# Copyright 2017-2020 The GPflow Contributors. All Rights Reserved.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
from typing import Callable, Optional, Tuple
import tensorflow as tf
from ..base import MeanAndVariance
from ..config import default_float, default_jitter
from ..utilities.ops import leading_transpose
[docs]def base_conditional(
Kmn: tf.Tensor,
Kmm: tf.Tensor,
Knn: tf.Tensor,
f: tf.Tensor,
*,
full_cov: bool = False,
q_sqrt: Optional[tf.Tensor] = None,
white: bool = False,
) -> MeanAndVariance:
r"""
Given a g1 and g2, and distribution p and q such that
p(g2) = N(g2; 0, Kmm)
p(g1) = N(g1; 0, Knn)
p(g1 | g2) = N(g1; Knm (Kmm⁻¹) g2, Knn - Knm (Kmm⁻¹) Kmn)
And
q(g2) = N(g2; f, q_sqrt q_sqrtᵀ)
This method computes the mean and (co)variance of
q(g1) = ∫ q(g2) p(g1 | g2)
:param Kmn: [M, ..., N]
:param Kmm: [M, M]
:param Knn: [..., N, N] or N
:param f: [M, R]
:param full_cov: bool
:param q_sqrt: If this is a Tensor, it must have shape [R, M, M] (lower
triangular) or [M, R] (diagonal)
:param white: bool
:return: [N, R] or [R, N, N]
"""
Lm = tf.linalg.cholesky(Kmm)
return base_conditional_with_lm(
Kmn=Kmn, Lm=Lm, Knn=Knn, f=f, full_cov=full_cov, q_sqrt=q_sqrt, white=white
)
[docs]def base_conditional_with_lm(
Kmn: tf.Tensor,
Lm: tf.Tensor,
Knn: tf.Tensor,
f: tf.Tensor,
*,
full_cov: bool = False,
q_sqrt: Optional[tf.Tensor] = None,
white: bool = False,
) -> MeanAndVariance:
r"""
Has the same functionality as the `base_conditional` function, except that instead of
`Kmm` this function accepts `Lm`, which is the Cholesky decomposition of `Kmm`.
This allows `Lm` to be precomputed, which can improve performance.
"""
# compute kernel stuff
num_func = tf.shape(f)[-1] # R
N = tf.shape(Kmn)[-1]
M = tf.shape(f)[-2]
# get the leading dims in Kmn to the front of the tensor
# if Kmn has rank two, i.e. [M, N], this is the identity op.
K = tf.rank(Kmn)
perm = tf.concat(
[
tf.reshape(tf.range(1, K - 1), [K - 2]), # leading dims (...)
tf.reshape(0, [1]), # [M]
tf.reshape(K - 1, [1]),
],
0,
) # [N]
Kmn = tf.transpose(Kmn, perm) # [..., M, N]
shape_constraints = [
(Kmn, [..., "M", "N"]),
(Lm, ["M", "M"]),
(Knn, [..., "N", "N"] if full_cov else [..., "N"]),
(f, ["M", "R"]),
]
if q_sqrt is not None:
shape_constraints.append(
(q_sqrt, (["M", "R"] if q_sqrt.shape.ndims == 2 else ["R", "M", "M"]))
)
tf.debugging.assert_shapes(
shape_constraints,
message="base_conditional() arguments "
"[Note that this check verifies the shape of an alternative "
"representation of Kmn. See the docs for the actual expected "
"shape.]",
)
leading_dims = tf.shape(Kmn)[:-2]
# Compute the projection matrix A
Lm = tf.broadcast_to(Lm, tf.concat([leading_dims, tf.shape(Lm)], 0)) # [..., M, M]
A = tf.linalg.triangular_solve(Lm, Kmn, lower=True) # [..., M, N]
# compute the covariance due to the conditioning
if full_cov:
fvar = Knn - tf.linalg.matmul(A, A, transpose_a=True) # [..., N, N]
cov_shape = tf.concat([leading_dims, [num_func, N, N]], 0)
fvar = tf.broadcast_to(tf.expand_dims(fvar, -3), cov_shape) # [..., R, N, N]
else:
fvar = Knn - tf.reduce_sum(tf.square(A), -2) # [..., N]
cov_shape = tf.concat([leading_dims, [num_func, N]], 0) # [..., R, N]
fvar = tf.broadcast_to(tf.expand_dims(fvar, -2), cov_shape) # [..., R, N]
# another backsubstitution in the unwhitened case
if not white:
A = tf.linalg.triangular_solve(tf.linalg.adjoint(Lm), A, lower=False)
