def _log_prob(self, x):
logits = self._logits_parameter_no_checks()
event_size = self._event_size(logits)
x = tf.cast(x, logits.dtype)
x = self._maybe_assert_valid_sample(x, dtype=logits.dtype)
# 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):
broadcast_shape = tf.broadcast_dynamic_shape(
tf.shape(logits), tf.shape(x))
logits = tf.broadcast_to(logits, broadcast_shape)
x = tf.broadcast_to(x, broadcast_shape)
logits_shape = tf.shape(tf.reduce_sum(logits, axis=-1))
logits_2d = tf.reshape(logits, [-1, event_size])
x_2d = tf.reshape(x, [-1, event_size])
ret = -tf.nn.softmax_cross_entropy_with_logits(
labels=tf.stop_gradient(x_2d), logits=logits_2d)
# Reshape back to user-supplied batch and sample dims prior to 2D reshape.
ret = tf.reshape(ret, logits_shape)
return ret
def _cdf(self, x):
with tf.control_dependencies(self._maybe_assert_valid_sample(x)):
concentration1 = tf.convert_to_tensor(self.concentration1)
concentration0 = tf.convert_to_tensor(self.concentration0)
shape = self._batch_shape_tensor(concentration1, concentration0)
concentration1 = tf.broadcast_to(concentration1, shape)
concentration0 = tf.broadcast_to(concentration0, shape)
return tf.math.betainc(concentration1, concentration0, x)
def _cdf(self, x):
x = self._maybe_assert_valid_sample(x)
logits = self._logits_parameter_no_checks()
total_count = tf.convert_to_tensor(self.total_count)
shape = self._batch_shape_tensor(logits_or_probs=logits,
total_count=total_count)
return tf.math.betainc(tf.broadcast_to(total_count, shape),
tf.broadcast_to(1. + x, shape),
tf.broadcast_to(tf.sigmoid(-logits), shape))
def _variance(self):
with tf.control_dependencies(self._runtime_assertions):
probs = self._marginal_hidden_probs()
# probs :: num_steps batch_shape num_states
means = self._observation_distribution.mean()
# means :: observation_batch_shape[:-1] num_states
# observation_event_shape
means_shape = tf.concat([
self.batch_shape_tensor(), [self._num_states],
self._observation_distribution.event_shape_tensor()
],
axis=0)
means = tf.broadcast_to(means, means_shape)
# means :: batch_shape num_states observation_event_shape
observation_event_shape = (
self._observation_distribution.event_shape_tensor())
batch_size = tf.reduce_prod(self.batch_shape_tensor())
flat_probs_shape = [self._num_steps, batch_size, self._num_states]
flat_means_shape = [
batch_size, 1, self._num_states,
tf.reduce_prod(observation_event_shape)
]
flat_probs = tf.reshape(probs, flat_probs_shape)
# flat_probs :: num_steps batch_size num_states
flat_means = tf.reshape(means, flat_means_shape)
# flat_means :: batch_size 1 num_states observation_event_size
flat_mean = tf.einsum("ijk,jmkl->jiml", flat_probs, flat_means)
# flat_mean :: batch_size num_steps 1 observation_event_size
variances = self._observation_distribution.variance()
variances = tf.broadcast_to(variances, means_shape)
# variances :: batch_shape num_states observation_event_shape
flat_variances = tf.reshape(variances, flat_means_shape)
# flat_variances :: batch_size 1 num_states observation_event_size
# For a mixture of n distributions with mixture probabilities
# p[i], and where the individual distributions have means and
# variances given by mean[i] and var[i], the variance of
# the mixture is given by:
#
# var = sum i=1..n p[i] * ((mean[i] - mean)**2 + var[i]**2)
flat_variance = tf.einsum("ijk,jikl->jil", flat_probs,
(flat_means - flat_mean)**2 +
flat_variances)
# flat_variance :: batch_size num_steps observation_event_size
unflat_mean_shape = tf.concat([
self.batch_shape_tensor(), [self._num_steps],
observation_event_shape
],
axis=0)
# returns :: batch_shape num_steps observation_event_shape
return tf.reshape(flat_variance, unflat_mean_shape)
def _log_prob(self, value):
with tf.control_dependencies(self._runtime_assertions):
# The argument `value` is a tensor of sequences of observations.
# `observation_batch_shape` is the shape of that tensor with the
# sequence part removed.
# `observation_batch_shape` is then broadcast to the full batch shape
# to give the `batch_shape` that defines the shape of the result.
observation_tensor_shape = tf.shape(value)
observation_batch_shape = observation_tensor_shape[:-1 - self.
