Source code for gpflow.posteriors

# Copyright 2016-2020 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.

import enum
from abc import ABC, abstractmethod
from dataclasses import dataclass
from typing import Optional, Tuple, Type, Union, cast

import tensorflow as tf
from check_shapes import (
    ErrorContext,
    Shape,
    check_shapes,
    get_shape,
    inherit_check_shapes,
    register_get_shape,
)

from . import covariances, kernels, mean_functions
from .base import MeanAndVariance, Module, RegressionData, TensorType
from .conditionals.util import (
    base_conditional,
    base_conditional_with_lm,
    expand_independent_outputs,
    fully_correlated_conditional,
    independent_interdomain_conditional,
    mix_latent_gp,
    separate_independent_conditional_implementation,
)
from .config import default_float, default_jitter
from .covariances import Kuf, Kuu
from .inducing_variables import (
    FallbackSeparateIndependentInducingVariables,
    FallbackSharedIndependentInducingVariables,
    InducingPoints,
    InducingVariables,
    SeparateIndependentInducingVariables,
    SharedIndependentInducingVariables,
)
from .kernels import Kernel
from .likelihoods import Gaussian
from .mean_functions import MeanFunction
from .utilities import Dispatcher, add_likelihood_noise_cov, assert_params_false
from .utilities.ops import eye


class _QDistribution(Module):
    """
    Base class for our parametrization of q(u) in the `AbstractPosterior`.
    Internal - do not rely on this outside of GPflow.
    """


class _DeltaDist(_QDistribution):
    @check_shapes(
        "q_mu: [M, L]",
    )
    def __init__(self, q_mu: TensorType) -> None:
        self.q_mu = q_mu

    @property
    def q_sqrt(self) -> Optional[tf.Tensor]:
        return None


class _DiagNormal(_QDistribution):
    @check_shapes(
        "q_mu: [M, L]",
        "q_sqrt: [M, L]",
    )
    def __init__(self, q_mu: TensorType, q_sqrt: TensorType) -> None:
        self.q_mu = q_mu
        self.q_sqrt = q_sqrt


class _MvNormal(_QDistribution):
    @check_shapes(
        "q_mu: [M, L]",
        "q_sqrt: [L, M, M]  # lower-triangular",
    )
    def __init__(self, q_mu: TensorType, q_sqrt: TensorType) -> None:
        self.q_mu = q_mu
        self.q_sqrt = q_sqrt


[docs]class PrecomputeCacheType(enum.Enum): """ - `PrecomputeCacheType.TENSOR` (or `"tensor"`): Precomputes the cached quantities and stores them as tensors (which allows differentiating through the prediction). This is the default. - `PrecomputeCacheType.VARIABLE` (or `"variable"`): Precomputes the cached quantities and stores them as variables, which allows for updating their values without changing the compute graph (relevant for AOT compilation). - `PrecomputeCacheType.NOCACHE` (or `"nocache"` or `None`): Avoids immediate cache computation. This is useful for avoiding extraneous computations when you only want to call the posterior's `fused_predict_f` method. """ TENSOR = "tensor" VARIABLE = "variable" NOCACHE = "nocache"
[docs]@dataclass class PrecomputedValue: value: tf.Tensor """ The precomputed value itself. """ axis_dynamic: Tuple[bool, ...] """ A tuple with one element per dimension of `value`. That element is `True` if that dimension of `value` might change size. """ def __post_init__(self) -> None: tf.debugging.assert_rank( self.value, len(self.axis_dynamic), "axis_dynamic must have one element per dimension of value.", )
[docs] @staticmethod @check_shapes( "alpha: [M_L_or_L_M_M...]", "Qinv: [M_M_or_L_M_M...]", ) def wrap_alpha_Qinv(alpha: TensorType, Qinv: TensorType) -> Tuple["PrecomputedValue", ...]: """ Wraps `alpha` and `Qinv` in `PrecomputedValue`\ s. """ one_dynamic = False L_dynamic = False M_dynamic = False # TODO(jesper): Support variable number of inducing points? alpha_rank = tf.rank(alpha) if alpha_rank == 2: alpha_dynamic: Tuple[bool, ...] = (M_dynamic, L_dynamic) elif alpha_rank == 3: alpha_dynamic = (L_dynamic, M_dynamic, one_dynamic) else: raise AssertionError(f"Unknown rank of alpha {alpha_rank}.") Qinv_rank = tf.rank(Qinv) if Qinv_rank == 2: Qinv_dynamic: Tuple[bool, ...] = (M_dynamic, M_dynamic) elif Qinv_rank == 3: Qinv_dynamic = (L_dynamic, M_dynamic, M_dynamic) else: raise AssertionError(f"Unknown rank of Qinv {Qinv_rank}.") return ( PrecomputedValue(alpha, alpha_dynamic), PrecomputedValue(Qinv, Qinv_dynamic), )
