def rmspe(labels, predictions, weights=None):
if context.executing_eagerly():
raise RuntimeError('rmspe is not supported '
'when eager execution is enabled.')
predictions, labels, weights = metrics_impl._remove_squeezable_dimensions(
predictions=predictions, labels=labels, weights=weights)
# The target has been take log1p, so take expm1 back
labels, predictions = math_ops.expm1(labels), math_ops.expm1(
predictions)
mspe, update_op = metrics_impl.mean(
math_ops.square((labels - predictions) / labels), weights)
rmspe = math_ops.sqrt(mspe)
rmspe_update_op = math_ops.sqrt(update_op)
return rmspe, rmspe_update_op
def pearson(logu, name=None):
"""The Pearson Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
The Pearson Csiszar-function is:
```none
f(u) = (u - 1)**2
```
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: `float`-like `Tensor` representing `log(u)` from above.
name: Python `str` name prefixed to Ops created by this function.
Returns:
pearson_of_u: `float`-like `Tensor` of the Csiszar-function evaluated at
`u = exp(logu)`.
"""
with ops.name_scope(name, "pearson", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
return math_ops.square(math_ops.expm1(logu))
def total_variation(logu, name=None):
"""The Total Variation Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
The Total-Variation Csiszar-function is:
```none
f(u) = 0.5 |u - 1|
```
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: `float`-like `Tensor` representing `log(u)` from above.
name: Python `str` name prefixed to Ops created by this function.
Returns:
total_variation_of_u: `float`-like `Tensor` of the Csiszar-function
evaluated at `u = exp(logu)`.
"""
with ops.name_scope(name, "total_variation", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
return 0.5 * math_ops.abs(math_ops.expm1(logu))
def pearson(logu, name=None):
"""The Pearson Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
The Pearson Csiszar-function is:
```none
f(u) = (u - 1)**2
```
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: Floating-type `Tensor` representing `log(u)` from above.
name: Python `str` name prefixed to Ops created by this function.
Returns:
pearson_of_u: Floating-type `Tensor` of the Csiszar-function evaluated at
`u = exp(logu)`.
"""
with ops.name_scope(name, "pearson", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
return math_ops.square(math_ops.expm1(logu))
def _inverse(self, y):
y = self._maybe_assert_valid_y(y)
if self.power == 0.:
return math_ops.log(y)
# If large y accuracy is an issue, consider using:
# (y**self.power - 1.) / self.power when y >> 1.
return math_ops.expm1(math_ops.log(y) * self.power) / self.power
def total_variation(logu, name=None):
"""The Total Variation Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
The Total-Variation Csiszar-function is:
```none
f(u) = 0.5 |u - 1|
```
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: Floating-type `Tensor` representing `log(u)` from above.
name: Python `str` name prefixed to Ops created by this function.
Returns:
total_variation_of_u: Floating-type `Tensor` of the Csiszar-function
evaluated at `u = exp(logu)`.
"""
with ops.name_scope(name, "total_variation", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
return 0.5 * math_ops.abs(math_ops.expm1(logu))
def _inverse(self, y):
y = self._maybe_assert_valid_y(y)
if self.power == 0.:
return math_ops.log(y)
# If large y accuracy is an issue, consider using:
# (y**self.power - 1.) / self.power when y >> 1.
return math_ops.expm1(math_ops.log(y) * self.power) / self.power
def chi_square(logu, name=None):
"""The chi-Square Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
The Chi-square Csiszar-function is:
```none
f(u) = u**2 - 1
```
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: Floating-type `Tensor` representing `log(u)` from above.
name: Python `str` name prefixed to Ops created by this function.
Returns:
chi_square_of_u: Floating-type `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
"""
with ops.name_scope(name, "chi_square", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
return math_ops.expm1(2. * logu)
def chi_square(logu, name=None):
"""The chi-Square Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
The Chi-square Csiszar-function is:
```none
f(u) = u**2 - 1
```
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: `float`-like `Tensor` representing `log(u)` from above.
name: Python `str` name prefixed to Ops created by this function.