_underlying_event_rank]
# value :: observation_batch_shape num_steps observation_event_shape
batch_shape = tf.broadcast_dynamic_shape(observation_batch_shape,
self.batch_shape_tensor())
log_init = tf.broadcast_to(
self._log_init,
tf.concat([batch_shape, [self._num_states]], axis=0))
# log_init :: batch_shape num_states
log_transition = self._log_trans
# `observation_event_shape` is the shape of each sequence of observations
# emitted by the model.
observation_event_shape = observation_tensor_shape[
-1 - self._underlying_event_rank:]
working_obs = tf.broadcast_to(
value, tf.concat([batch_shape, observation_event_shape],
axis=0))
# working_obs :: batch_shape observation_event_shape
r = self._underlying_event_rank
# Move index into sequence of observations to front so we can apply
# tf.foldl
working_obs = distribution_util.move_dimension(
working_obs, -1 - r, 0)[..., tf.newaxis]
# working_obs :: num_steps batch_shape underlying_event_shape
observation_probs = (
self._observation_distribution.log_prob(working_obs))
def forward_step(log_prev_step, log_prob_observation):
return _log_vector_matrix(
log_prev_step, log_transition) + log_prob_observation
fwd_prob = tf.foldl(forward_step,
observation_probs,
initializer=log_init)
# fwd_prob :: batch_shape num_states
log_prob = tf.reduce_logsumexp(fwd_prob, axis=-1)
# log_prob :: batch_shape
return log_prob
def _swap_m_with_i(vecs, m, i):
"""Swaps `m` and `i` on axis -1. (Helper for pivoted_cholesky.)
Given a batch of int64 vectors `vecs`, scalar index `m`, and compatibly shaped
per-vector indices `i`, this function swaps elements `m` and `i` in each
vector. For the use-case below, these are permutation vectors.
Args:
vecs: Vectors on which we perform the swap, int64 `Tensor`.
m: Scalar int64 `Tensor`, the index into which the `i`th element is going.
i: Batch int64 `Tensor`, shaped like vecs.shape[:-1] + [1]; the index into
which the `m`th element is going.
Returns:
vecs: The updated vectors.
"""
vecs = tf.convert_to_tensor(vecs, dtype=tf.int64, name='vecs')
m = tf.convert_to_tensor(m, dtype=tf.int64, name='m')
i = tf.convert_to_tensor(i, dtype=tf.int64, name='i')
trailing_elts = tf.broadcast_to(
tf.range(m + 1,
prefer_static.shape(vecs, out_type=tf.int64)[-1]),
prefer_static.shape(vecs[..., m + 1:]))
trailing_elts = tf.where(tf.equal(trailing_elts, i),
tf.gather(vecs, [m], axis=-1), vecs[..., m + 1:])
# TODO(bjp): Could we use tensor_scatter_nd_update?
vecs_shape = vecs.shape
vecs = tf.concat([
vecs[..., :m],
tf.gather(vecs, i, batch_dims=int(prefer_static.rank(vecs)) - 1),
trailing_elts
],
axis=-1)
tensorshape_util.set_shape(vecs, vecs_shape)
return vecs
def _marginal_hidden_probs(self):
"""Compute marginal pdf for each individual observable."""
initial_log_probs = tf.broadcast_to(
self._log_init,
tf.concat([self.batch_shape_tensor(), [self._num_states]], axis=0))
# initial_log_probs :: batch_shape num_states
def _scan_multiple_steps():
"""Perform `scan` operation when `num_steps` > 1."""
transition_log_probs = self._log_trans
def forward_step(log_probs, _):
return _log_vector_matrix(log_probs, transition_log_probs)
dummy_index = tf.zeros(self._num_steps - 1, dtype=tf.float32)
forward_log_probs = tf.scan(forward_step,
dummy_index,
initializer=initial_log_probs,
name="forward_log_probs")
return tf.concat([[initial_log_probs], forward_log_probs], axis=0)
forward_log_probs = prefer_static.cond(
self._num_steps > 1, _scan_multiple_steps,
lambda: initial_log_probs[tf.newaxis, ...])