[docs]@register_get_shape(PrecomputedValue) def get_precomputed_value_shape(shaped: PrecomputedValue, context: ErrorContext) -> Shape: return get_shape(shaped.value, context)
def _validate_precompute_cache_type( value: Union[None, PrecomputeCacheType, str] ) -> PrecomputeCacheType: if value is None: return PrecomputeCacheType.NOCACHE elif isinstance(value, PrecomputeCacheType): return value elif isinstance(value, str): return PrecomputeCacheType(value.lower()) else: raise ValueError( f"{value} is not a valid PrecomputeCacheType." " Valid options: 'tensor', 'variable', 'nocache' (or None)." )
[docs]class AbstractPosterior(Module, ABC): @check_shapes( "X_data: [N_D_or_M_D_P...]", ) def __init__( self, kernel: Kernel, X_data: Union[tf.Tensor, InducingVariables], cache: Optional[Tuple[tf.Tensor, ...]] = None, mean_function: Optional[mean_functions.MeanFunction] = None, ) -> None: """ Users should use `create_posterior` to create instances of concrete subclasses of this AbstractPosterior class instead of calling this constructor directly. For `create_posterior` to be able to correctly instantiate subclasses, developers need to ensure their subclasses don't change the constructor signature. """ super().__init__() self.kernel = kernel self.X_data = X_data self.cache = cache self.mean_function = mean_function self._precompute_cache: Optional[PrecomputeCacheType] = None @check_shapes( "Xnew: [batch..., D]", "mean: [batch..., Q]", "return: [batch..., Q]", ) def _add_mean_function(self, Xnew: TensorType, mean: TensorType) -> tf.Tensor: if self.mean_function is None: return mean else: return mean + self.mean_function(Xnew) @abstractmethod def _precompute(self) -> Tuple[PrecomputedValue, ...]: """ Precompute a cache. The result of this method will later be passed to `_conditional_with_precompute` as the `cache` argument. """
[docs] @check_shapes( "Xnew: [batch..., N, D]", "return[0]: [batch..., N, P]", "return[1]: [batch..., N, P, N, P] if full_cov and full_output_cov", "return[1]: [batch..., P, N, N] if full_cov and (not full_output_cov)", "return[1]: [batch..., N, P, P] if (not full_cov) and full_output_cov", "return[1]: [batch..., N, P] if (not full_cov) and (not full_output_cov)", ) def fused_predict_f( self, Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False ) -> MeanAndVariance: """ Computes predictive mean and (co)variance at Xnew, including mean_function Does not make use of caching """ mean, cov = self._conditional_fused( Xnew, full_cov=full_cov, full_output_cov=full_output_cov ) return self._add_mean_function(Xnew, mean), cov
@abstractmethod @check_shapes( "Xnew: [batch..., N, D]", "return[0]: [batch..., N, P]", "return[1]: [batch..., N, P, N, P] if full_cov and full_output_cov", "return[1]: [batch..., P, N, N] if full_cov and (not full_output_cov)", "return[1]: [batch..., N, P, P] if (not full_cov) and full_output_cov", "return[1]: [batch..., N, P] if (not full_cov) and (not full_output_cov)", ) def _conditional_fused( self, Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False ) -> MeanAndVariance: """ Computes predictive mean and (co)variance at Xnew, *excluding* mean_function Does not make use of caching """
[docs] @check_shapes( "Xnew: [batch..., N, D]", "return[0]: [batch..., N, P]", "return[1]: [batch..., N, P, N, P] if full_cov and full_output_cov", "return[1]: [batch..., P, N, N] if full_cov and (not full_output_cov)", "return[1]: [batch..., N, P, P] if (not full_cov) and full_output_cov", "return[1]: [batch..., N, P] if (not full_cov) and (not full_output_cov)", ) def predict_f( self, Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False ) -> MeanAndVariance: """ Computes predictive mean and (co)variance at Xnew, including mean_function. Relies on precomputed alpha and Qinv (see _precompute method) """ if self.cache is None: raise ValueError( "Cache has not been precomputed yet. Call update_cache first or use fused_predict_f" ) mean, cov = self._conditional_with_precompute( self.cache, Xnew, full_cov=full_cov, full_output_cov=full_output_cov ) return self._add_mean_function(Xnew, mean), cov
