def _sample_3d(self, n, seed=None):
"""Specialized inversion sampler for 3D."""
seed = SeedStream(seed, salt='von_mises_fisher_3d')
u_shape = tf.concat([[n], self._batch_shape_tensor()], axis=0)
z = tf.random.uniform(u_shape, seed=seed(), dtype=self.dtype)
# TODO(bjp): Higher-order odd dim analytic CDFs are available in [1], could
# be bisected for bounded sampling runtime (i.e. not rejection sampling).
# [1]: Inversion sampler via: https://ieeexplore.ieee.org/document/7347705/
# The inversion is: u = 1 + log(z + (1-z)*exp(-2*kappa)) / kappa
# We must protect against both kappa and z being zero.
safe_conc = tf.where(self.concentration > 0, self.concentration,
tf.ones_like(self.concentration))
safe_z = tf.where(z > 0, z, tf.ones_like(z))
safe_u = 1 + tf.reduce_logsumexp(
[tf.math.log(safe_z),
tf.math.log1p(-safe_z) - 2 * safe_conc],
axis=0) / safe_conc
# Limit of the above expression as kappa->0 is 2*z-1
u = tf.where(self.concentration > tf.zeros_like(safe_u), safe_u,
2 * z - 1)
# Limit of the expression as z->0 is -1.
u = tf.where(tf.equal(z, 0), -tf.ones_like(u), u)
if not self._allow_nan_stats:
u = tf.debugging.check_numerics(u, 'u in _sample_3d')
return u[..., tf.newaxis]
def _survival_function(self, y):
low = self._low
high = self._high
# Recall the promise:
# survival_function(y) := P[Y > y]
# = 0, if y >= high,
# = 1, if y < low,
# = P[X > y], otherwise.
# P[Y > j] = P[ceiling(Y) > j] since mass is only at integers, not in
# between.
j = tf.math.ceil(y)
# P[X > j], used when low < X < high.
result_so_far = self.distribution.survival_function(j)
# Re-define values at the cutoffs.
if low is not None:
result_so_far = tf.where(j < low, tf.ones_like(result_so_far),
result_so_far)
if high is not None:
result_so_far = tf.where(j >= high, tf.zeros_like(result_so_far),
result_so_far)
return result_so_far
def _variance(self):
df = tf.convert_to_tensor(self.df)
scale = tf.convert_to_tensor(self.scale)
# We need to put the tf.where inside the outer tf.where to ensure we never
# hit a NaN in the gradient.
denom = tf.where(df > 2., df - 2., tf.ones_like(df))
# Abs(scale) superfluous.
var = (tf.ones(self._batch_shape_tensor(df=df, scale=scale),
dtype=self.dtype)
* tf.square(scale) * df / denom)
# When 1 < df <= 2, variance is infinite.
result_where_defined = tf.where(
df > 2.,
var,
dtype_util.as_numpy_dtype(self.dtype)(np.inf))
if self.allow_nan_stats:
return tf.where(
df > 1.,
result_where_defined,
dtype_util.as_numpy_dtype(self.dtype)(np.nan))
else:
return distribution_util.with_dependencies([
assert_util.assert_less(
tf.ones([], dtype=self.dtype),
df,
message='variance not defined for components of df <= 1'),
], result_where_defined)
def assert_finite(x, data=None, summarize=None, message=None, name=None):
"""Assert all elements of `x` are finite.
Args:
x: Numeric `Tensor`.
data: The tensors to print out if the condition is False. Defaults to
error message and first few entries of `x`.
summarize: Print this many entries of each tensor.
message: A string to prefix to the default message.
name: A name for this operation (optional).
Defaults to "assert_finite".
Returns:
Op raising `InvalidArgumentError` unless `x` has specified rank or lower.
If static checks determine `x` has correct rank, a `no_op` is returned.
Raises:
ValueError: If static checks determine `x` has wrong rank.