return tf.exp(forward_log_probs)
def _mean(self, **kwargs):
if not self.bijector.is_constant_jacobian:
raise NotImplementedError("mean is not implemented for non-affine "
"bijectors")
distribution_kwargs, bijector_kwargs = self._kwargs_split_fn(kwargs)
x = self.distribution.mean(**distribution_kwargs)
if self._is_maybe_batch_override or self._is_maybe_event_override:
# A batch (respectively event) shape override is only allowed if the batch
# (event) shape of the base distribution is [], so concatenating all the
# shapes does the right thing.
new_shape = prefer_static.concat([
prefer_static.ones_like(self._override_batch_shape),
self.distribution.batch_shape_tensor(),
prefer_static.ones_like(self._override_event_shape),
self.distribution.event_shape_tensor(),
], 0)
x = tf.reshape(x, new_shape)
new_shape = prefer_static.concat(
[self.batch_shape_tensor(),
self.event_shape_tensor()], 0)
x = tf.broadcast_to(x, new_shape)
y = self.bijector.forward(x, **bijector_kwargs)
sample_shape = tf.convert_to_tensor([],
dtype=tf.int32,
name="sample_shape")
y = self._set_sample_static_shape(y, sample_shape)
return y
def _sample_n(self, n, seed=None):
seed = SeedStream(seed, "beta")
concentration1 = tf.convert_to_tensor(self.concentration1)
concentration0 = tf.convert_to_tensor(self.concentration0)
shape = self._batch_shape_tensor(concentration1, concentration0)
expanded_concentration1 = tf.broadcast_to(concentration1, shape)
expanded_concentration0 = tf.broadcast_to(concentration0, shape)
gamma1_sample = tf.random.gamma(shape=[n],
alpha=expanded_concentration1,
dtype=self.dtype,
seed=seed())
gamma2_sample = tf.random.gamma(shape=[n],
alpha=expanded_concentration0,
dtype=self.dtype,
seed=seed())
beta_sample = gamma1_sample / (gamma1_sample + gamma2_sample)
return beta_sample
def _cdf(self, x):
df = tf.convert_to_tensor(self.df)
# Take Abs(scale) to make subsequent where work correctly.
y = (x - self.loc) / tf.abs(self.scale)
x_t = df / (y**2. + df)
neg_cdf = 0.5 * tf.math.betainc(
0.5 * tf.broadcast_to(df, prefer_static.shape(x_t)), 0.5, x_t)
return tf.where(y < 0., neg_cdf, 1. - neg_cdf)
def cholesky_concat(chol, cols, name=None):
"""Concatenates `chol @ chol.T` with additional rows and columns.
This operation is conceptually identical to:
```python
def cholesky_concat_slow(chol, cols): # cols shaped (n + m) x m = z x m
mat = tf.matmul(chol, chol, adjoint_b=True) # batch of n x n
# Concat columns.
mat = tf.concat([mat, cols[..., :tf.shape(mat)[-2], :]], axis=-1) # n x z
# Concat rows.
mat = tf.concat([mat, tf.linalg.matrix_transpose(cols)], axis=-2) # z x z
return tf.linalg.cholesky(mat)
```
but whereas `cholesky_concat_slow` would cost `O(z**3)` work,
`cholesky_concat` only costs `O(z**2 + m**3)` work.
The resulting (implicit) matrix must be symmetric and positive definite.
Thus, the bottom right `m x m` must be self-adjoint, and we do not require a
separate `rows` argument (which can be inferred from `conj(cols.T)`).
Args:
chol: Cholesky decomposition of `mat = chol @ chol.T`.
cols: The new columns whose first `n` rows we would like concatenated to the
right of `mat = chol @ chol.T`, and whose conjugate transpose we would
like concatenated to the bottom of `concat(mat, cols[:n,:])`. A `Tensor`
with final dims `(n+m, m)`. The first `n` rows are the top right rectangle
(their conjugate transpose forms the bottom left), and the bottom `m x m`
is self-adjoint.
name: Optional name for this op.
Returns:
chol_concat: The Cholesky decomposition of:
```
[ [ mat cols[:n, :] ]
[ conj(cols.T) ] ]
```
"""
with tf.name_scope(name or 'cholesky_extend'):
dtype = dtype_util.common_dtype([chol, cols], dtype_hint=tf.float32)
chol = tf.convert_to_tensor(chol, name='chol', dtype=dtype)
cols = tf.convert_to_tensor(cols, name='cols', dtype=dtype)
n = prefer_static.shape(chol)[-1]
mat_nm, mat_mm = cols[..., :n, :], cols[..., n:, :]
solved_nm = linear_operator_util.matrix_triangular_solve_with_broadcast(
chol, mat_nm)
lower_right_mm = tf.linalg.cholesky(
mat_mm - tf.matmul(solved_nm, solved_nm, adjoint_a=True))
lower_left_mn = tf.math.conj(tf.linalg.matrix_transpose(solved_nm))
out_batch = prefer_static.shape(solved_nm)[:-2]
chol = tf.broadcast_to(
chol,
tf.concat([out_batch, prefer_static.shape(chol)[-2:]], axis=0))
top_right_zeros_nm = tf.zeros_like(solved_nm)
return tf.concat([
tf.concat([chol, top_right_zeros_nm], axis=-1),
tf.concat([lower_left_mn, lower_right_mm], axis=-1)
],
axis=-2)
def _sample_n(self, n, seed=None):