@abstractmethod @check_shapes( "Xnew: [batch..., N, D]", "return[0]: [batch..., N, P]", "return[1]: [batch..., N, P, N, P] if full_cov and full_output_cov", "return[1]: [batch..., P, N, N] if full_cov and (not full_output_cov)", "return[1]: [batch..., N, P, P] if (not full_cov) and full_output_cov", "return[1]: [batch..., N, P] if (not full_cov) and (not full_output_cov)", ) def _conditional_with_precompute( self, cache: Tuple[tf.Tensor, ...], Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False, ) -> MeanAndVariance: """ Computes predictive mean and (co)variance at Xnew, *excluding* mean_function. Relies on cached alpha and Qinv. """
[docs] def update_cache(self, precompute_cache: Optional[PrecomputeCacheType] = None) -> None: """ Sets the cache depending on the value of `precompute_cache` to a `tf.Tensor`, `tf.Variable`, or clears the cache. If `precompute_cache` is not given, the setting defaults to the most-recently-used one. """ if precompute_cache is None: if self._precompute_cache is None: raise ValueError( "You must pass precompute_cache explicitly" " (the cache had not been updated before)." ) precompute_cache = self._precompute_cache else: self._precompute_cache = precompute_cache if precompute_cache is PrecomputeCacheType.NOCACHE: self.cache = None elif precompute_cache is PrecomputeCacheType.TENSOR: self.cache = tuple(c.value for c in self._precompute()) elif precompute_cache is PrecomputeCacheType.VARIABLE: cache = self._precompute() if self.cache is not None and all(isinstance(c, tf.Variable) for c in self.cache): # re-use existing variables for cache_var, c in zip(self.cache, cache): cache_var.assign(c.value) else: # create variables shapes = [ [None if d else s for d, s in zip(c.axis_dynamic, tf.shape(c.value))] for c in cache ] self.cache = tuple( tf.Variable(c.value, trainable=False, shape=s) for c, s in zip(cache, shapes) )
[docs]class GPRPosterior(AbstractPosterior): @check_shapes( "data[0]: [N, D]", "data[1]: [N, Q]", ) def __init__( self, kernel: Kernel, data: RegressionData, likelihood: Gaussian, mean_function: MeanFunction, *, precompute_cache: Optional[PrecomputeCacheType], ) -> None: X, Y = data super().__init__(kernel, X, mean_function=mean_function) self.Y_data = Y self.likelihood = likelihood if precompute_cache is not None: self.update_cache(precompute_cache) @inherit_check_shapes def _conditional_with_precompute( self, cache: Tuple[tf.Tensor, ...], Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False, ) -> MeanAndVariance: """ Computes predictive mean and (co)variance at Xnew, *excluding* mean_function. Relies on cached alpha and Qinv. """ assert_params_false(self._conditional_with_precompute, full_output_cov=full_output_cov) err, Lm = cache Knn = self.kernel(Xnew, full_cov=full_cov) Kmn = self.kernel(self.X_data, Xnew) return base_conditional_with_lm( Kmn=Kmn, Lm=Lm, Knn=Knn, f=err, full_cov=full_cov, q_sqrt=None, white=False, ) @check_shapes( "return[0]: [M, D]", "return[1]: [M, M]", ) def _precompute(self) -> Tuple[PrecomputedValue, ...]: assert self.mean_function is not None X_data = cast(tf.Tensor, self.X_data) err = self.Y_data - self.mean_function(X_data) Kmm = self.kernel(X_data) Kmm_plus_s = add_likelihood_noise_cov(Kmm, self.likelihood, X_data) Lm = tf.linalg.cholesky(Kmm_plus_s) D = err.shape[1] M = X_data.shape[0] D_dynamic = D is None M_dynamic = M is None return ( PrecomputedValue(err, (M_dynamic, D_dynamic)), PrecomputedValue(Lm, (M_dynamic, M_dynamic)), ) @inherit_check_shapes def _conditional_fused( self, Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False ) -> MeanAndVariance: """ Computes predictive mean and (co)variance at Xnew, *excluding* mean_function Does not make use of caching """ temp_cache = tuple(c.value for c in self._precompute()) return self._conditional_with_precompute(temp_cache, Xnew, full_cov, full_output_cov)
