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 _log_prob(self, x, **kwargs):
batch_ndims = prefer_static.rank_from_shape(
self.distribution.batch_shape_tensor,
self.distribution.batch_shape)
extra_sample_ndims = prefer_static.rank_from_shape(self.sample_shape)
event_ndims = prefer_static.rank_from_shape(
self.distribution.event_shape_tensor,
self.distribution.event_shape)
ndims = prefer_static.rank(x)
# (1) Expand x's dims.
d = ndims - batch_ndims - extra_sample_ndims - event_ndims
x = tf.reshape(x,
shape=tf.pad(
tf.shape(x),
paddings=[[prefer_static.maximum(0, -d), 0]],
constant_values=1))
sample_ndims = prefer_static.maximum(0, d)
# (2) Transpose x's dims.
sample_dims = prefer_static.range(0, sample_ndims)
batch_dims = prefer_static.range(sample_ndims,
sample_ndims + batch_ndims)
extra_sample_dims = prefer_static.range(
sample_ndims + batch_ndims,
sample_ndims + batch_ndims + extra_sample_ndims)
event_dims = prefer_static.range(
sample_ndims + batch_ndims + extra_sample_ndims, ndims)
perm = prefer_static.concat(
[sample_dims, extra_sample_dims, batch_dims, event_dims], axis=0)
x = tf.transpose(a=x, perm=perm)
# (3) Compute x's log_prob.
lp = self.distribution.log_prob(x, **kwargs)
# (4) Make the final reduction in x.
axis = prefer_static.range(sample_ndims,
sample_ndims + extra_sample_ndims)
return tf.reduce_sum(lp, axis=axis)
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
def _maybe_rotate_dims(self, x, rotate_right=False):
"""Helper which rolls left event_dims left or right event_dims right."""
needs_rotation_const = tf.get_static_value(self._needs_rotation)
if needs_rotation_const is not None and not needs_rotation_const:
return x
ndims = prefer_static.rank(x)
n = (ndims -
self._rotate_ndims) if rotate_right else self._rotate_ndims
perm = prefer_static.concat(
[prefer_static.range(n, ndims),
prefer_static.range(0, n)], axis=0)
return tf.transpose(a=x, perm=perm)
def __init__(self,
samples,
event_ndims=0,
validate_args=False,
allow_nan_stats=True,
name='Empirical'):
"""Initialize `Empirical` distributions.
Args:
samples: Numeric `Tensor` of shape [B1, ..., Bk, S, E1, ..., En]`,
`k, n >= 0`. Samples or batches of samples on which the distribution
is based. The first `k` dimensions index into a batch of independent
distributions. Length of `S` dimension determines number of samples
in each multiset. The last `n` dimension represents samples for each
distribution. n is specified by argument event_ndims.
event_ndims: Python `int32`, default `0`. number of dimensions for each
event. When `0` this distribution has scalar samples. When `1` this
distribution has vector-like samples.
validate_args: Python `bool`, default `False`. When `True` distribution
parameters are checked for validity despite possibly degrading runtime
performance. When `False` invalid inputs may silently render incorrect
outputs.
allow_nan_stats: Python `bool`, default `True`. When `True`, statistics
(e.g., mean, mode, variance) use the value `NaN` to indicate the
result is undefined. When `False`, an exception is raised if one or
more of the statistic's batch members are undefined.
name: Python `str` name prefixed to Ops created by this class.
Raises:
ValueError: if the rank of `samples` is statically known and less than
event_ndims + 1.
"""
parameters = dict(locals())
with tf.name_scope(name):
self._samples = tensor_util.convert_nonref_to_tensor(samples)
dtype = dtype_util.common_dtype(
[self._samples], dtype_hint=self._samples.dtype)
self._event_ndims = event_ndims
# Note: this tf.rank call affects the graph, but is ok in `__init__`
# because we don't expect shapes (or ranks) to be runtime-variable, nor
# ever need to differentiate with respect to them.
samples_rank = prefer_static.rank(self.samples)
self._samples_axis = samples_rank - self._event_ndims - 1
super(Empirical, self).__init__(
dtype=dtype,
reparameterization_type=reparameterization.FULLY_REPARAMETERIZED,
validate_args=validate_args,
allow_nan_stats=allow_nan_stats,
parameters=parameters,
name=name)
def _inverse(self, y):
ndims = prefer_static.rank(y)
shifted_y = tf.pad(
tf.slice(
y, tf.zeros(ndims, dtype=tf.int32),
prefer_static.shape(y) -
tf.one_hot(ndims + self.axis, ndims, dtype=tf.int32)
), # Remove the last entry of y in the chosen dimension.
paddings=tf.one_hot(
tf.one_hot(ndims + self.axis, ndims, on_value=0, off_value=-1),
2,
dtype=tf.int32
) # Insert zeros at the beginning of the chosen dimension.