# Like with the univariate Student's t, sampling can be implemented as a
# ratio of samples from a multivariate gaussian with the appropriate
# covariance matrix and a sample from the chi-squared distribution.
seed = SeedStream(seed, salt='multivariate t')
loc = tf.broadcast_to(self.loc, self._sample_shape())
mvn = mvn_linear_operator.MultivariateNormalLinearOperator(
loc=tf.zeros_like(loc), scale=self.scale)
normal_samp = mvn.sample(n, seed=seed())
df = tf.broadcast_to(self.df, self.batch_shape_tensor())
chi2 = chi2_lib.Chi2(df=df)
chi2_samp = chi2.sample(n, seed=seed())
return (
self._loc +
normal_samp * tf.math.rsqrt(chi2_samp / self._df)[..., tf.newaxis])
def _sample_n(self, n, seed=None):
del seed # unused
loc = tf.convert_to_tensor(self.loc)
return tf.broadcast_to(
loc,
tf.concat([[n],
self._batch_shape_tensor(loc=loc),
self._event_shape_tensor(loc=loc)],
axis=0))
def _variance(self):
probs = self._categorical.probs_parameter()
outcomes = tf.broadcast_to(
self.outcomes, shape=dist_util.prefer_static_shape(probs))
if dtype_util.is_integer(outcomes.dtype):
if self._validate_args:
outcomes = dist_util.embed_check_integer_casting_closed(
outcomes, target_dtype=probs.dtype)
outcomes = tf.cast(outcomes, dtype=probs.dtype)
square_d = tf.math.squared_difference(
outcomes, self._mean(probs)[..., tf.newaxis])
return tf.reduce_sum(probs * square_d, axis=-1)
def _covariance(self):
if distribution_util.is_diagonal_scale(self.scale):
mvn_cov = tf.linalg.diag(tf.square(self.scale.diag_part()))
else:
mvn_cov = self.scale.matmul(self.scale.to_dense(),
adjoint_arg=True)
cov_shape = tf.concat(
[self._sample_shape(),
self._event_shape_tensor()], -1)
mvn_cov = tf.broadcast_to(mvn_cov, cov_shape)
return self._std_var_helper(mvn_cov, 'covariance', 2, lambda x: x)
def _variance(self):
if distribution_util.is_diagonal_scale(self.scale):
mvn_var = tf.square(self.scale.diag_part())
elif (isinstance(self.scale, tf.linalg.LinearOperatorLowRankUpdate)
and self.scale.is_self_adjoint):
mvn_var = tf.linalg.diag_part(
self.scale.matmul(self.scale.to_dense()))
else:
mvn_var = tf.linalg.diag_part(
self.scale.matmul(self.scale.to_dense(), adjoint_arg=True))
mvn_var = tf.broadcast_to(mvn_var, self._sample_shape())
return self._std_var_helper(mvn_var, 'variance', 1, lambda x: x)
def _mean(self):
mean = tf.broadcast_to(self.loc, self._sample_shape())
if self.allow_nan_stats:
return tf.where(self.df[..., tf.newaxis] > 1., mean,
dtype_util.as_numpy_dtype(self.dtype)(np.nan))
else:
with tf.control_dependencies([
assert_util.assert_less(
tf.cast(1., self.dtype),
self.df,
message='Mean not defined for components of df <= 1.'),
]):
return tf.identity(mean)
def _stddev(self):
if distribution_util.is_diagonal_scale(self.scale):
mvn_std = tf.abs(self.scale.diag_part())
elif (isinstance(self.scale, tf.linalg.LinearOperatorLowRankUpdate)
and self.scale.is_self_adjoint):
mvn_std = tf.sqrt(
tf.linalg.diag_part(self.scale.matmul(self.scale.to_dense())))
else:
mvn_std = tf.sqrt(
tf.linalg.diag_part(
self.scale.matmul(self.scale.to_dense(),
adjoint_arg=True)))
mvn_std = tf.broadcast_to(mvn_std, self._sample_shape())
return self._std_var_helper(mvn_std, 'standard deviation', 1, tf.sqrt)
def _entropy(self):
df = tf.broadcast_to(self.df, self.batch_shape_tensor())
num_dims = tf.cast(self.event_shape_tensor()[0], self.dtype)
def _lbeta(concentration0, concentration1):
return (tf.math.lgamma(concentration1) +
tf.math.lgamma(concentration0) -
tf.math.lgamma(concentration0 + concentration1))
shape_factor = self._scale.log_abs_determinant()
beta_factor = num_dims / 2. * (
tf.math.log(df) + np.log(np.pi)) - tf.math.lgamma(
num_dims / 2.) + _lbeta(num_dims / 2., df / 2.)