[docs]class SGPRPosterior(AbstractPosterior): """ This class represents posteriors which can be derived from SGPR models to compute faster predictions on unseen points. """ @check_shapes( "data[0]: [N, D]", "data[1]: [N, Q]", "inducing_variable: [M, D, 1]", ) def __init__( self, kernel: Kernel, data: RegressionData, inducing_variable: InducingPoints, likelihood: Gaussian, num_latent_gps: int, mean_function: MeanFunction, *, precompute_cache: Optional[PrecomputeCacheType], ) -> None: X, Y = data super().__init__(kernel, X, mean_function=mean_function) self.Y_data = Y self.likelihood = likelihood self.inducing_variable = inducing_variable self.num_latent_gps = num_latent_gps if precompute_cache is not None: self.update_cache(precompute_cache) @inherit_check_shapes def _conditional_with_precompute( self, cache: Tuple[tf.Tensor, ...], Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False, ) -> MeanAndVariance: """ Computes predictive mean and (co)variance at Xnew, *excluding* mean_function. Relies on cached alpha and Qinv. """ assert_params_false(self._conditional_with_precompute, full_output_cov=full_output_cov) L, LB, c = cache Kus = Kuf(self.inducing_variable, self.kernel, Xnew) tmp1 = tf.linalg.triangular_solve(L, Kus, lower=True) tmp2 = tf.linalg.triangular_solve(LB, tmp1, lower=True) mean = tf.linalg.matmul(tmp2, c, transpose_a=True) if full_cov: var = ( self.kernel(Xnew) + tf.linalg.matmul(tmp2, tmp2, transpose_a=True) - tf.linalg.matmul(tmp1, tmp1, transpose_a=True) ) var = tf.tile(var[None, ...], [self.num_latent_gps, 1, 1]) # [P, N, N] else: var = ( self.kernel(Xnew, full_cov=False) + tf.reduce_sum(tf.square(tmp2), 0) - tf.reduce_sum(tf.square(tmp1), 0) ) var = tf.tile(var[:, None], [1, self.num_latent_gps]) return mean, var @check_shapes( "return[0]: [M, M]", "return[1]: [M, M]", "return[2]: [M, D]", ) def _precompute(self) -> Tuple[PrecomputedValue, ...]: assert self.mean_function is not None X_data = cast(tf.Tensor, self.X_data) num_inducing = self.inducing_variable.num_inducing err = self.Y_data - self.mean_function(X_data) kuf = Kuf(self.inducing_variable, self.kernel, X_data) kuu = Kuu(self.inducing_variable, self.kernel, jitter=default_jitter()) sigma_sq = tf.squeeze(self.likelihood.variance_at(X_data), axis=-1) sigma = tf.sqrt(sigma_sq) L = tf.linalg.cholesky(kuu) # cache alpha, qinv A = tf.linalg.triangular_solve(L, kuf / sigma, lower=True) B = tf.linalg.matmul(A, A, transpose_b=True) + tf.eye( num_inducing, dtype=default_float() ) # cache qinv LB = tf.linalg.cholesky(B) # cache alpha Aerr = tf.linalg.matmul(A, err / sigma[..., None]) c = tf.linalg.triangular_solve(LB, Aerr, lower=True) D = err.shape[1] M = X_data.shape[0] D_dynamic = D is None M_dynamic = M is None return ( PrecomputedValue(L, (M_dynamic, M_dynamic)), PrecomputedValue(LB, (M_dynamic, M_dynamic)), PrecomputedValue(c, (M_dynamic, D_dynamic)), ) @inherit_check_shapes def _conditional_fused( self, Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False ) -> MeanAndVariance: """ Compute the mean and variance of the latent function at some new points Xnew. Does not make use of caching """ temp_cache = tuple(c.value for c in self._precompute()) return self._conditional_with_precompute(temp_cache, Xnew, full_cov, full_output_cov)
[docs]class VGPPosterior(AbstractPosterior): @check_shapes( "X: [N, D]", "q_mu: [N, P]", "q_sqrt: [N_P_or_P_N_N...]", ) def __init__( self, kernel: Kernel, X: tf.Tensor, q_mu: tf.Tensor, q_sqrt: tf.Tensor, mean_function: Optional[mean_functions.MeanFunction] = None, white: bool = True, *, precompute_cache: Optional[PrecomputeCacheType], ) -> None: super().