"""
with tf.name_scope(name or 'assert_finite'):
x_ = tf.get_static_value(x)
if x_ is not None:
if ~np.all(np.isfinite(x_)):
raise ValueError(message)
return x
assertion = tf1.assert_equal(tf.math.is_finite(x),
tf.ones_like(x, tf.bool),
data=data,
summarize=summarize,
message=message)
with tf.control_dependencies([assertion]):
return tf.identity(x)
def _covariance(self):
# Derivation: https://sachinruk.github.io/blog/von-Mises-Fisher/
event_dim = tf.compat.dimension_value(self.event_shape[0])
if event_dim is None:
raise ValueError(
'event shape must be statically known for _bessel_ive')
# TODO(bjp): Enable this; numerically unstable.
if event_dim > 2:
raise ValueError(
'vMF covariance is numerically unstable for dim>2')
concentration = self.concentration[..., tf.newaxis]
safe_conc = tf.where(concentration > 0, concentration,
tf.ones_like(concentration))
h = (_bessel_ive(event_dim / 2, safe_conc) /
_bessel_ive(event_dim / 2 - 1, safe_conc))
intermediate = (
tf.matmul(self.mean_direction[..., :, tf.newaxis],
self.mean_direction[..., tf.newaxis, :]) *
(1 - event_dim * h / safe_conc - h**2)[..., tf.newaxis])
cov = tf.linalg.set_diag(
intermediate,
tf.linalg.diag_part(intermediate) + (h / safe_conc))
return tf.where(
concentration[..., tf.newaxis] > tf.zeros_like(cov), cov,
tf.linalg.eye(event_dim, batch_shape=self.batch_shape_tensor()) /
event_dim)
def _cdf(self, y):
low = self._low
high = self._high
# Recall the promise:
# cdf(y) := P[Y <= y]
# = 1, if y >= high,
# = 0, if y < low,
# = P[X <= y], otherwise.
# P[Y <= j] = P[floor(Y) <= j] since mass is only at integers, not in
# between.
j = tf.floor(y)
# P[X <= j], used when low < X < high.
result_so_far = self.distribution.cdf(j)
# Re-define values at the cutoffs.
if low is not None:
result_so_far = tf.where(j < low, tf.zeros_like(result_so_far),
result_so_far)
if high is not None:
result_so_far = tf.where(j >= high, tf.ones_like(result_so_far),
result_so_far)
return result_so_far
def _extract_log_probs(num_states, dist):
"""Tabulate log probabilities from a batch of distributions."""
states = tf.reshape(
tf.range(num_states),
tf.concat([[num_states],
tf.ones_like(dist.batch_shape_tensor())],
axis=0))
return distribution_util.move_dimension(dist.log_prob(states), 0, -1)
def _prob(self, x):
low = tf.convert_to_tensor(self.low)
high = tf.convert_to_tensor(self.high)
return tf.where(
tf.math.is_nan(x),
x,
tf.where(
# This > is only sound for continuous uniform
(x < low) | (x > high),
tf.zeros_like(x),
tf.ones_like(x) / self._range(low=low, high=high)))
def _expand_base_distribution_mean(self):
"""Ensures `self.distribution.mean()` has `[batch, event]` shape."""
single_draw_shape = concat_vectors(self.batch_shape_tensor(),
self.event_shape_tensor())
m = tf.reshape(
self.distribution.mean(), # A scalar.
shape=tf.ones_like(single_draw_shape, dtype=tf.int32))
m = tf.tile(m, multiples=single_draw_shape)
tensorshape_util.set_shape(
m, tensorshape_util.concatenate(self.batch_shape,
self.event_shape))
return m
def _log_prob(self, x):
temperature = tf.convert_to_tensor(self.temperature)
logits = self._logits_parameter_no_checks()
x = self._assert_valid_sample(x)
# broadcast logits or x if need be.
if (not tensorshape_util.is_fully_defined(x.shape)
or not tensorshape_util.is_fully_defined(logits.shape)
or x.shape != logits.shape):
logits = tf.ones_like(x, dtype=logits.dtype) * logits
x = tf.ones_like(logits, dtype=x.dtype) * x
# compute the normalization constant
k = tf.cast(self._event_size(logits), x.dtype)
log_norm_const = (tf.math.lgamma(k) +
(k - 1.) * tf.math.log(temperature))
# compute the unnormalized density
log_softmax = tf.math.log_softmax(logits -
x * temperature[..., tf.newaxis])
log_unnorm_prob = tf.reduce_sum(log_softmax, axis=[-1], keepdims=False)
# combine unnormalized density with normalization constant
return log_norm_const + log_unnorm_prob
def _bessel_ive(v, z, cache=None):
"""Computes I_v(z)*exp(-abs(z)) using a recurrence relation, where z > 0."""