Returns:
chi_square_of_u: `float`-like `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
"""
with ops.name_scope(name, "chi_square", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
return math_ops.expm1(2. * logu)
def t_power(logu, t, self_normalized=False, name=None):
"""The T-Power Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
When `self_normalized = True` the T-Power Csiszar-function is:
```none
f(u) = s [ u**t - 1 - t(u - 1) ]
s = { -1 0 < t < 1
{ +1 otherwise
```
When `self_normalized = False` the `- t(u - 1)` term is omitted.
This is similar to the `amari_alpha` Csiszar-function, with the associated
divergence being the same up to factors depending only on `t`.
Args:
logu: `float`-like `Tensor` representing `log(u)` from above.
t: `Tensor` of same `dtype` as `logu` and broadcastable shape.
self_normalized: Python `bool` indicating whether `f'(u=1)=0`.
name: Python `str` name prefixed to Ops created by this function.
Returns:
t_power_of_u: `float`-like `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
"""
with ops.name_scope(name, "t_power", [logu, t]):
logu = ops.convert_to_tensor(logu, name="logu")
t = ops.convert_to_tensor(t, dtype=logu.dtype.base_dtype, name="t")
fu = math_ops.expm1(t * logu)
if self_normalized:
fu -= t * math_ops.expm1(logu)
fu *= array_ops.where(math_ops.logical_and(0. < t, t < 1.),
-array_ops.ones_like(t),
array_ops.ones_like(t))
return fu
def _cdf(self, x):
if self.validate_args:
x = distribution_util.embed_check_nonnegative_integer_form(x)
else:
# Whether or not x is integer-form, the following is well-defined.
# However, scipy takes the floor, so we do too.
x = math_ops.floor(x)
x *= array_ops.ones_like(self.probs)
return array_ops.where(
x < 0., array_ops.zeros_like(x), -math_ops.expm1(
(1. + x) * math_ops.log1p(-self.probs)))
def _cdf(self, counts):
if self.validate_args:
# We set `check_integer=False` since the CDF is defined on whole real
# line.
counts = math_ops.floor(
distribution_util.embed_check_nonnegative_discrete(
counts, check_integer=False))
counts *= array_ops.ones_like(self.probs)
return array_ops.where(
counts < 0., array_ops.zeros_like(counts), -math_ops.expm1(
(counts + 1) * math_ops.log1p(-self.probs)))
def t_power(logu, t, self_normalized=False, name=None):
"""The T-Power Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
When `self_normalized = True` the T-Power Csiszar-function is:
```none
f(u) = s [ u**t - 1 - t(u - 1) ]
s = { -1 0 < t < 1
{ +1 otherwise
```
When `self_normalized = False` the `- t(u - 1)` term is omitted.
This is similar to the `amari_alpha` Csiszar-function, with the associated
divergence being the same up to factors depending only on `t`.
Args:
logu: `float`-like `Tensor` representing `log(u)` from above.
t: `Tensor` of same `dtype` as `logu` and broadcastable shape.
self_normalized: Python `bool` indicating whether `f'(u=1)=0`.
name: Python `str` name prefixed to Ops created by this function.
Returns:
t_power_of_u: `float`-like `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
"""
with ops.name_scope(name, "t_power", [logu, t]):
logu = ops.convert_to_tensor(logu, name="logu")
t = ops.convert_to_tensor(t, dtype=logu.dtype.base_dtype, name="t")
fu = math_ops.expm1(t * logu)
if self_normalized:
fu -= t * math_ops.expm1(logu)
fu *= array_ops.where(math_ops.logical_and(0. < t, t < 1.),
-array_ops.ones_like(t), array_ops.ones_like(t))
return fu
def _cdf(self, x):
if self.validate_args:
x = distribution_util.embed_check_nonnegative_integer_form(x)
else:
# Whether or not x is integer-form, the following is well-defined.
# However, scipy takes the floor, so we do too.
x = math_ops.floor(x)
x *= array_ops.ones_like(self.probs)
return array_ops.where(
x < 0.,
array_ops.zeros_like(x),
-math_ops.expm1((1. + x) * math_ops.log1p(-self.probs)))
def softplus_inverse(x, name=None):
"""Computes the inverse softplus, i.e., x = softplus_inverse(softplus(x)).
Mathematically this op is equivalent to:
```none
softplus_inverse = log(exp(x) - 1.)
```
Args:
x: `Tensor`. Non-negative (not enforced), floating-point.
name: A name for the operation (optional).
Returns:
`Tensor`. Has the same type/shape as input `x`.