digamma_factor = (num_dims + df) / 2. * (tf.math.digamma(
(num_dims + df) / 2.) - tf.math.digamma(df / 2.))
return shape_factor + beta_factor + digamma_factor
def _fn(self, **kwargs):
"""Implements summary statistic, eg, mean, stddev, mode."""
x = getattr(self.distribution, attr)(**kwargs)
shape = prefer_static.concat([
self.distribution.batch_shape_tensor(),
prefer_static.ones(prefer_static.rank_from_shape(
self.sample_shape),
dtype=self.sample_shape.dtype),
self.distribution.event_shape_tensor(),
],
axis=0)
x = tf.reshape(x, shape=shape)
shape = prefer_static.concat([
self.distribution.batch_shape_tensor(),
self.sample_shape,
self.distribution.event_shape_tensor(),
],
axis=0)
return tf.broadcast_to(x, shape)
def _mean(self):
with tf.control_dependencies(self._runtime_assertions):
probs = self._marginal_hidden_probs()
# probs :: num_steps batch_shape num_states
means = self._observation_distribution.mean()
# means :: observation_batch_shape[:-1] num_states
# observation_event_shape
means_shape = tf.concat([
self.batch_shape_tensor(), [self._num_states],
self._observation_distribution.event_shape_tensor()
],
axis=0)
means = tf.broadcast_to(means, means_shape)
# means :: batch_shape num_states observation_event_shape
observation_event_shape = (
self._observation_distribution.event_shape_tensor())
batch_size = tf.reduce_prod(self.batch_shape_tensor())
flat_probs_shape = [self._num_steps, batch_size, self._num_states]
flat_means_shape = [
batch_size, self._num_states,
tf.reduce_prod(observation_event_shape)
]
flat_probs = tf.reshape(probs, flat_probs_shape)
# flat_probs :: num_steps batch_size num_states
flat_means = tf.reshape(means, flat_means_shape)
# flat_means :: batch_size num_states observation_event_size
flat_mean = tf.einsum("ijk,jkl->jil", flat_probs, flat_means)
# flat_mean :: batch_size num_steps observation_event_size
unflat_mean_shape = tf.concat([
self.batch_shape_tensor(), [self._num_steps],
observation_event_shape
],
axis=0)
# returns :: batch_shape num_steps observation_event_shape
return tf.reshape(flat_mean, unflat_mean_shape)
def lu_reconstruct(lower_upper, perm, validate_args=False, name=None):
"""The inverse LU decomposition, `X == lu_reconstruct(*tf.linalg.lu(X))`.
Args:
lower_upper: `lu` as returned by `tf.linalg.lu`, i.e., if
`matmul(P, matmul(L, U)) = X` then `lower_upper = L + U - eye`.
perm: `p` as returned by `tf.linag.lu`, i.e., if
`matmul(P, matmul(L, U)) = X` then `perm = argmax(P)`.
validate_args: Python `bool` indicating whether arguments should be checked
for correctness.
Default value: `False` (i.e., don't validate arguments).
name: Python `str` name given to ops managed by this object.
Default value: `None` (i.e., 'lu_reconstruct').
Returns:
x: The original input to `tf.linalg.lu`, i.e., `x` as in,
`lu_reconstruct(*tf.linalg.lu(x))`.
#### Examples
```python
import numpy as np
from tensorflow_probability.python.internal.backend import jax as tf
import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.jax
x = [[[3., 4], [1, 2]],
[[7., 8], [3, 4]]]
x_reconstructed = tfp.math.lu_reconstruct(*tf.linalg.lu(x))
tf.assert_near(x, x_reconstructed)
# ==> True
```
"""
with tf.name_scope(name or 'lu_reconstruct'):
lower_upper = tf.convert_to_tensor(lower_upper,
dtype_hint=tf.float32,
name='lower_upper')
perm = tf.convert_to_tensor(perm, dtype_hint=tf.int32, name='perm')
assertions = lu_reconstruct_assertions(lower_upper, perm,
validate_args)
if assertions:
with tf.control_dependencies(assertions):
lower_upper = tf.identity(lower_upper)
perm = tf.identity(perm)
shape = tf.shape(lower_upper)
lower = tf.linalg.set_diag(
tf.linalg.band_part(lower_upper, num_lower=-1, num_upper=0),
tf.ones(shape[:-1], dtype=lower_upper.dtype))
upper = tf.linalg.band_part(lower_upper, num_lower=0, num_upper=-1)
x = tf.matmul(lower, upper)
if (tensorshape_util.rank(lower_upper.shape) is None
or tensorshape_util.rank(lower_upper.shape) != 2):