# TODO(b/67497980): Switch to a more numerically faithful implementation.
z = tf.convert_to_tensor(z)
wrap = lambda result: tf.debugging.check_numerics(result, 'besseli{}'.
format(v))
if float(v) >= 2:
raise ValueError(
'Evaluating bessel_i by recurrence becomes imprecise for large v')
cache = cache or {}
safe_z = tf.where(z > 0, z, tf.ones_like(z))
if v in cache:
return wrap(cache[v])
if v == 0:
cache[v] = tf.math.bessel_i0e(z)
elif v == 1:
cache[v] = tf.math.bessel_i1e(z)
elif v == 0.5:
# sinh(x)*exp(-abs(x)), sinh(x) = (e^x - e^{-x}) / 2
sinhe = lambda x: (tf.exp(x - tf.abs(x)) - tf.exp(-x - tf.abs(x))) / 2
cache[v] = (
np.sqrt(2 / np.pi) * sinhe(z) *
tf.where(z > 0, tf.math.rsqrt(safe_z), tf.ones_like(safe_z)))
elif v == -0.5:
# cosh(x)*exp(-abs(x)), cosh(x) = (e^x + e^{-x}) / 2
coshe = lambda x: (tf.exp(x - tf.abs(x)) + tf.exp(-x - tf.abs(x))) / 2
cache[v] = (
np.sqrt(2 / np.pi) * coshe(z) *
tf.where(z > 0, tf.math.rsqrt(safe_z), tf.ones_like(safe_z)))
if v <= 1:
return wrap(cache[v])
# Recurrence relation:
cache[v] = (_bessel_ive(v - 2, z, cache) -
(2 * (v - 1)) * _bessel_ive(v - 1, z, cache) / z)
return wrap(cache[v])
def _mean(self):
# Derivation: https://sachinruk.github.io/blog/von-Mises-Fisher/
event_dim = tf.compat.dimension_value(self.event_shape[0])
if event_dim is None:
raise ValueError(
'event shape must be statically known for _bessel_ive')
safe_conc = tf.where(self.concentration > 0, self.concentration,
tf.ones_like(self.concentration))
safe_mean = self.mean_direction * (
_bessel_ive(event_dim / 2, safe_conc) /
_bessel_ive(event_dim / 2 - 1, safe_conc))[..., tf.newaxis]
return tf.where(
self.concentration[..., tf.newaxis] > tf.zeros_like(safe_mean),
safe_mean, tf.zeros_like(safe_mean))
def _log_normalization(self):
"""Computes the log-normalizer of the distribution."""
event_dim = tf.compat.dimension_value(self.event_shape[0])
if event_dim is None:
raise ValueError('vMF _log_normalizer currently only supports '
'statically known event shape')
safe_conc = tf.where(self.concentration > 0, self.concentration,
tf.ones_like(self.concentration))
safe_lognorm = ((event_dim / 2 - 1) * tf.math.log(safe_conc) -
(event_dim / 2) * np.log(2 * np.pi) - tf.math.log(
_bessel_ive(event_dim / 2 - 1, safe_conc)) -
tf.abs(safe_conc))
log_nsphere_surface_area = (
np.log(2.) + (event_dim / 2) * np.log(np.pi) -
tf.math.lgamma(tf.cast(event_dim / 2, self.dtype)))
return tf.where(self.concentration > 0, -safe_lognorm,
log_nsphere_surface_area)
def _broadcast_event_and_samples(event, samples, event_ndims):
"""Broadcasts the event or samples."""
# This is the shape of self.samples, without the samples axis, i.e. the shape
# of the result of a call to dist.sample(). This way we can broadcast it with
# event to get a properly-sized event, then add the singleton dim back at
# -event_ndims - 1.
samples_shape = tf.concat(
[
tf.shape(samples)[:-event_ndims - 1],
tf.shape(samples)[tf.rank(samples) - event_ndims:]
],
axis=0)
event = event * tf.ones(samples_shape, dtype=event.dtype)
event = tf.expand_dims(event, axis=-event_ndims - 1)
samples = samples * tf.ones_like(event, dtype=samples.dtype)
return event, samples
def _bdtr(k, n, p):
"""The binomial cumulative distribution function.
Args:
k: floating point `Tensor`.
n: floating point `Tensor`.
p: floating point `Tensor`.
Returns:
`sum_{j=0}^k p^j (1 - p)^(n - j)`.