"""
with ops.name_scope(name, "softplus_inverse", values=[x]):
x = ops.convert_to_tensor(x, name="x")
# We begin by deriving a more numerically stable softplus_inverse:
# x = softplus(y) = Log[1 + exp{y}], (which means x > 0).
# ==> exp{x} = 1 + exp{y} (1)
# ==> y = Log[exp{x} - 1] (2)
# = Log[(exp{x} - 1) / exp{x}] + Log[exp{x}]
# = Log[(1 - exp{-x}) / 1] + Log[exp{x}]
# = Log[1 - exp{-x}] + x (3)
# (2) is the "obvious" inverse, but (3) is more stable than (2) for large x.
# For small x (e.g. x = 1e-10), (3) will become -inf since 1 - exp{-x} will
# be zero. To fix this, we use 1 - exp{-x} approx x for small x > 0.
#
# In addition to the numerically stable derivation above, we clamp
# small/large values to be congruent with the logic in:
# tensorflow/core/kernels/softplus_op.h
#
# Finally, we set the input to one whenever the input is too large or too
# small. This ensures that no unchosen codepath is +/- inf. This is
# necessary to ensure the gradient doesn't get NaNs. Recall that the
# gradient of `where` behaves like `pred*pred_true + (1-pred)*pred_false`
# thus an `inf` in an unselected path results in `0*inf=nan`. We are careful
# to overwrite `x` with ones only when we will never actually use this
# value. Note that we use ones and not zeros since `log(expm1(0.)) = -inf`.
threshold = np.log(np.finfo(x.dtype.as_numpy_dtype).eps) + 2.
is_too_small = math_ops.less(x, np.exp(threshold))
is_too_large = math_ops.greater(x, -threshold)
too_small_value = math_ops.log(x)
too_large_value = x
# This `where` will ultimately be a NOP because we won't select this
# codepath whenever we used the surrogate `ones_like`.
x = array_ops.where(math_ops.logical_or(is_too_small, is_too_large),
array_ops.ones_like(x), x)
y = x + math_ops.log(-math_ops.expm1(-x)) # == log(expm1(x))
return array_ops.where(
is_too_small, too_small_value,
array_ops.where(is_too_large, too_large_value, y))
def _inverse_log_det_jacobian(self, y):
# Could also do:
# ildj = math_ops.reduce_sum(y - distribution_util.softplus_inverse(y),
# axis=event_dims)
# but the following is more numerically stable. Ie,
# Y = Log[1 + exp{X}] ==> X = Log[exp{Y} - 1]
# ==> dX/dY = exp{Y} / (exp{Y} - 1)
# = 1 / (1 - exp{-Y}),
# which is the most stable for large Y > 0. For small Y, we use
# 1 - exp{-Y} approx Y.
return -math_ops.reduce_sum(math_ops.log(-math_ops.expm1(-y)),
axis=self._event_dims_tensor(y))
def softplus_inverse(x, name=None):
"""Computes the inverse softplus, i.e., x = softplus_inverse(softplus(x)).
Mathematically this op is equivalent to:
```none
softplus_inverse = log(exp(x) - 1.)
```
Args:
x: `Tensor`. Non-negative (not enforced), floating-point.
name: A name for the operation (optional).
Returns:
`Tensor`. Has the same type/shape as input `x`.
"""
with ops.name_scope(name, "softplus_inverse", values=[x]):
x = ops.convert_to_tensor(x, name="x")
# We begin by deriving a more numerically stable softplus_inverse:
# x = softplus(y) = Log[1 + exp{y}], (which means x > 0).
# ==> exp{x} = 1 + exp{y} (1)
# ==> y = Log[exp{x} - 1] (2)
# = Log[(exp{x} - 1) / exp{x}] + Log[exp{x}]
# = Log[(1 - exp{-x}) / 1] + Log[exp{x}]
# = Log[1 - exp{-x}] + x (3)
# (2) is the "obvious" inverse, but (3) is more stable than (2) for large x.
# For small x (e.g. x = 1e-10), (3) will become -inf since 1 - exp{-x} will
# be zero. To fix this, we use 1 - exp{-x} approx x for small x > 0.
#
# In addition to the numerically stable derivation above, we clamp
# small/large values to be congruent with the logic in:
# tensorflow/core/kernels/softplus_op.h
#
# Finally, we set the input to one whenever the input is too large or too
# small. This ensures that no unchosen codepath is +/- inf. This is
# necessary to ensure the gradient doesn't get NaNs. Recall that the
# gradient of `where` behaves like `pred*pred_true + (1-pred)*pred_false`
# thus an `inf` in an unselected path results in `0*inf=nan`. We are careful
# to overwrite `x` with ones only when we will never actually use this
# value. Note that we use ones and not zeros since `log(expm1(0.)) = -inf`.
threshold = np.log(np.finfo(x.dtype.as_numpy_dtype).eps) + 2.