# We either don't know the batch rank or there are >0 batch dims.
batch_size = tf.reduce_prod(shape[:-2])
d = shape[-1]
x = tf.reshape(x, [batch_size, d, d])
perm = tf.reshape(perm, [batch_size, d])
perm = tf.map_fn(tf.math.invert_permutation, perm)
batch_indices = tf.broadcast_to(
tf.range(batch_size)[:, tf.newaxis], [batch_size, d])
x = tf.gather_nd(x, tf.stack([batch_indices, perm], axis=-1))
x = tf.reshape(x, shape)
else:
x = tf.gather(x, tf.math.invert_permutation(perm))
x.set_shape(lower_upper.shape)
return x
def _mode(self):
loc = tf.convert_to_tensor(self.loc)
return tf.broadcast_to(loc, self._batch_shape_tensor(loc=loc))
def _stddev(self):
scale = tf.convert_to_tensor(self.scale)
return tf.broadcast_to(scale * np.pi / np.sqrt(3),
self._batch_shape_tensor(scale=scale))
def _entropy(self):
scale = tf.convert_to_tensor(self.scale)
return tf.broadcast_to(np.log(2.) + 1 + tf.math.log(scale),
self._batch_shape_tensor(scale=scale))
def _mode(self):
return tf.broadcast_to(self.loc, self._sample_shape())
def _stddev(self):
scale = tf.convert_to_tensor(self.scale)
return tf.broadcast_to(np.sqrt(2.) * scale,
self._batch_shape_tensor(scale=scale))
def _mode(self): scale = tf.convert_to_tensor(self.scale) return tf.broadcast_to(scale, self._batch_shape_tensor(scale=scale))
def pivoted_cholesky(matrix, max_rank, diag_rtol=1e-3, name=None):
"""Computes the (partial) pivoted cholesky decomposition of `matrix`.
The pivoted Cholesky is a low rank approximation of the Cholesky decomposition
of `matrix`, i.e. as described in [(Harbrecht et al., 2012)][1]. The
currently-worst-approximated diagonal element is selected as the pivot at each
iteration. This yields from a `[B1...Bn, N, N]` shaped `matrix` a `[B1...Bn,
N, K]` shaped rank-`K` approximation `lr` such that `lr @ lr.T ~= matrix`.
Note that, unlike the Cholesky decomposition, `lr` is not triangular even in
a rectangular-matrix sense. However, under a permutation it could be made
triangular (it has one more zero in each column as you move to the right).
Such a matrix can be useful as a preconditioner for conjugate gradient
optimization, i.e. as in [(Wang et al. 2019)][2], as matmuls and solves can be
cheaply done via the Woodbury matrix identity, as implemented by
`tf.linalg.LinearOperatorLowRankUpdate`.
Args:
matrix: Floating point `Tensor` batch of symmetric, positive definite
matrices.
max_rank: Scalar `int` `Tensor`, the rank at which to truncate the
approximation.
diag_rtol: Scalar floating point `Tensor` (same dtype as `matrix`). If the
errors of all diagonal elements of `lr @ lr.T` are each lower than
`element * diag_rtol`, iteration is permitted to terminate early.
name: Optional name for the op.
Returns:
lr: Low rank pivoted Cholesky approximation of `matrix`.
#### References
[1]: H Harbrecht, M Peters, R Schneider. On the low-rank approximation by the
pivoted Cholesky decomposition. _Applied numerical mathematics_,
62(4):428-440, 2012.
[2]: K. A. Wang et al. Exact Gaussian Processes on a Million Data Points.