)
return y - shifted_y
def _forward(self, x):
x = tf.convert_to_tensor(x, name='x')
batch_shape = prefer_static.shape(x)[:-1]
# Pad zeros on the top row and right column.
y = fill_triangular.FillTriangular().forward(x)
rank = prefer_static.rank(y)
paddings = tf.concat([
tf.zeros(shape=(rank - 2, 2), dtype=tf.int32),
tf.constant([[1, 0], [0, 1]], dtype=tf.int32)
],
axis=0)
y = tf.pad(y, paddings)
# Set diagonal to 1s.
n = prefer_static.shape(y)[-1]
diag = tf.ones(tf.concat([batch_shape, [n]], axis=-1), dtype=x.dtype)
y = tf.linalg.set_diag(y, diag)
# Normalize each row to have Euclidean (L2) norm 1.
y /= tf.norm(y, axis=-1)[..., tf.newaxis]
return y
def fill_triangular(x, upper=False, name=None):
"""Creates a (batch of) triangular matrix from a vector of inputs.
Created matrix can be lower- or upper-triangular. (It is more efficient to
create the matrix as upper or lower, rather than transpose.)
Triangular matrix elements are filled in a clockwise spiral. See example,
below.
If `x.shape` is `[b1, b2, ..., bB, d]` then the output shape is
`[b1, b2, ..., bB, n, n]` where `n` is such that `d = n(n+1)/2`, i.e.,
`n = int(np.sqrt(0.25 + 2. * m) - 0.5)`.
Example:
```python
fill_triangular([1, 2, 3, 4, 5, 6])
# ==> [[4, 0, 0],
# [6, 5, 0],
# [3, 2, 1]]
fill_triangular([1, 2, 3, 4, 5, 6], upper=True)
# ==> [[1, 2, 3],
# [0, 5, 6],
# [0, 0, 4]]
```
The key trick is to create an upper triangular matrix by concatenating `x`
and a tail of itself, then reshaping.
Suppose that we are filling the upper triangle of an `n`-by-`n` matrix `M`
from a vector `x`. The matrix `M` contains n**2 entries total. The vector `x`
contains `n * (n+1) / 2` entries. For concreteness, we'll consider `n = 5`
(so `x` has `15` entries and `M` has `25`). We'll concatenate `x` and `x` with
the first (`n = 5`) elements removed and reversed:
```python
x = np.arange(15) + 1
xc = np.concatenate([x, x[5:][::-1]])
# ==> array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 14, 13,
# 12, 11, 10, 9, 8, 7, 6])
# (We add one to the arange result to disambiguate the zeros below the
# diagonal of our upper-triangular matrix from the first entry in `x`.)
# Now, when reshapedlay this out as a matrix:
y = np.reshape(xc, [5, 5])
# ==> array([[ 1, 2, 3, 4, 5],
# [ 6, 7, 8, 9, 10],
# [11, 12, 13, 14, 15],
# [15, 14, 13, 12, 11],
# [10, 9, 8, 7, 6]])
# Finally, zero the elements below the diagonal:
y = np.triu(y, k=0)
# ==> array([[ 1, 2, 3, 4, 5],
# [ 0, 7, 8, 9, 10],
# [ 0, 0, 13, 14, 15],
# [ 0, 0, 0, 12, 11],
# [ 0, 0, 0, 0, 6]])
```
From this example we see that the resuting matrix is upper-triangular, and
contains all the entries of x, as desired. The rest is details:
- If `n` is even, `x` doesn't exactly fill an even number of rows (it fills
`n / 2` rows and half of an additional row), but the whole scheme still
works.
- If we want a lower triangular matrix instead of an upper triangular,
we remove the first `n` elements from `x` rather than from the reversed
`x`.
For additional comparisons, a pure numpy version of this function can be found
in `distribution_util_test.py`, function `_fill_triangular`.
Args:
x: `Tensor` representing lower (or upper) triangular elements.
upper: Python `bool` representing whether output matrix should be upper
triangular (`True`) or lower triangular (`False`, default).
name: Python `str`. The name to give this op.
Returns:
tril: `Tensor` with lower (or upper) triangular elements filled from `x`.
Raises:
ValueError: if `x` cannot be mapped to a triangular matrix.