is_too_small = math_ops.less(x, np.exp(threshold))
is_too_large = math_ops.greater(x, -threshold)
too_small_value = math_ops.log(x)
too_large_value = x
# This `where` will ultimately be a NOP because we won't select this
# codepath whenever we used the surrogate `ones_like`.
x = array_ops.where(math_ops.logical_or(is_too_small, is_too_large),
array_ops.ones_like(x), x)
y = x + math_ops.log(-math_ops.expm1(-x)) # == log(expm1(x))
return array_ops.where(is_too_small, too_small_value,
array_ops.where(is_too_large, too_large_value, y))
def _inverse_and_inverse_log_det_jacobian(self, y):
y = self._maybe_assert_valid_y(y)
event_dims = self._event_dims_tensor(y)
if self.power == 0.:
x = math_ops.log(y)
ildj = -math_ops.reduce_sum(x, axis=event_dims)
return x, ildj
# TODO(jvdillon): If large y accuracy is an issue, consider using
# (y**self.power - 1.) / self.power when y >> 1.
x = math_ops.expm1(math_ops.log(y) * self.power) / self.power
ildj = (self.power - 1.) * math_ops.reduce_sum(math_ops.log(y),
axis=event_dims)
return x, ildj
def _cdf(self, counts):
if self.validate_args:
# We set `check_integer=False` since the CDF is defined on whole real
# line.
counts = math_ops.floor(
distribution_util.embed_check_nonnegative_discrete(
counts, check_integer=False))
counts *= array_ops.ones_like(self.probs)
return array_ops.where(
counts < 0.,
array_ops.zeros_like(counts),
-math_ops.expm1(
(counts + 1) * math_ops.log1p(-self.probs)))
def modified_gan(logu, self_normalized=False, name=None):
"""The Modified-GAN Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
When `self_normalized = True` the modified-GAN (Generative/Adversarial
Network) Csiszar-function is:
```none
f(u) = log(1 + u) - log(u) + 0.5 (u - 1)
```
When `self_normalized = False` the `0.5 (u - 1)` is omitted.
The unmodified GAN Csiszar-function is identical to Jensen-Shannon (with
`self_normalized = False`).
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: Floating-type `Tensor` representing `log(u)` from above.
self_normalized: Python `bool` indicating whether `f'(u=1)=0`. When
`f'(u=1)=0` the implied Csiszar f-Divergence remains non-negative even
when `p, q` are unnormalized measures.
name: Python `str` name prefixed to Ops created by this function.
Returns:
chi_square_of_u: Floating-type `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
"""
with ops.name_scope(name, "chi_square", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
y = nn_ops.softplus(logu) - logu
if self_normalized:
y += 0.5 * math_ops.expm1(logu)
return y
def log1p_abs(logu, name=None):
"""The log1p-abs Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
The Log1p-Abs Csiszar-function is:
```none
f(u) = u**(sign(u-1)) - 1
```
This function is so-named because it was invented from the following recipe.
Choose a convex function g such that g(0)=0 and solve for f:
```none
log(1 + f(u)) = g(log(u)).
<=>
f(u) = exp(g(log(u))) - 1
```
That is, the graph is identically `g` when y-axis is `log1p`-domain and x-axis
is `log`-domain.
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: Floating-type `Tensor` representing `log(u)` from above.
name: Python `str` name prefixed to Ops created by this function.
Returns:
log1p_abs_of_u: Floating-type `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
"""
with ops.name_scope(name, "log1p_abs", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
return math_ops.expm1(math_ops.abs(logu))
def log1p_abs(logu, name=None):
"""The log1p-abs Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
The Log1p-Abs Csiszar-function is:
```none
f(u) = u**(sign(u-1)) - 1
```
This function is so-named because it was invented from the following recipe.