"""
# Trick for getting safe backprop/gradients into n, k when
# betainc(a = 0, ..) = nan
# Write:
# where(unsafe, safe_output, betainc(where(unsafe, safe_input, input)))
ones = tf.ones_like(n - k)
k_eq_n = tf.equal(k, n)
safe_dn = tf.where(k_eq_n, ones, n - k)
dk = tf.math.betainc(a=safe_dn, b=k + 1, x=1 - p)
return tf.where(k_eq_n, ones, dk)
def _stddev(self):
with tf.control_dependencies(self._assertions):
distribution_means = [d.mean() for d in self.components]
distribution_devs = [d.stddev() for d in self.components]
cat_probs = self._cat_probs(log_probs=False)
stacked_means = tf.stack(distribution_means, axis=-1)
stacked_devs = tf.stack(distribution_devs, axis=-1)
cat_probs = [self._expand_to_event_rank(c_p) for c_p in cat_probs]
broadcasted_cat_probs = (tf.stack(cat_probs, axis=-1) *
tf.ones_like(stacked_means))
batched_dev = distribution_util.mixture_stddev(
tf.reshape(broadcasted_cat_probs,
[-1, len(self.components)]),
tf.reshape(stacked_means, [-1, len(self.components)]),
tf.reshape(stacked_devs, [-1, len(self.components)]))
# I.e. re-shape to list(batch_shape) + list(event_shape).
return tf.reshape(batched_dev,
tf.shape(broadcasted_cat_probs)[:-1])
def _cdf(self, x):
low = tf.convert_to_tensor(self.low)
high = tf.convert_to_tensor(self.high)
peak = tf.convert_to_tensor(self.peak)
interval_length = high - low
# Due to the PDF being not smooth at the peak, we have to treat each side
# somewhat differently. The PDF is two line segments, and thus we get
# quadratics here for the CDF.
result_inside_interval = tf.where(
(x >= low) & (x <= peak),
# (x - low) ** 2 / ((high - low) * (peak - low))
tf.math.squared_difference(x, low) / (interval_length *
(peak - low)),
# 1 - (high - x) ** 2 / ((high - low) * (high - peak))
1. - tf.math.squared_difference(high, x) / (interval_length *
(high - peak)))
# We now add that the left tail is 0 and the right tail is 1.
result_if_not_big = tf.where(x < low, tf.zeros_like(x),
result_inside_interval)
return tf.where(x >= high, tf.ones_like(x), result_if_not_big)
def _broadcast_cat_event_and_params(event, params, base_dtype):
"""Broadcasts the event or distribution parameters."""
if dtype_util.is_integer(event.dtype):
pass
elif dtype_util.is_floating(event.dtype):
# When `validate_args=True` we've already ensured int/float casting
# is closed.
event = tf.cast(event, dtype=tf.int32)
else:
raise TypeError('`value` should have integer `dtype` or '
'`self.dtype` ({})'.format(base_dtype))
shape_known_statically = (
tensorshape_util.rank(params.shape) is not None
and tensorshape_util.is_fully_defined(params.shape[:-1])
and tensorshape_util.is_fully_defined(event.shape))
if not shape_known_statically or params.shape[:-1] != event.shape:
params = params * tf.ones_like(event[..., tf.newaxis],
dtype=params.dtype)
params_shape = tf.shape(params)[:-1]
event = event * tf.ones(params_shape, dtype=event.dtype)
if tensorshape_util.rank(params.shape) is not None:
tensorshape_util.set_shape(event, params.shape[:-1])
return event, params
def _sample_n(self, n, seed=None):
seed = SeedStream(seed, 'dirichlet_multinomial')
concentration = tf.convert_to_tensor(self._concentration)
total_count = tf.convert_to_tensor(self._total_count)
n_draws = tf.cast(total_count, dtype=tf.int32)
k = self._event_shape_tensor(concentration)[0]
alpha = tf.math.multiply(
tf.ones_like(total_count[..., tf.newaxis]),
concentration,
name='alpha')
unnormalized_logits = tf.math.log(
tf.random.gamma(
shape=[n],
alpha=alpha,
dtype=self.dtype,
seed=seed()))
x = multinomial.draw_sample(
1, k, unnormalized_logits, n_draws, self.dtype, seed())
final_shape = tf.concat(
[[n], self._batch_shape_tensor(concentration, total_count), [k]], 0)
return tf.reshape(x, final_shape)
def _mode(self):
return self.loc * tf.ones_like(self.scale)
def _entropy(self):
h = np.log(2 * np.pi) + tf.math.log(self.scale)
return h * tf.ones_like(self.loc)
def _entropy(self): log_normalization = 0.5 * np.log(2. * np.pi) + tf.math.log(self.scale) entropy = 0.5 + log_normalization return entropy * tf.ones_like(self.loc)
def _stddev(self):
return self.scale * tf.ones_like(self.loc) * np.pi / np.sqrt(6)
def _stddev(self): return self.scale * tf.ones_like(self.loc)
def reduce_weighted_logsumexp(logx,
w=None,
axis=None,
keep_dims=False,
return_sign=False,
name=None):
"""Computes `log(abs(sum(weight * exp(elements across tensor dimensions))))`.
