(x) - event_ndims dimensions.
name
The name to give this op.
**kwargs
Named arguments forwarded to subclass implementation.
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.
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 rank(y) - event_ndims dimensions.
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 self.dtype is specified and y.dtype is not
self.dtype.
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