Choose a convex function g such that g(0)=0 and solve for f:
```none
log(1 + f(u)) = g(log(u)).
<=>
f(u) = exp(g(log(u))) - 1
```
That is, the graph is identically `g` when y-axis is `log1p`-domain and x-axis
is `log`-domain.
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: `float`-like `Tensor` representing `log(u)` from above.
name: Python `str` name prefixed to Ops created by this function.
Returns:
log1p_abs_of_u: `float`-like `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
"""
with ops.name_scope(name, "log1p_abs", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
return math_ops.expm1(math_ops.abs(logu))
def modified_gan(logu, self_normalized=False, name=None):
"""The Modified-GAN Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
When `self_normalized = True` the modified-GAN (Generative/Adversarial
Network) Csiszar-function is:
```none
f(u) = log(1 + u) - log(u) + 0.5 (u - 1)
```
When `self_normalized = False` the `0.5 (u - 1)` is omitted.
The unmodified GAN Csiszar-function is identical to Jensen-Shannon (with
`self_normalized = False`).
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: `float`-like `Tensor` representing `log(u)` from above.
self_normalized: Python `bool` indicating whether `f'(u=1)=0`. When
`f'(u=1)=0` the implied Csiszar f-Divergence remains non-negative even
when `p, q` are unnormalized measures.
name: Python `str` name prefixed to Ops created by this function.
Returns:
chi_square_of_u: `float`-like `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
"""
with ops.name_scope(name, "chi_square", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
y = nn_ops.softplus(logu) - logu
if self_normalized:
y += 0.5 * math_ops.expm1(logu)
return y
def jeffreys(logu, name=None):
"""The Jeffreys Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
The Jeffreys Csiszar-function is:
```none
f(u) = 0.5 ( u log(u) - log(u) )
= 0.5 kl_forward + 0.5 kl_reverse
= symmetrized_csiszar_function(kl_reverse)
= symmetrized_csiszar_function(kl_forward)
```
This Csiszar-function induces a symmetric f-Divergence, i.e.,
`D_f[p, q] = D_f[q, p]`.
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: Floating-type `Tensor` representing `log(u)` from above.
name: Python `str` name prefixed to Ops created by this function.
Returns:
jeffreys_of_u: Floating-type `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
"""
with ops.name_scope(name, "jeffreys", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
return 0.5 * math_ops.expm1(logu) * logu
def jeffreys(logu, name=None):
"""The Jeffreys Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
The Jeffreys Csiszar-function is:
```none
f(u) = 0.5 ( u log(u) - log(u) )
= 0.5 kl_forward + 0.5 kl_reverse
= symmetrized_csiszar_function(kl_reverse)
= symmetrized_csiszar_function(kl_forward)
```
This Csiszar-function induces a symmetric f-Divergence, i.e.,
`D_f[p, q] = D_f[q, p]`.
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for `|logu| >> 0`.
Args:
logu: `float`-like `Tensor` representing `log(u)` from above.
name: Python `str` name prefixed to Ops created by this function.
Returns:
jeffreys_of_u: `float`-like `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
"""
with ops.name_scope(name, "jeffreys", [logu]):
logu = ops.convert_to_tensor(logu, name="logu")
return 0.5 * math_ops.expm1(logu) * logu
def amari_alpha(logu, alpha=1., self_normalized=False, name=None):
"""The Amari-alpha Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
When `self_normalized = True`, the Amari-alpha Csiszar-function is:
```none
f(u) = { -log(u) + (u - 1), alpha = 0
{ u log(u) - (u - 1), alpha = 1
{ [(u**alpha - 1) - alpha (u - 1)] / (alpha (alpha - 1)), otherwise
```
When `self_normalized = False` the `(u - 1)` terms are omitted.
Warning: when `alpha != 0` and/or `self_normalized = True` this function makes
non-log-space calculations and may therefore be numerically unstable for
`|logu| >> 0`.
For more information, see:
A. Cichocki and S. Amari. "Families of Alpha-Beta-and GammaDivergences:
Flexible and Robust Measures of Similarities." Entropy, vol. 12, no. 6, pp.
1532-1568, 2010.