If all weights `w` are known to be positive, it is more efficient to directly
use `reduce_logsumexp`, i.e., `tf.reduce_logsumexp(logx + tf.log(w))` is more
efficient than `du.reduce_weighted_logsumexp(logx, w)`.
Reduces `input_tensor` along the dimensions given in `axis`.
Unless `keep_dims` is true, the rank of the tensor is reduced by 1 for each
entry in `axis`. If `keep_dims` is true, the reduced dimensions
are retained with length 1.
If `axis` has no entries, all dimensions are reduced, and a
tensor with a single element is returned.
This function is more numerically stable than log(sum(w * exp(input))). It
avoids overflows caused by taking the exp of large inputs and underflows
caused by taking the log of small inputs.
For example:
```python
x = tf.constant([[0., 0, 0],
[0, 0, 0]])
w = tf.constant([[-1., 1, 1],
[1, 1, 1]])
du.reduce_weighted_logsumexp(x, w)
# ==> log(-1*1 + 1*1 + 1*1 + 1*1 + 1*1 + 1*1) = log(4)
du.reduce_weighted_logsumexp(x, w, axis=0)
# ==> [log(-1+1), log(1+1), log(1+1)]
du.reduce_weighted_logsumexp(x, w, axis=1)
# ==> [log(-1+1+1), log(1+1+1)]
du.reduce_weighted_logsumexp(x, w, axis=1, keep_dims=True)
# ==> [[log(-1+1+1)], [log(1+1+1)]]
du.reduce_weighted_logsumexp(x, w, axis=[0, 1])
# ==> log(-1+5)
```
Args:
logx: The tensor to reduce. Should have numeric type.
w: The weight tensor. Should have numeric type identical to `logx`.
axis: The dimensions to reduce. If `None` (the default), reduces all
dimensions. Must be in the range `[-rank(input_tensor),
rank(input_tensor))`.
keep_dims: If true, retains reduced dimensions with length 1.
return_sign: If `True`, returns the sign of the result.
name: A name for the operation (optional).
Returns:
lswe: The `log(abs(sum(weight * exp(x))))` reduced tensor.
sign: (Optional) The sign of `sum(weight * exp(x))`.
"""
with tf.name_scope(name or 'reduce_weighted_logsumexp'):
logx = tf.convert_to_tensor(logx, name='logx')
if w is None:
lswe = tf.reduce_logsumexp(logx, axis=axis, keepdims=keep_dims)
if return_sign:
sgn = tf.ones_like(lswe)
return lswe, sgn
return lswe
w = tf.convert_to_tensor(w, dtype=logx.dtype, name='w')
log_absw_x = logx + tf.math.log(tf.abs(w))
max_log_absw_x = tf.reduce_max(log_absw_x, axis=axis, keepdims=True)
# If the largest element is `-inf` or `inf` then we don't bother subtracting
# off the max. We do this because otherwise we'd get `inf - inf = NaN`. That
# this is ok follows from the fact that we're actually free to subtract any
# value we like, so long as we add it back after taking the `log(sum(...))`.
max_log_absw_x = tf.where(
tf.math.is_inf(max_log_absw_x),
tf.zeros([], max_log_absw_x.dtype),
max_log_absw_x)
wx_over_max_absw_x = (tf.sign(w) * tf.exp(log_absw_x - max_log_absw_x))
sum_wx_over_max_absw_x = tf.reduce_sum(
wx_over_max_absw_x, axis=axis, keepdims=keep_dims)
if not keep_dims:
max_log_absw_x = tf.squeeze(max_log_absw_x, axis)
sgn = tf.sign(sum_wx_over_max_absw_x)
lswe = max_log_absw_x + tf.math.log(sgn * sum_wx_over_max_absw_x)
if return_sign:
return lswe, sgn
return lswe
def draw_sample(num_samples, num_classes, logits, num_trials, dtype, seed):
"""Sample a multinomial.