"""
with tf.name_scope(name or 'fill_triangular'):
x = tf.convert_to_tensor(x, name='x')
m = tf.compat.dimension_value(
tensorshape_util.with_rank_at_least(x.shape, 1)[-1])
if m is not None:
# Formula derived by solving for n: m = n(n+1)/2.
m = np.int32(m)
n = np.sqrt(0.25 + 2. * m) - 0.5
if n != np.floor(n):
raise ValueError(
'Input right-most shape ({}) does not '
'correspond to a triangular matrix.'.format(m))
n = np.int32(n)
static_final_shape = tensorshape_util.concatenate(
x.shape[:-1], [n, n])
else:
m = tf.shape(x)[-1]
# For derivation, see above. Casting automatically lops off the 0.5, so we
# omit it. We don't validate n is an integer because this has
# graph-execution cost; an error will be thrown from the reshape, below.
n = tf.cast(tf.sqrt(0.25 + tf.cast(2 * m, dtype=tf.float32)),
dtype=tf.int32)
static_final_shape = tensorshape_util.concatenate(
tensorshape_util.with_rank_at_least(x.shape, 1)[:-1],
[None, None])
# Try it out in numpy:
# n = 3
# x = np.arange(n * (n + 1) / 2)
# m = x.shape[0]
# n = np.int32(np.sqrt(.25 + 2 * m) - .5)
# x_tail = x[(m - (n**2 - m)):]
# np.concatenate([x_tail, x[::-1]], 0).reshape(n, n) # lower
# # ==> array([[3, 4, 5],
# [5, 4, 3],
# [2, 1, 0]])
# np.concatenate([x, x_tail[::-1]], 0).reshape(n, n) # upper
# # ==> array([[0, 1, 2],
# [3, 4, 5],
# [5, 4, 3]])
#
# Note that we can't simply do `x[..., -(n**2 - m):]` because this doesn't
# correctly handle `m == n == 1`. Hence, we do nonnegative indexing.
# Furthermore observe that:
# m - (n**2 - m)
# = n**2 / 2 + n / 2 - (n**2 - n**2 / 2 + n / 2)
# = 2 (n**2 / 2 + n / 2) - n**2
# = n**2 + n - n**2
# = n
ndims = prefer_static.rank(x)
if upper:
x_list = [x, tf.reverse(x[..., n:], axis=[ndims - 1])]
else:
x_list = [x[..., n:], tf.reverse(x, axis=[ndims - 1])]
new_shape = (tensorshape_util.as_list(static_final_shape)
if tensorshape_util.is_fully_defined(static_final_shape)
else tf.concat([tf.shape(x)[:-1], [n, n]], axis=0))
x = tf.reshape(tf.concat(x_list, axis=-1), new_shape)
x = tf.linalg.band_part(x,
num_lower=(0 if upper else -1),
num_upper=(-1 if upper else 0))
tensorshape_util.set_shape(x, static_final_shape)
return x
def fill_triangular_inverse(x, upper=False, name=None):
"""Creates a vector from a (batch of) triangular matrix.
The vector is created from the lower-triangular or upper-triangular portion
depending on the value of the parameter `upper`.
If `x.shape` is `[b1, b2, ..., bB, n, n]` then the output shape is
`[b1, b2, ..., bB, d]` where `d = n (n + 1) / 2`.
Example:
```python
fill_triangular_inverse(
[[4, 0, 0],
[6, 5, 0],
[3, 2, 1]])
# ==> [1, 2, 3, 4, 5, 6]
fill_triangular_inverse(
[[1, 2, 3],
[0, 5, 6],
[0, 0, 4]], upper=True)
# ==> [1, 2, 3, 4, 5, 6]
```
Args:
x: `Tensor` representing lower (or upper) triangular elements.
upper: Python `bool` representing whether output matrix should be upper
triangular (`True`) or lower triangular (`False`, default).
name: Python `str`. The name to give this op.
Returns:
flat_tril: (Batch of) vector-shaped `Tensor` representing vectorized lower
(or upper) triangular elements from `x`.
"""
with tf.name_scope(name or 'fill_triangular_inverse'):
x = tf.convert_to_tensor(x, name='x')
n = tf.compat.dimension_value(
tensorshape_util.with_rank_at_least(x.shape, 2)[-1])
if n is not None:
n = np.int32(n)
m = np.int32((n * (n + 1)) // 2)
static_final_shape = tensorshape_util.concatenate(
x.shape[:-2], [m])
else:
n = tf.shape(x)[-1]
m = (n * (n + 1)) // 2
static_final_shape = tensorshape_util.concatenate(
tensorshape_util.with_rank_at_least(x.shape, 2)[:-2], [None])
ndims = prefer_static.rank(x)
if upper:
initial_elements = x[..., 0, :]
triangular_portion = x[..., 1:, :]
else:
initial_elements = tf.reverse(x[..., -1, :], axis=[ndims - 2])
triangular_portion = x[..., :-1, :]
rotated_triangular_portion = tf.reverse(tf.reverse(triangular_portion,
axis=[ndims - 1]),
axis=[ndims - 2])
consolidated_matrix = triangular_portion + rotated_triangular_portion
end_sequence = tf.reshape(
consolidated_matrix,
tf.concat([tf.shape(x)[:-2], [n * (n - 1)]], axis=0))
y = tf.concat([initial_elements, end_sequence[..., :m - n]], axis=-1)
tensorshape_util.set_shape(y, static_final_shape)
return y