# construct the conditional mean
f_shape = tf.concat([leading_dims, [M, num_func]], 0) # [..., M, R]
f = tf.broadcast_to(f, f_shape) # [..., M, R]
fmean = tf.linalg.matmul(A, f, transpose_a=True) # [..., N, R]
if q_sqrt is not None:
q_sqrt_dims = q_sqrt.shape.ndims
if q_sqrt_dims == 2:
LTA = A * tf.expand_dims(tf.transpose(q_sqrt), 2) # [R, M, N]
elif q_sqrt_dims == 3:
L = tf.linalg.band_part(q_sqrt, -1, 0) # force lower triangle # [R, M, M]
L_shape = tf.shape(L)
L = tf.broadcast_to(L, tf.concat([leading_dims, L_shape], 0))
shape = tf.concat([leading_dims, [num_func, M, N]], axis=0)
A_tiled = tf.broadcast_to(tf.expand_dims(A, -3), shape)
LTA = tf.linalg.matmul(L, A_tiled, transpose_a=True) # [R, M, N]
else: # pragma: no cover
raise ValueError("Bad dimension for q_sqrt: %s" % str(q_sqrt.shape.ndims))
if full_cov:
fvar = fvar + tf.linalg.matmul(LTA, LTA, transpose_a=True) # [R, N, N]
else:
fvar = fvar + tf.reduce_sum(tf.square(LTA), -2) # [R, N]
if not full_cov:
fvar = tf.linalg.adjoint(fvar) # [N, R]
shape_constraints = [
(Kmn, [..., "M", "N"]), # tensor included again for N dimension
(f, [..., "M", "R"]), # tensor included again for R dimension
(fmean, [..., "N", "R"]),
(fvar, [..., "R", "N", "N"] if full_cov else [..., "N", "R"]),
]
tf.debugging.assert_shapes(shape_constraints, message="base_conditional() return values")
return fmean, fvar
[docs]def sample_mvn(
mean: tf.Tensor, cov: tf.Tensor, full_cov: bool, num_samples: Optional[int] = None
) -> tf.Tensor:
"""
Returns a sample from a D-dimensional Multivariate Normal distribution
:param mean: [..., N, D]
:param cov: [..., N, D] or [..., N, D, D]
:param full_cov: if `True` return a "full" covariance matrix, otherwise a "diag":
- "full": cov holds the full covariance matrix (without jitter)
- "diag": cov holds the diagonal elements of the covariance matrix
:return: sample from the MVN of shape [..., (S), N, D], S = num_samples
"""
shape_constraints = [
(mean, [..., "N", "D"]),
(cov, [..., "N", "D", "D"] if full_cov else [..., "N", "D"]),
]
tf.debugging.assert_shapes(shape_constraints, message="sample_mvn() arguments")
mean_shape = tf.shape(mean)
S = num_samples if num_samples is not None else 1
D = mean_shape[-1]
leading_dims = mean_shape[:-2]
if not full_cov:
# mean: [..., N, D] and cov [..., N, D]
eps_shape = tf.concat([leading_dims, [S], mean_shape[-2:]], 0)
eps = tf.random.normal(eps_shape, dtype=default_float()) # [..., S, N, D]
samples = mean[..., None, :, :] + tf.sqrt(cov)[..., None, :, :] * eps # [..., S, N, D]
else:
# mean: [..., N, D] and cov [..., N, D, D]
jittermat = (
tf.eye(D, batch_shape=mean_shape[:-1], dtype=default_float()) * default_jitter()
) # [..., N, D, D]
eps_shape = tf.concat([mean_shape, [S]], 0)
eps = tf.random.normal(eps_shape, dtype=default_float()) # [..., N, D, S]
chol = tf.linalg.cholesky(cov + jittermat) # [..., N, D, D]
samples = mean[..., None] + tf.linalg.matmul(chol, eps) # [..., N, D, S]
samples = leading_transpose(samples, [..., -1, -3, -2]) # [..., S, N, D]
shape_constraints = [
(mean, [..., "N", "D"]),
(samples, [..., "S", "N", "D"]),
]
tf.debugging.assert_shapes(shape_constraints, message="sample_mvn() return values")
if num_samples is None:
return tf.squeeze(samples, axis=-3) # [..., N, D]
return samples # [..., S, N, D]
[docs]def expand_independent_outputs(fvar: tf.Tensor, full_cov: bool, full_output_cov: bool) -> tf.Tensor:
"""
Reshapes fvar to the correct shape, specified by `full_cov` and `full_output_cov`.