The batch shape is given by broadcasting num_trials with
remove_last_dimension(logits).
Args:
num_samples: Python int or singleton integer Tensor: number of multinomial
samples to draw.
num_classes: Python int or singleton integer Tensor: number of classes.
logits: Floating Tensor with last dimension k, of (unnormalized) logit
probabilities per class.
num_trials: Tensor of number of categorical trials each multinomial consists
of. num_trials[..., tf.newaxis] must broadcast with logits.
dtype: dtype at which to emit samples.
seed: Random seed.
Returns:
samples: Tensor of given dtype and shape [n] + batch_shape + [k].
"""
with tf.name_scope('draw_sample'):
# broadcast the num_trials and logits to same shape
num_trials = tf.ones_like(logits[..., 0],
dtype=num_trials.dtype) * num_trials
logits = tf.ones_like(num_trials[..., tf.newaxis],
dtype=logits.dtype) * logits
# flatten the total_count and logits
# flat_logits has shape [B1B2...Bm, num_classes]
flat_logits = tf.reshape(logits, [-1, num_classes])
flat_num_trials = num_samples * tf.reshape(num_trials,
[-1]) # [B1B2...Bm]
# Computes each logits and num_trials situation by map_fn.
# Using just one batch tf.random.categorical call doesn't work because that
# requires num_trials to be the same across all members of the batch of
# logits. This restriction makes sense for tf.random.categorical because
# for it, num_trials is part of the returned shape. However, the
# multinomial sampler does not need that restriction, because it sums out
# exactly that dimension.
# One possibility would be to draw a batch categorical whose sample count is
# max(num_trials) and mask out the excess ones. However, if the elements of
# num_trials vary widely, this can be wasteful of memory.
# TODO(b/123763054, b/112152209): Revisit the possibility of writing this
# with a batch categorical followed by batch unsorted_segment_sum, once both
# of those work and are memory-efficient enough.
def _sample_one_batch_member(args):
logits, num_cat_samples = args[0], args[1] # [K], []
# x has shape [1, num_cat_samples = num_samples * num_trials]
x = tf.random.categorical(logits[tf.newaxis, ...],
num_cat_samples,
seed=seed)
x = tf.reshape(x, shape=[num_samples,
-1]) # [num_samples, num_trials]
x = tf.one_hot(
x, depth=num_classes) # [num_samples, num_trials, num_classes]
x = tf.reduce_sum(x, axis=-2) # [num_samples, num_classes]
return tf.cast(x, dtype=dtype)
if seed is not None:
# Force parallel_iterations to 1 to ensure reproducibility
# b/139210489
x = tf.map_fn(
_sample_one_batch_member,
[flat_logits, flat_num_trials],
dtype=dtype, # [B1B2...Bm, num_samples, num_classes]
parallel_iterations=1)
else:
# Invoke default parallel_iterations behavior
x = tf.map_fn(_sample_one_batch_member,
[flat_logits, flat_num_trials],
dtype=dtype) # [B1B2...Bm, num_samples, num_classes]
# reshape the results to proper shape
x = tf.transpose(a=x, perm=[1, 0, 2])
final_shape = tf.concat(
[[num_samples], tf.shape(num_trials), [num_classes]], axis=0)
x = tf.reshape(x, final_shape)
return x
def _entropy(self):
# Use broadcasting rules to calculate the full broadcast sigma.
scale = self.scale * tf.ones_like(self.loc)
return 1. + tf.math.log(scale) + np.euler_gamma
def _mode(self):
return tf.ones_like(self.power, dtype=self.dtype)
def _sample_n(self, n, seed=None):
with tf.control_dependencies(self._runtime_assertions):
strm = SeedStream(seed, salt="HiddenMarkovModel")
num_states = self._num_states
batch_shape = self.batch_shape_tensor()
batch_size = tf.reduce_prod(batch_shape)
# The batch sizes of the underlying initial distributions and
# transition distributions might not match the batch size of
# the HMM distribution.
# As a result we need to ask for more samples from the
# underlying distributions and then reshape the results into
# the correct batch size for the HMM.
init_repeat = (
tf.reduce_prod(self.batch_shape_tensor()) // tf.reduce_prod(
self._initial_distribution.batch_shape_tensor()))
init_state = self._initial_distribution.sample(n * init_repeat,
seed=strm())
init_state = tf.reshape(init_state, [n, batch_size])
# init_state :: n batch_size
transition_repeat = (
tf.reduce_prod(self.batch_shape_tensor()) // tf.reduce_prod(
self._transition_distribution.batch_shape_tensor()[:-1]))
def generate_step(state, _):
"""Take a single step in Markov chain."""