_arXiv preprint arXiv:1903.08114_, 2019. https://arxiv.org/abs/1903.08114
"""
with tf.name_scope(name or 'pivoted_cholesky'):
dtype = dtype_util.common_dtype([matrix, diag_rtol],
dtype_hint=tf.float32)
matrix = tf.convert_to_tensor(matrix, name='matrix', dtype=dtype)
if tensorshape_util.rank(matrix.shape) is None:
raise NotImplementedError(
'Rank of `matrix` must be known statically')
max_rank = tf.convert_to_tensor(max_rank,
name='max_rank',
dtype=tf.int64)
max_rank = tf.minimum(
max_rank,
prefer_static.shape(matrix, out_type=tf.int64)[-1])
diag_rtol = tf.convert_to_tensor(diag_rtol,
dtype=dtype,
name='diag_rtol')
matrix_diag = tf.linalg.diag_part(matrix)
# matrix is P.D., therefore all matrix_diag > 0, so we don't need abs.
orig_error = tf.reduce_max(matrix_diag, axis=-1)
def cond(m, pchol, perm, matrix_diag):
"""Condition for `tf.while_loop` continuation."""
del pchol
del perm
error = tf.linalg.norm(matrix_diag, ord=1, axis=-1)
max_err = tf.reduce_max(error / orig_error)
return (m < max_rank) & (tf.equal(m, 0) | (max_err > diag_rtol))
batch_dims = tensorshape_util.rank(matrix.shape) - 2
def batch_gather(params, indices, axis=-1):
return tf.gather(params, indices, axis=axis, batch_dims=batch_dims)
def body(m, pchol, perm, matrix_diag):
"""Body of a single `tf.while_loop` iteration."""
# Here is roughly a numpy, non-batched version of what's going to happen.
# (See also Algorithm 1 of Harbrecht et al.)
# 1: maxi = np.argmax(matrix_diag[perm[m:]]) + m
# 2: maxval = matrix_diag[perm][maxi]
# 3: perm[m], perm[maxi] = perm[maxi], perm[m]
# 4: row = matrix[perm[m]][perm[m + 1:]]
# 5: row -= np.sum(pchol[:m][perm[m + 1:]] * pchol[:m][perm[m]]], axis=-2)
# 6: pivot = np.sqrt(maxval); row /= pivot
# 7: row = np.concatenate([[[pivot]], row], -1)
# 8: matrix_diag[perm[m:]] -= row**2
# 9: pchol[m, perm[m:]] = row
# Find the maximal position of the (remaining) permuted diagonal.
# Steps 1, 2 above.
permuted_diag = batch_gather(matrix_diag, perm[..., m:])
maxi = tf.argmax(permuted_diag, axis=-1,
output_type=tf.int64)[..., tf.newaxis]
maxval = batch_gather(permuted_diag, maxi)
maxi = maxi + m
maxval = maxval[..., 0]
# Update perm: Swap perm[...,m] with perm[...,maxi]. Step 3 above.
perm = _swap_m_with_i(perm, m, maxi)
# Step 4.
row = batch_gather(matrix, perm[..., m:m + 1], axis=-2)
row = batch_gather(row, perm[..., m + 1:])
# Step 5.
prev_rows = pchol[..., :m, :]
prev_rows_perm_m_onward = batch_gather(prev_rows, perm[...,
m + 1:])
prev_rows_pivot_col = batch_gather(prev_rows, perm[..., m:m + 1])
row -= tf.reduce_sum(prev_rows_perm_m_onward * prev_rows_pivot_col,
axis=-2)[..., tf.newaxis, :]
# Step 6.
pivot = tf.sqrt(maxval)[..., tf.newaxis, tf.newaxis]
# Step 7.
row = tf.concat([pivot, row / pivot], axis=-1)
# TODO(b/130899118): Pad grad fails with int64 paddings.
# Step 8.
paddings = tf.concat([
tf.zeros([prefer_static.rank(pchol) - 1, 2], dtype=tf.int32),
[[tf.cast(m, tf.int32), 0]]
],
axis=0)
diag_update = tf.pad(row**2, paddings=paddings)[..., 0, :]
reverse_perm = _invert_permutation(perm)
matrix_diag -= batch_gather(diag_update, reverse_perm)
# Step 9.
row = tf.pad(row, paddings=paddings)
# TODO(bjp): Defer the reverse permutation all-at-once at the end?
row = batch_gather(row, reverse_perm)
pchol_shape = pchol.shape
pchol = tf.concat([pchol[..., :m, :], row, pchol[..., m + 1:, :]],
axis=-2)
tensorshape_util.set_shape(pchol, pchol_shape)
return m + 1, pchol, perm, matrix_diag
m = np.int64(0)
pchol = tf.zeros_like(matrix[..., :max_rank, :])
matrix_shape = prefer_static.shape(matrix, out_type=tf.int64)
perm = tf.broadcast_to(prefer_static.range(matrix_shape[-1]),
matrix_shape[:-1])
_, pchol, _, _ = tf.while_loop(cond=cond,
body=body,
loop_vars=(m, pchol, perm, matrix_diag))
pchol = tf.linalg.matrix_transpose(pchol)
tensorshape_util.set_shape(
pchol, tensorshape_util.concatenate(matrix_diag.shape, [None]))
return pchol
def lu_solve(lower_upper, perm, rhs, validate_args=False, name=None):
"""Solves systems of linear eqns `A X = RHS`, given LU factorizations.