Args:
logu: Floating-type `Tensor` representing `log(u)` from above.
alpha: Floating-type Python scalar. (See Mathematical Details for meaning.)
self_normalized: Python `bool` indicating whether `f'(u=1)=0`. When
`f'(u=1)=0` the implied Csiszar f-Divergence remains non-negative even
when `p, q` are unnormalized measures.
name: Python `str` name prefixed to Ops created by this function.
Returns:
amari_alpha_of_u: Floating-type `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
Raises:
TypeError: if `alpha` is `None` or a `Tensor`.
TypeError: if `self_normalized` is `None` or a `Tensor`.
"""
with ops.name_scope(name, "amari_alpha", [logu]):
if alpha is None or contrib_framework.is_tensor(alpha):
raise TypeError("`alpha` cannot be `None` or `Tensor` type.")
if self_normalized is None or contrib_framework.is_tensor(self_normalized):
raise TypeError("`self_normalized` cannot be `None` or `Tensor` type.")
logu = ops.convert_to_tensor(logu, name="logu")
if alpha == 0.:
f = -logu
elif alpha == 1.:
f = math_ops.exp(logu) * logu
else:
f = math_ops.expm1(alpha * logu) / (alpha * (alpha - 1.))
if not self_normalized:
return f
if alpha == 0.:
return f + math_ops.expm1(logu)
elif alpha == 1.:
return f - math_ops.expm1(logu)
else:
return f - math_ops.expm1(logu) / (alpha - 1.)
def _forward(self, x): x = self._maybe_assert_valid_x(x) return -math_ops.expm1(-((x / self.scale) ** self.concentration))
def _forward(self, x): x = self._maybe_assert_valid_x(x) return -math_ops.expm1(-((x / self.scale) ** self.concentration))
def amari_alpha(logu, alpha=1., self_normalized=False, name=None):
"""The Amari-alpha Csiszar-function in log-space.
A Csiszar-function is a member of,
```none
F = { f:R_+ to R : f convex }.
```
When `self_normalized = True`, the Amari-alpha Csiszar-function is:
```none
f(u) = { -log(u) + (u - 1), alpha = 0
{ u log(u) - (u - 1), alpha = 1
{ [(u**alpha - 1) - alpha (u - 1)] / (alpha (alpha - 1)), otherwise
```
When `self_normalized = False` the `(u - 1)` terms are omitted.
Warning: when `alpha != 0` and/or `self_normalized = True` this function makes
non-log-space calculations and may therefore be numerically unstable for
`|logu| >> 0`.
For more information, see:
A. Cichocki and S. Amari. "Families of Alpha-Beta-and GammaDivergences:
Flexible and Robust Measures of Similarities." Entropy, vol. 12, no. 6, pp.
1532-1568, 2010.
Args:
logu: `float`-like `Tensor` representing `log(u)` from above.
alpha: `float`-like Python scalar. (See Mathematical Details for meaning.)
self_normalized: Python `bool` indicating whether `f'(u=1)=0`. When
`f'(u=1)=0` the implied Csiszar f-Divergence remains non-negative even
when `p, q` are unnormalized measures.
name: Python `str` name prefixed to Ops created by this function.
Returns:
amari_alpha_of_u: `float`-like `Tensor` of the Csiszar-function evaluated
at `u = exp(logu)`.
Raises:
TypeError: if `alpha` is `None` or a `Tensor`.
TypeError: if `self_normalized` is `None` or a `Tensor`.
"""
with ops.name_scope(name, "amari_alpha", [logu]):
if alpha is None or contrib_framework.is_tensor(alpha):
raise TypeError("`alpha` cannot be `None` or `Tensor` type.")
if self_normalized is None or contrib_framework.is_tensor(
self_normalized):
raise TypeError(
"`self_normalized` cannot be `None` or `Tensor` type.")
logu = ops.convert_to_tensor(logu, name="logu")
if alpha == 0.:
f = -logu
elif alpha == 1.:
f = math_ops.exp(logu) * logu
else:
f = math_ops.expm1(alpha * logu) / (alpha * (alpha - 1.))
if not self_normalized:
return f
if alpha == 0.:
return f + math_ops.expm1(logu)
elif alpha == 1.:
return f - math_ops.expm1(logu)
else:
return f - math_ops.expm1(logu) / (alpha - 1.)