gen = self._transition_distribution.sample(n *
transition_repeat,
seed=strm())
# gen :: (n * transition_repeat) transition_batch
new_states = tf.reshape(gen, [n, batch_size, num_states])
# new_states :: n batch_size num_states
old_states_one_hot = tf.one_hot(state,
num_states,
dtype=tf.int32)
# old_states :: n batch_size num_states
return tf.reduce_sum(old_states_one_hot * new_states, axis=-1)
def _scan_multiple_steps():
"""Take multiple steps with tf.scan."""
dummy_index = tf.zeros(self._num_steps - 1, dtype=tf.float32)
if seed is not None:
# Force parallel_iterations to 1 to ensure reproducibility
# b/139210489
hidden_states = tf.scan(generate_step,
dummy_index,
initializer=init_state,
parallel_iterations=1)
else:
# Invoke default parallel_iterations behavior
hidden_states = tf.scan(generate_step,
dummy_index,
initializer=init_state)
# TODO(b/115618503): add/use prepend_initializer to tf.scan
return tf.concat([[init_state], hidden_states], axis=0)
hidden_states = prefer_static.cond(
self._num_steps > 1, _scan_multiple_steps,
lambda: init_state[tf.newaxis, ...])
hidden_one_hot = tf.one_hot(
hidden_states,
num_states,
dtype=self._observation_distribution.dtype)
# hidden_one_hot :: num_steps n batch_size num_states
# The observation distribution batch size might not match
# the required batch size so as with the initial and
# transition distributions we generate more samples and
# reshape.
observation_repeat = (batch_size // tf.reduce_prod(
self._observation_distribution.batch_shape_tensor()[:-1]))
possible_observations = self._observation_distribution.sample(
[self._num_steps, observation_repeat * n], seed=strm())
inner_shape = self._observation_distribution.event_shape
# possible_observations :: num_steps (observation_repeat * n)
# observation_batch[:-1] num_states inner_shape
possible_observations = tf.reshape(
possible_observations,
tf.concat([[self._num_steps, n], batch_shape, [num_states],
inner_shape],
axis=0))
# possible_observations :: steps n batch_size num_states inner_shape
hidden_one_hot = tf.reshape(
hidden_one_hot,
tf.concat([[self._num_steps, n], batch_shape, [num_states],
tf.ones_like(inner_shape)],
axis=0))
# hidden_one_hot :: steps n batch_size num_states "inner_shape"
observations = tf.reduce_sum(hidden_one_hot *
possible_observations,
axis=-1 - tf.size(inner_shape))
# observations :: steps n batch_size inner_shape
observations = distribution_util.move_dimension(
observations, 0, 1 + tf.size(batch_shape))
# returned :: n batch_shape steps inner_shape
return observations
def _sample_n(self, num_samples, seed=None, name=None):
"""Returns a Tensor of samples from an LKJ distribution.
Args:
num_samples: Python `int`. The number of samples to draw.
seed: Python integer seed for RNG
name: Python `str` name prefixed to Ops created by this function.
Returns:
samples: A Tensor of correlation matrices with shape `[n, B, D, D]`,
where `B` is the shape of the `concentration` parameter, and `D`
is the `dimension`.
Raises:
ValueError: If `dimension` is negative.
"""
if self.dimension < 0:
raise ValueError(
'Cannot sample negative-dimension correlation matrices.')