__init__(kernel, X, mean_function=mean_function) self.q_mu = q_mu self.q_sqrt = q_sqrt self.white = white if precompute_cache is not None: self.update_cache(precompute_cache) @inherit_check_shapes def _conditional_with_precompute( self, cache: Tuple[tf.Tensor, ...], Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False, ) -> MeanAndVariance: assert_params_false(self._conditional_with_precompute, full_output_cov=full_output_cov) (Lm,) = cache Kmn = self.kernel(self.X_data, Xnew) # [M, ..., N] Knn = self.kernel( Xnew, full_cov=full_cov ) # [..., N] (full_cov = False) or [..., N, N] (True) return base_conditional_with_lm( Kmn=Kmn, Lm=Lm, Knn=Knn, f=self.q_mu, full_cov=full_cov, q_sqrt=self.q_sqrt, white=self.white, ) @check_shapes( "return[0]: [M, M]", ) def _precompute(self) -> Tuple[PrecomputedValue, ...]: X_data = cast(tf.Tensor, self.X_data) Kmm = self.kernel(X_data) + eye( tf.shape(X_data)[-2], value=default_jitter(), dtype=X_data.dtype ) # [..., M, M] Lm = tf.linalg.cholesky(Kmm) M = X_data.shape[0] M_dynamic = M is None return (PrecomputedValue(Lm, (M_dynamic, M_dynamic)),) @inherit_check_shapes def _conditional_fused( self, Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False ) -> MeanAndVariance: temp_cache = tuple(c.value for c in self._precompute()) return self._conditional_with_precompute(temp_cache, Xnew, full_cov, full_output_cov)
[docs]class BasePosterior(AbstractPosterior): @check_shapes( "inducing_variable: [M, D, broadcast P]", "q_mu: [N, P]", "q_sqrt: [N_P_or_P_N_N...]", ) def __init__( self, kernel: Kernel, inducing_variable: InducingVariables, q_mu: tf.Tensor, q_sqrt: tf.Tensor, whiten: bool = True, mean_function: Optional[mean_functions.MeanFunction] = None, *, precompute_cache: Optional[PrecomputeCacheType], ): super().__init__(kernel, inducing_variable, mean_function=mean_function) self.whiten = whiten self._set_qdist(q_mu, q_sqrt) if precompute_cache is not None: self.update_cache(precompute_cache) @property # type: ignore[misc] @check_shapes( "return: [N, P]", ) def q_mu(self) -> tf.Tensor: return self._q_dist.q_mu @property # type: ignore[misc] @check_shapes( "return: [N_P_or_P_N_N...]", ) def q_sqrt(self) -> tf.Tensor: return self._q_dist.q_sqrt @check_shapes( "q_mu: [N, P]", "q_sqrt: [N_P_or_P_N_N...]", ) def _set_qdist(self, q_mu: TensorType, q_sqrt: TensorType) -> None: if q_sqrt is None: self._q_dist = _DeltaDist(q_mu) elif len(q_sqrt.shape) == 2: # q_diag self._q_dist = _DiagNormal(q_mu, q_sqrt) else: self._q_dist = _MvNormal(q_mu, q_sqrt) @check_shapes( "return[0]: [M_L_or_L_M_M...]", "return[1]: [L, M, M]", ) def _precompute(self) -> Tuple[PrecomputedValue, ...]: Kuu = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter()) # [(R), M, M] q_mu = self._q_dist.q_mu if Kuu.shape.ndims == 4: ML = tf.reduce_prod(tf.shape(Kuu)[:2]) Kuu = tf.reshape(Kuu, [ML, ML]) if Kuu.shape.ndims == 3: q_mu = tf.linalg.adjoint(self._q_dist.q_mu)[..., None] # [..., R, M, 1] L = tf.linalg.cholesky(Kuu) if not self.whiten: # alpha = Kuu⁻¹ q_mu alpha = tf.linalg.cholesky_solve(L, q_mu) else: # alpha = L⁻ᵀ q_mu alpha = tf.linalg.triangular_solve(L, q_mu, adjoint=True) # predictive mean = Kfu alpha # predictive variance = Kff - Kfu Qinv Kuf # S = q_sqrt q_sqrtᵀ I = tf.eye(tf.shape(L)[-1], dtype=L.dtype) if isinstance(self._q_dist, _DeltaDist): B = I else: if not self.whiten: # Qinv = Kuu⁻¹ - Kuu⁻¹ S Kuu⁻¹ # = Kuu⁻¹ - L⁻ᵀ L⁻¹ S L⁻ᵀ L⁻¹ # = L⁻ᵀ (I - L⁻¹ S L⁻ᵀ) L⁻¹ # = L⁻ᵀ B L⁻¹ if isinstance(self._q_dist, _DiagNormal): q_sqrt = tf.linalg.diag(tf.linalg.adjoint(self._q_dist.q_sqrt)) elif isinstance(self._q_dist, _MvNormal): q_sqrt = self._q_dist.q_sqrt Linv_qsqrt = tf.linalg.triangular_solve(L, q_sqrt) Linv_cov_u_LinvT = tf.matmul(Linv_qsqrt, Linv_qsqrt, transpose_b=True) else: if isinstance(self._q_dist, _DiagNormal): Linv_cov_u_LinvT = tf.linalg.diag(tf.linalg.adjoint(self._q_dist.q_sqrt ** 2)) elif isinstance(self._q_dist, _MvNormal): q_sqrt = self._q_dist.q_sqrt Linv_cov_u_LinvT = tf.matmul(q_sqrt, q_sqrt, transpose_b=True) # Qinv = Kuu⁻¹ - L⁻ᵀ S L⁻¹ # Linv = (L⁻¹ I) = solve(L, I) # Kinv = Linvᵀ @ Linv B = I - Linv_cov_u_LinvT LinvT_B = tf.linalg.triangular_solve(L, B, adjoint=True) B_Linv = tf.linalg.adjoint(LinvT_B) Qinv = tf.linalg.triangular_solve(L, B_Linv, adjoint=True) M, L = tf.unstack(tf.shape(self._q_dist.q_mu), num=2) Qinv = tf.broadcast_to(Qinv, [L, M, M]) return PrecomputedValue.wrap_alpha_Qinv(alpha, Qinv)