:param fvar: has shape [N, P] (full_cov = False) or [P, N, N] (full_cov = True).
:return:
1. full_cov: True and full_output_cov: True
fvar [N, P, N, P]
2. full_cov: True and full_output_cov: False
fvar [P, N, N]
3. full_cov: False and full_output_cov: True
fvar [N, P, P]
4. full_cov: False and full_output_cov: False
fvar [N, P]
"""
if full_cov and full_output_cov:
fvar = tf.linalg.diag(tf.transpose(fvar)) # [N, N, P, P]
fvar = tf.transpose(fvar, [0, 2, 1, 3]) # [N, P, N, P]
if not full_cov and full_output_cov:
fvar = tf.linalg.diag(fvar) # [N, P, P]
if full_cov and not full_output_cov:
pass # [P, N, N]
if not full_cov and not full_output_cov:
pass # [N, P]
return fvar
[docs]def independent_interdomain_conditional(
Kmn: tf.Tensor,
Kmm: tf.Tensor,
Knn: tf.Tensor,
f: tf.Tensor,
*,
full_cov: bool = False,
full_output_cov: bool = False,
q_sqrt: Optional[tf.Tensor] = None,
white: bool = False,
) -> MeanAndVariance:
"""
The inducing outputs live in the g-space (R^L).
Interdomain conditional calculation.
:param Kmn: [M, L, N, P]
:param Kmm: [L, M, M]
:param Knn: [N, P] or [N, P, P] or [P, N, N] or [N, P, N, P]
:param f: data matrix, [M, L]
:param q_sqrt: [L, M, M] or [M, L]
:param full_cov: calculate covariance between inputs
:param full_output_cov: calculate covariance between outputs
:param white: use whitened representation
:return:
- mean: [N, P]
- variance: [N, P], [N, P, P], [P, N, N], [N, P, N, P]
"""
M, L, N, P = tf.unstack(tf.shape(Kmn), num=Kmn.shape.ndims, axis=0)
shape_constraints = [
(Kmn, ["M", "L", "N", "P"]),
(Kmm, ["L", "M", "M"]),
(f, ["M", "L"]),
]
if q_sqrt is not None:
shape_constraints.append(
(q_sqrt, ["M", "L"] if q_sqrt.shape.ndims == 2 else ["L", "M", "M"])
)
Lm = tf.linalg.cholesky(Kmm) # [L, M, M]
# Compute the projection matrix A
Kmn = tf.reshape(tf.transpose(Kmn, (1, 0, 2, 3)), (L, M, N * P))
A = tf.linalg.triangular_solve(Lm, Kmn, lower=True) # [L, M, M] \ [L, M, N*P] -> [L, M, N*P]
Ar = tf.reshape(A, (L, M, N, P))
# compute the covariance due to the conditioning
if full_cov and full_output_cov:
fvar = Knn - tf.tensordot(Ar, Ar, [[0, 1], [0, 1]]) # [N, P, N, P]
intended_cov_shape = ["N", "P", "N", "P"]
elif full_cov and not full_output_cov:
At = tf.reshape(tf.transpose(Ar), (P, N, M * L)) # [P, N, L]
fvar = Knn - tf.linalg.matmul(At, At, transpose_b=True) # [P, N, N]
intended_cov_shape = ["P", "N", "N"]
elif not full_cov and full_output_cov:
At = tf.reshape(tf.transpose(Ar, [2, 3, 1, 0]), (N, P, M * L)) # [N, P, L]
fvar = Knn - tf.linalg.matmul(At, At, transpose_b=True) # [N, P, P]
intended_cov_shape = ["N", "P", "P"]
elif not full_cov and not full_output_cov:
fvar = Knn - tf.reshape(tf.reduce_sum(tf.square(A), [0, 1]), (N, P)) # Knn: [N, P]
intended_cov_shape = ["N", "P"]
# another backsubstitution in the unwhitened case
if not white:
A = tf.linalg.triangular_solve(
Lm, A, adjoint=True
) # [L, M, M] \ [L, M, N*P] -> [L, M, N*P]
Ar = tf.reshape(A, (L, M, N, P))
fmean = tf.tensordot(Ar, f, [[1, 0], [0, 1]]) # [N, P]
if q_sqrt is not None:
if q_sqrt.shape.ndims == 3:
Lf = tf.linalg.band_part(q_sqrt, -1, 0) # [L, M, M]
LTA = tf.linalg.matmul(
Lf, A, transpose_a=True
) # [L, M, M] * [L, M, P] -> [L, M, P]
else: # q_sqrt [M, L]
LTA = A * tf.transpose(q_sqrt)[..., None] # [L, M, P]
if full_cov and full_output_cov:
LTAr = tf.reshape(LTA, (L * M, N * P))
fvar = fvar + tf.reshape(tf.linalg.matmul(LTAr, LTAr, transpose_a=True), (N, P, N, P))
elif full_cov and not full_output_cov:
LTAr = tf.transpose(tf.reshape(LTA, (L * M, N, P)), [2, 0, 1]) # [P, M, N]
fvar = fvar + tf.linalg.matmul(LTAr, LTAr, transpose_a=True) # [P, N, N]
elif not full_cov and full_output_cov:
LTAr = tf.transpose(tf.reshape(LTA, (L * M, N, P)), [1, 0, 2]) # [N, M, P]
fvar = fvar + tf.linalg.matmul(LTAr, LTAr, transpose_a=True) # [N, P, P]
elif not full_cov and not full_output_cov:
fvar = fvar + tf.reshape(tf.reduce_sum(tf.square(LTA), (0, 1)), (N, P))
shape_constraints.extend(
[
(Knn, intended_cov_shape),
(fmean, ["N", "P"]),
(fvar, intended_cov_shape),
]
)
tf.debugging.assert_shapes(shape_constraints, message="independent_interdomain_conditional()")
return fmean, fvar
[docs]def rollaxis_left(A: tf.Tensor, num_rolls: int) -> tf.Tensor:
"""Roll the tensor `A` backwards `num_rolls` times."""
assert num_rolls > 0
rank = tf.rank(A)
perm = tf.concat([num_rolls + tf.range(rank - num_rolls), tf.range(num_rolls)], 0)
return tf.transpose(A, perm)
[docs]def rollaxis_right(A: tf.Tensor, num_rolls: int) -> tf.Tensor:
"""Roll the tensor `A` forward `num_rolls` times."""
assert num_rolls > 0
rank = tf.rank(A)
perm = tf.concat([rank - num_rolls + tf.range(num_rolls), tf.range(rank - num_rolls)], 0)
return tf.transpose(A, perm)
[docs]def mix_latent_gp(
W: tf.Tensor, g_mean: tf.Tensor, g_var: tf.Tensor, full_cov: bool, full_output_cov: bool
) -> MeanAndVariance:
r"""Takes the mean and variance of an uncorrelated L-dimensional latent GP
and returns the mean and the variance of the mixed GP, `f = W g`,
where both f and g are GPs, with W having a shape [P, L]
:param W: [P, L]
:param g_mean: [..., N, L]
:param g_var: [..., N, L] (full_cov = False) or [L, ..., N, N] (full_cov = True)
:return: f_mean and f_var, shape depends on `full_cov` and `full_output_cov`
"""
shape_constraints = [
(W, ["P", "L"]),
(g_mean, [..., "N", "L"]),
]
if not full_cov:
shape_constraints.append((g_var, [..., "N", "L"]))
else:
# NOTE(awav) cannot assert g_var shape here because of the inner "leading"
# dimensions, see https://github.com/GPflow/GPflow/issues/1296
pass
f_mean = tf.tensordot(g_mean, W, [[-1], [-1]]) # [..., N, P]
if full_cov and full_output_cov: # g_var is [L, ..., N, N]
# this branch is practically never taken
g_var = rollaxis_left(g_var, 1) # [..., N, N, L]
shape_constraints.append((g_var, [..., "N", "N", "L"]))
g_var = tf.expand_dims(g_var, axis=-2) # [..., N, N, 1, L]
g_var_W = g_var * W # [..., N, P, L]
f_var = tf.tensordot(g_var_W, W, [[-1], [-1]]) # [..., N, N, P, P]
f_var = leading_transpose(f_var, [..., -4, -2, -3, -1]) # [..., N, P, N, P]
intended_cov_shape = [..., "N", "P", "N", "P"]