# Notation below: B is the batch shape, i.e., tf.shape(concentration)
seed = SeedStream(seed, 'sample_lkj')
with tf.name_scope('sample_lkj' or name):
concentration = tf.convert_to_tensor(self.concentration)
if not dtype_util.is_floating(concentration.dtype):
raise TypeError(
'The concentration argument should have floating type, not '
'{}'.format(dtype_util.name(concentration.dtype)))
concentration = _replicate(num_samples, concentration)
concentration_shape = tf.shape(concentration)
if self.dimension <= 1:
# For any dimension <= 1, there is only one possible correlation matrix.
shape = tf.concat(
[concentration_shape, [self.dimension, self.dimension]],
axis=0)
return tf.ones(shape=shape, dtype=concentration.dtype)
beta_conc = concentration + (self.dimension - 2.) / 2.
beta_dist = beta.Beta(concentration1=beta_conc,
concentration0=beta_conc)
# Note that the sampler below deviates from [1], by doing the sampling in
# cholesky space. This does not change the fundamental logic of the
# sampler, but does speed up the sampling.
# This is the correlation coefficient between the first two dimensions.
# This is also `r` in reference [1].
corr12 = 2. * beta_dist.sample(seed=seed()) - 1.
# Below we construct the Cholesky of the initial 2x2 correlation matrix,
# which is of the form:
# [[1, 0], [r, sqrt(1 - r**2)]], where r is the correlation between the
# first two dimensions.
# This is the top-left corner of the cholesky of the final sample.
first_row = tf.concat([
tf.ones_like(corr12)[..., tf.newaxis],
tf.zeros_like(corr12)[..., tf.newaxis]
],
axis=-1)
second_row = tf.concat([
corr12[..., tf.newaxis],
tf.sqrt(1 - corr12**2)[..., tf.newaxis]
],
axis=-1)
chol_result = tf.concat([
first_row[..., tf.newaxis, :], second_row[..., tf.newaxis, :]
],
axis=-2)
for n in range(2, self.dimension):
# Loop invariant: on entry, result has shape B + [n, n]
beta_conc = beta_conc - 0.5
# norm is y in reference [1].
norm = beta.Beta(concentration1=n / 2.,
concentration0=beta_conc).sample(seed=seed())
# distance shape: B + [1] for broadcast
distance = tf.sqrt(norm)[..., tf.newaxis]
# direction is u in reference [1].
# direction shape: B + [n]
direction = _uniform_unit_norm(n, concentration_shape,
concentration.dtype, seed)
# raw_correlation is w in reference [1].
raw_correlation = distance * direction # shape: B + [n]
# This is the next row in the cholesky of the result,
# which differs from the construction in reference [1].
# In the reference, the new row `z` = chol_result @ raw_correlation^T
# = C @ raw_correlation^T (where as short hand we use C = chol_result).
# We prove that the below equation is the right row to add to the
# cholesky, by showing equality with reference [1].
# Let S be the sample constructed so far, and let `z` be as in
# reference [1]. Then at this iteration, the new sample S' will be
# [[S z^T]
# [z 1]]
# In our case we have the cholesky decomposition factor C, so
# we want our new row x (same size as z) to satisfy:
# [[S z^T] [[C 0] [[C^T x^T] [[CC^T Cx^T]
# [z 1]] = [x k]] [0 k]] = [xC^t xx^T + k**2]]
# Since C @ raw_correlation^T = z = C @ x^T, and C is invertible,
# we have that x = raw_correlation. Also 1 = xx^T + k**2, so k
# = sqrt(1 - xx^T) = sqrt(1 - |raw_correlation|**2) = sqrt(1 -
# distance**2).
new_row = tf.concat(
[raw_correlation,
tf.sqrt(1. - norm[..., tf.newaxis])],
axis=-1)
# Finally add this new row, by growing the cholesky of the result.
chol_result = tf.concat([
chol_result,
tf.zeros_like(chol_result[..., 0][..., tf.newaxis])
],
axis=-1)
chol_result = tf.concat(
[chol_result, new_row[..., tf.newaxis, :]], axis=-2)
if self.input_output_cholesky:
return chol_result
result = tf.matmul(chol_result, chol_result, transpose_b=True)
# The diagonal for a correlation matrix should always be ones. Due to
# numerical instability the matmul might not achieve that, so manually set
# these to ones.
result = tf.linalg.set_diag(
result, tf.ones(shape=tf.shape(result)[:-1],
dtype=result.dtype))
# This sampling algorithm can produce near-PSD matrices on which standard
# algorithms such as `tf.cholesky` or `tf.linalg.self_adjoint_eigvals`
# fail. Specifically, as documented in b/116828694, around 2% of trials
# of 900,000 5x5 matrices (distributed according to 9 different
# concentration parameter values) contained at least one matrix on which
# the Cholesky decomposition failed.
return result