Note: this function does not verify the implied matrix is actually invertible
nor is this condition checked even when `validate_args=True`.
Args:
lower_upper: `lu` as returned by `tf.linalg.lu`, i.e., if
`matmul(P, matmul(L, U)) = X` then `lower_upper = L + U - eye`.
perm: `p` as returned by `tf.linag.lu`, i.e., if
`matmul(P, matmul(L, U)) = X` then `perm = argmax(P)`.
rhs: Matrix-shaped float `Tensor` representing targets for which to solve;
`A X = RHS`. To handle vector cases, use:
`lu_solve(..., rhs[..., tf.newaxis])[..., 0]`.
validate_args: Python `bool` indicating whether arguments should be checked
for correctness. Note: this function does not verify the implied matrix is
actually invertible, even when `validate_args=True`.
Default value: `False` (i.e., don't validate arguments).
name: Python `str` name given to ops managed by this object.
Default value: `None` (i.e., 'lu_solve').
Returns:
x: The `X` in `A @ X = RHS`.
#### Examples
```python
import numpy as np
from tensorflow_probability.python.internal.backend import jax as tf
import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.jax
x = [[[1., 2],
[3, 4]],
[[7, 8],
[3, 4]]]
inv_x = tfp.math.lu_solve(*tf.linalg.lu(x), rhs=tf.eye(2))
tf.assert_near(tf.matrix_inverse(x), inv_x)
# ==> True
```
"""
with tf.name_scope(name or 'lu_solve'):
lower_upper = tf.convert_to_tensor(lower_upper,
dtype_hint=tf.float32,
name='lower_upper')
perm = tf.convert_to_tensor(perm, dtype_hint=tf.int32, name='perm')
rhs = tf.convert_to_tensor(rhs,
dtype_hint=lower_upper.dtype,
name='rhs')
assertions = _lu_solve_assertions(lower_upper, perm, rhs,
validate_args)
if assertions:
with tf.control_dependencies(assertions):
lower_upper = tf.identity(lower_upper)
perm = tf.identity(perm)
rhs = tf.identity(rhs)
if (tensorshape_util.rank(rhs.shape) == 2
and tensorshape_util.rank(perm.shape) == 1):
# Both rhs and perm have scalar batch_shape.
permuted_rhs = tf.gather(rhs, perm, axis=-2)
else:
# Either rhs or perm have non-scalar batch_shape or we can't determine
# this information statically.
rhs_shape = tf.shape(rhs)
broadcast_batch_shape = tf.broadcast_dynamic_shape(
rhs_shape[:-2],
tf.shape(perm)[:-1])
d, m = rhs_shape[-2], rhs_shape[-1]
rhs_broadcast_shape = tf.concat([broadcast_batch_shape, [d, m]],
axis=0)
# Tile out rhs.
broadcast_rhs = tf.broadcast_to(rhs, rhs_broadcast_shape)
broadcast_rhs = tf.reshape(broadcast_rhs, [-1, d, m])
# Tile out perm and add batch indices.
broadcast_perm = tf.broadcast_to(perm, rhs_broadcast_shape[:-1])
broadcast_perm = tf.reshape(broadcast_perm, [-1, d])
broadcast_batch_size = tf.reduce_prod(broadcast_batch_shape)
broadcast_batch_indices = tf.broadcast_to(
tf.range(broadcast_batch_size)[:, tf.newaxis],
[broadcast_batch_size, d])
broadcast_perm = tf.stack(
[broadcast_batch_indices, broadcast_perm], axis=-1)
permuted_rhs = tf.gather_nd(broadcast_rhs, broadcast_perm)
permuted_rhs = tf.reshape(permuted_rhs, rhs_broadcast_shape)
lower = tf.linalg.set_diag(
tf.linalg.band_part(lower_upper, num_lower=-1, num_upper=0),
tf.ones(tf.shape(lower_upper)[:-1], dtype=lower_upper.dtype))
return linear_operator_util.matrix_triangular_solve_with_broadcast(
lower_upper, # Only upper is accessed.
linear_operator_util.matrix_triangular_solve_with_broadcast(
lower, permuted_rhs),
lower=False)