elif full_cov and not full_output_cov: # g_var is [L, ..., N, N]
# this branch is practically never taken
f_var = tf.tensordot(g_var, W ** 2, [[0], [-1]]) # [..., N, N, P]
f_var = leading_transpose(f_var, [..., -1, -3, -2]) # [..., P, N, N]
intended_cov_shape = [..., "P", "N", "N"]
elif not full_cov and full_output_cov: # g_var is [..., N, L]
g_var = tf.expand_dims(g_var, axis=-2) # [..., N, 1, L]
g_var_W = g_var * W # [..., N, P, L]
f_var = tf.tensordot(g_var_W, W, [[-1], [-1]]) # [..., N, P, P]
intended_cov_shape = [..., "N", "P", "P"]
elif not full_cov and not full_output_cov: # g_var is [..., N, L]
W_squared = W ** 2 # [P, L]
f_var = tf.tensordot(g_var, W_squared, [[-1], [-1]]) # [..., N, P]
intended_cov_shape = [..., "N", "P"]
shape_constraints.extend(
[
(f_mean, [..., "N", "P"]),
(f_var, intended_cov_shape),
]
)
tf.debugging.assert_shapes(shape_constraints, message="mix_latent_gp()")
return f_mean, f_var
[docs]def separate_independent_conditional_implementation(
Kmns: tf.Tensor,
Kmms: tf.Tensor,
Knns: tf.Tensor,
f: tf.Tensor,
*,
full_cov: bool = False,
q_sqrt: Optional[tf.Tensor] = None,
white: bool = False,
) -> MeanAndVariance:
"""Multi-output GP with independent GP priors.
Number of latent processes equals the number of outputs (L = P).
The covariance matrices used to calculate the conditional have the following shape:
- Kuu: [P, M, M]
- Kuf: [P, M, N]
- Kff: [P, N] or [P, N, N]
Further reference
-----------------
- See `gpflow.conditionals._conditional` for a detailed explanation of
conditional in the single-output case.
- See the multioutput notebook for more information about the multioutput framework.
- See above for the parameters and the return value.
"""
fs = tf.transpose(f)[:, :, None] # [P, M, 1]
# [P, 1, M, M] or [P, M, 1]
base_conditional_args_to_map: Tuple[tf.Tensor, ...]
single_gp_conditional: Callable[[Tuple[tf.Tensor, ...]], MeanAndVariance]
if q_sqrt is not None:
q_sqrts = (
tf.transpose(q_sqrt)[:, :, None] if q_sqrt.shape.ndims == 2 else q_sqrt[:, None, :, :]
)
base_conditional_args_to_map = (Kmms, Kmns, Knns, fs, q_sqrts)
def single_gp_conditional(
t: Tuple[tf.Tensor, ...]
) -> MeanAndVariance: # pragma: no cover - tf.map_fn is invisible to codecov
Kmm, Kmn, Knn, f, q_sqrt = t
return base_conditional(Kmn, Kmm, Knn, f, full_cov=full_cov, q_sqrt=q_sqrt, white=white)
else:
base_conditional_args_to_map = (Kmms, Kmns, Knns, fs)
def single_gp_conditional(
t: Tuple[tf.Tensor, ...]
) -> MeanAndVariance: # pragma: no cover - tf.map_fn is invisible to codecov
Kmm, Kmn, Knn, f = t
return base_conditional(Kmn, Kmm, Knn, f, full_cov=full_cov, q_sqrt=q_sqrt, white=white)
rmu, rvar = tf.map_fn(
single_gp_conditional, base_conditional_args_to_map, (default_float(), default_float())
) # [P, N, 1], [P, 1, N, N] or [P, N, 1]
fmu = rollaxis_left(tf.squeeze(rmu, axis=-1), 1) # [N, P]
if full_cov:
fvar = tf.squeeze(rvar, axis=-3) # [..., 0, :, :] # [P, N, N]
else:
fvar = rollaxis_left(tf.squeeze(rvar, axis=-1), 1) # [N, P]
return fmu, fvar