[docs]class IndependentPosterior(BasePosterior): @check_shapes( "mean: [batch..., N, P]", "cov: [batch..., P, N, N] if full_cov", "cov: [batch..., N, P] if not full_cov", "return[0]: [batch..., N, P]", "return[1]: [batch..., N, P, N, P] if full_cov and full_output_cov", "return[1]: [batch..., N, P, P] if (not full_cov) and full_output_cov", "return[1]: [batch..., P, N, N] if full_cov and (not full_output_cov)", "return[1]: [batch..., N, P] if (not full_cov) and (not full_output_cov)", ) def _post_process_mean_and_cov( self, mean: TensorType, cov: TensorType, full_cov: bool, full_output_cov: bool ) -> MeanAndVariance: return mean, expand_independent_outputs(cov, full_cov, full_output_cov) @check_shapes( "Xnew: [N, D]", "return: [broadcast P, N, N] if full_cov", "return: [broadcast P, N] if (not full_cov)", ) def _get_Kff(self, Xnew: TensorType, full_cov: bool) -> tf.Tensor: # TODO: this assumes that Xnew has shape [N, D] and no leading dims if isinstance(self.kernel, (kernels.SeparateIndependent, kernels.IndependentLatent)): # NOTE calling kernel(Xnew, full_cov=full_cov, full_output_cov=False) directly would # return # if full_cov: [P, N, N] -- this is what we want # else: [N, P] instead of [P, N] as we get from the explicit stack below Kff = tf.stack([k(Xnew, full_cov=full_cov) for k in self.kernel.kernels], axis=0) elif isinstance(self.kernel, kernels.MultioutputKernel): # effectively, SharedIndependent path Kff = self.kernel.kernel(Xnew, full_cov=full_cov) # NOTE calling kernel(Xnew, full_cov=full_cov, full_output_cov=False) directly would # return # if full_cov: [P, N, N] instead of [N, N] # else: [N, P] instead of [N] else: # standard ("single-output") kernels Kff = self.kernel(Xnew, full_cov=full_cov) # [N, N] if full_cov else [N] return Kff @inherit_check_shapes def _conditional_with_precompute( self, cache: Tuple[tf.Tensor, ...], Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False, ) -> MeanAndVariance: # Qinv: [L, M, M] # alpha: [M, L] alpha, Qinv = cache Kuf = covariances.Kuf(self.X_data, self.kernel, Xnew) # [(R), M, N] Kff = self._get_Kff(Xnew, full_cov) mean = tf.matmul(Kuf, alpha, transpose_a=True) if Kuf.shape.ndims == 3: mean = tf.linalg.adjoint(tf.squeeze(mean, axis=-1)) if full_cov: Kfu_Qinv_Kuf = tf.matmul(Kuf, Qinv @ Kuf, transpose_a=True) cov = Kff - Kfu_Qinv_Kuf else: # [Aᵀ B]_ij = Aᵀ_ik B_kj = A_ki B_kj # TODO check whether einsum is faster now? Kfu_Qinv_Kuf = tf.reduce_sum(Kuf * tf.matmul(Qinv, Kuf), axis=-2) cov = Kff - Kfu_Qinv_Kuf cov = tf.linalg.adjoint(cov) return self._post_process_mean_and_cov(mean, cov, full_cov, full_output_cov)
[docs]class IndependentPosteriorSingleOutput(IndependentPosterior): # could almost be the same as IndependentPosteriorMultiOutput ... @inherit_check_shapes def _conditional_fused( self, Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False ) -> MeanAndVariance: # same as IndependentPosteriorMultiOutput, Shared~/Shared~ branch, except for following # line: Knn = self.kernel(Xnew, full_cov=full_cov) Kmm = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter()) # [M, M] Kmn = covariances.Kuf(self.X_data, self.kernel, Xnew) # [M, N] fmean, fvar = base_conditional( Kmn, Kmm, Knn, self.q_mu, full_cov=full_cov, q_sqrt=self.q_sqrt, white=self.whiten ) # [N, P], [P, N, N] or [N, P] return self._post_process_mean_and_cov(fmean, fvar, full_cov, full_output_cov)
[docs]class IndependentPosteriorMultiOutput(IndependentPosterior): @inherit_check_shapes def _conditional_fused( self, Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False ) -> MeanAndVariance: if isinstance(self.X_data, SharedIndependentInducingVariables) and isinstance( self.kernel, kernels.SharedIndependent ): # same as IndependentPosteriorSingleOutput except for following line Knn = self.kernel.kernel(Xnew, full_cov=full_cov) # we don't call self.kernel() directly as that would do unnecessary tiling Kmm = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter()) # [M, M] Kmn = covariances.Kuf(self.X_data, self.kernel, Xnew) # [M, N] fmean, fvar = base_conditional( Kmn, Kmm, Knn, self.q_mu, full_cov=full_cov, q_sqrt=self.q_sqrt, white=self.whiten ) # [N, P], [P, N, N] or [N, P] else: # this is the messy thing with tf.map_fn, cleaned up by the # st/clean_up_broadcasting_conditionals branch # Following are: [P, M, M] - [P, M, N] - [P, N](x N) Kmms = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter()) # [P, M, M] Kmns = covariances.Kuf(self.X_data, self.kernel, Xnew) # [P, M, N] if isinstance(self.kernel, kernels.Combination): kernel_list = self.kernel.kernels else: kernel_list = [self.kernel.kernel] * len(self.X_data.inducing_variable_list) Knns = tf.stack( [k.K(Xnew) if full_cov else k.K_diag(Xnew) for k in kernel_list], axis=0 ) fmean, fvar = separate_independent_conditional_implementation( Kmns, Kmms, Knns, self.q_mu, q_sqrt=self.q_sqrt, full_cov=full_cov, white=self.whiten, ) return self._post_process_mean_and_cov(fmean, fvar, full_cov, full_output_cov)
[docs]class LinearCoregionalizationPosterior(IndependentPosteriorMultiOutput): @check_shapes( "mean: [batch..., N, L]", "cov: [batch..., L, N, N] if full_cov", "cov: [batch..., N, L] if not full_cov", "return[0]: [batch..., N, P]", "return[1]: [batch..., N, P, N, P] if full_cov and full_output_cov", "return[1]: [batch..., N, P, P] if (not full_cov) and full_output_cov", "return[1]: [batch..., P, N, N] if full_cov and (not full_output_cov)", "return[1]: [batch..., N, P] if (not full_cov) and (not full_output_cov)", ) def _post_process_mean_and_cov( self, mean: TensorType, cov: TensorType, full_cov: bool, full_output_cov: bool ) -> MeanAndVariance: cov = expand_independent_outputs(cov, full_cov, full_output_cov=False) mean, cov = mix_latent_gp(self.kernel.W, mean, cov, full_cov, full_output_cov) return mean, cov
[docs]class FullyCorrelatedPosterior(BasePosterior): @inherit_check_shapes def _conditional_with_precompute( self, cache: Tuple[tf.Tensor, ...], Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False, ) -> MeanAndVariance: # TODO: this assumes that Xnew has shape [N, D] and no leading dims # Qinv: [L, M, M] # alpha: [M, L] alpha, Qinv = cache Kuf = covariances.Kuf(self.X_data, self.kernel, Xnew) assert Kuf.shape.ndims == 4 M, L, N, K = tf.unstack(tf.shape(Kuf), num=Kuf.shape.ndims, axis=0) Kuf = tf.reshape(Kuf, (M * L, N * K)) kernel: kernels.MultioutputKernel = self.kernel Kff = kernel(Xnew, full_cov=full_cov, full_output_cov=full_output_cov) # full_cov=True and full_output_cov=True: [N, P, N, P] # full_cov=True and full_output_cov=False: [P, N, N] # full_cov=False and full_output_cov=True: [N, P, P] # full_cov=False and full_output_cov=False: [N, P] if full_cov == full_output_cov: new_shape = (N * K, N * K) if full_cov else (N * K,) Kff = tf.reshape(Kff, new_shape) N = tf.shape(Xnew)[0] K = tf.shape(Kuf)[-1] // N mean = tf.matmul(Kuf, alpha, transpose_a=True) if Kuf.shape.ndims == 3: mean = tf.linalg.adjoint(tf.squeeze(mean, axis=-1)) if not full_cov and not full_output_cov: # fully diagonal case in both inputs and outputs # [Aᵀ B]_ij = Aᵀ_ik B_kj = A_ki B_kj # TODO check whether einsum is faster now? Kfu_Qinv_Kuf = tf.reduce_sum(Kuf * tf.matmul(Qinv, Kuf), axis=-2) else: Kfu_Qinv_Kuf = tf.matmul(Kuf, Qinv @ Kuf, transpose_a=True) if not (full_cov and full_output_cov): # diagonal in either inputs or outputs new_shape = tf.concat([tf.shape(Kfu_Qinv_Kuf)[:-2], (N, K, N, K)], axis=0) Kfu_Qinv_Kuf = tf.reshape(Kfu_Qinv_Kuf, new_shape) if full_cov: # diagonal in outputs: move outputs to end tmp = tf.linalg.diag_part(tf.einsum("...ijkl->...ikjl", Kfu_Qinv_Kuf)) elif full_output_cov: # diagonal in inputs: move inputs to end tmp = tf.linalg.diag_part(tf.einsum("...ijkl->...jlik", Kfu_Qinv_Kuf)) Kfu_Qinv_Kuf = tf.einsum("...ijk->...kij", tmp) # move diagonal dim to [-3] cov = Kff - Kfu_Qinv_Kuf if not full_cov and not full_output_cov: cov = tf.linalg.adjoint(cov) mean = tf.reshape(mean, (N, K)) if full_cov == full_output_cov: cov_shape = (N, K, N, K) if full_cov else (N, K) else: cov_shape = (K, N, N) if full_cov else (N, K, K) cov = tf.reshape(cov, cov_shape) return mean, cov @inherit_check_shapes def _conditional_fused( self, Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False ) -> MeanAndVariance: Kmm = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter()) # [M, L, M, L] Kmn = covariances.Kuf(self.X_data, self.kernel, Xnew) # [M, L, N, P] kernel: kernels.MultioutputKernel = self.kernel Knn = kernel( Xnew, full_cov=full_cov, full_output_cov=full_output_cov ) # [N, P](x N)x P or [N, P](x P) M, L, N, K = tf.unstack(tf.shape(Kmn), num=Kmn.shape.ndims, axis=0) Kmm = tf.reshape(Kmm, (M * L, M * L)) if full_cov == full_output_cov: Kmn = tf.reshape(Kmn, (M * L, N * K)) Knn = tf.reshape(Knn, (N * K, N * K)) if full_cov else tf.reshape(Knn, (N * K,)) mean, cov = base_conditional( Kmn, Kmm, Knn, self.q_mu, full_cov=full_cov, q_sqrt=self.q_sqrt, white=self.whiten ) # [K, 1], [1, K](x NK) mean = tf.reshape(mean, (N, K)) cov = tf.reshape(cov, (N, K, N, K) if full_cov else (N, K)) else: Kmn = tf.reshape(Kmn, (M * L, N, K)) mean, cov = fully_correlated_conditional( Kmn, Kmm, Knn, self.q_mu, full_cov=full_cov, full_output_cov=full_output_cov, q_sqrt=self.q_sqrt, white=self.whiten, ) return mean, cov
[docs]class FallbackIndependentLatentPosterior(FullyCorrelatedPosterior): # XXX @inherit_check_shapes def _conditional_fused( self, Xnew: TensorType, full_cov: bool = False, full_output_cov: bool = False ) -> MeanAndVariance: Kmm = covariances.Kuu(self.X_data, self.kernel, jitter=default_jitter()) # [L, M, M] Kmn = covariances.Kuf(self.X_data, self.kernel, Xnew) # [M, L, N, P] kernel: kernels.IndependentLatent = self.kernel Knn = kernel( Xnew, full_cov=full_cov, full_output_cov=full_output_cov ) # [N, P](x N)x P or [N, P](x P) return independent_interdomain_conditional( Kmn, Kmm, Knn, self.q_mu, full_cov=full_cov, full_output_cov=full_output_cov, q_sqrt=self.q_sqrt, white=self.whiten, )
get_posterior_class = Dispatcher("get_posterior_class")
[docs]@get_posterior_class.register(kernels.Kernel, InducingVariables) def _get_posterior_base_case( kernel: Kernel, inducing_variable: InducingVariables ) -> Type[BasePosterior]: # independent single output return IndependentPosteriorSingleOutput
[docs]@get_posterior_class.register(kernels.MultioutputKernel, InducingPoints) def _get_posterior_fully_correlated_mo( kernel: Kernel, inducing_variable: InducingVariables ) -> Type[BasePosterior]: return FullyCorrelatedPosterior
[docs]@get_posterior_class.register( (kernels.SharedIndependent, kernels.SeparateIndependent), (SeparateIndependentInducingVariables, SharedIndependentInducingVariables), ) def _get_posterior_independent_mo( kernel: Kernel, inducing_variable: InducingVariables ) -> Type[BasePosterior]: # independent multi-output return IndependentPosteriorMultiOutput
[docs]@get_posterior_class.register( kernels.IndependentLatent, (FallbackSeparateIndependentInducingVariables, FallbackSharedIndependentInducingVariables), ) def _get_posterior_independentlatent_mo_fallback( kernel: Kernel, inducing_variable: InducingVariables ) -> Type[BasePosterior]: return FallbackIndependentLatentPosterior
[docs]@get_posterior_class.register( kernels.LinearCoregionalization, (SeparateIndependentInducingVariables, SharedIndependentInducingVariables), ) def _get_posterior_linearcoregionalization_mo_efficient( kernel: Kernel, inducing_variable: InducingVariables ) -> Type[BasePosterior]: # Linear mixing---efficient multi-output return LinearCoregionalizationPosterior
[docs]def create_posterior( kernel: Kernel, inducing_variable: InducingVariables, q_mu: TensorType, q_sqrt: TensorType, whiten: bool, mean_function: Optional[MeanFunction] = None, precompute_cache: Union[PrecomputeCacheType, str, None] = PrecomputeCacheType.TENSOR, ) -> BasePosterior: posterior_class = get_posterior_class(kernel, inducing_variable) precompute_cache = _validate_precompute_cache_type(precompute_cache) return posterior_class( # type: ignore[no-any-return] kernel, inducing_variable, q_mu, q_sqrt, whiten, mean_function, precompute_cache=precompute_cache, )