-
Notifications
You must be signed in to change notification settings - Fork 0
/
gp_utilities.py
254 lines (224 loc) · 11.5 KB
/
gp_utilities.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
'''
Author: Edoardo Caldarelli
Affiliation: Institut de Robòtica i Informàtica Industrial, CSIC-UPC
email: ecaldarelli@iri.upc.edu
October 2023
'''
from check_shapes import check_shapes, inherit_check_shapes
from gpflow.base import InputData, MeanAndVariance, RegressionData, TensorData, TensorType, Parameter
from gpflow.likelihoods import Gaussian
from gpflow.mean_functions import MeanFunction
from gpflow.utilities import assert_params_false, positive
import tensorflow as tf
import gpflow
from gpflow.kernels.base import ActiveDims
import numpy as np
from typing import Optional
class SquaredExponentialRFF(gpflow.kernels.Kernel):
@check_shapes(
"variance: [broadcast n_active_dims]",
)
def __init__(
self, variance: TensorType = 1.0, lengthscales: TensorType = 1.0, n_features=100, active_dims: Optional[ActiveDims] = None
) -> None:
"""
:param variance: the (initial) value for the variance parameter(s),
to induce ARD behaviour this must be initialised as an array the same
length as the number of active dimensions e.g. [1., 1., 1.]
:param active_dims: a slice or list specifying which columns of X are used
"""
super().__init__(active_dims)
self.variance = Parameter(variance, transform=positive(), name='kernel_var')
self.lengthscales = Parameter(lengthscales, transform=positive(), name='kernel_lth')
self.n_features = n_features
self._validate_ard_active_dims(self.variance)
input_dim = 1 if self.lengthscales.shape.ndims == 0 else self.lengthscales.shape.ndims
omegas = tf.random.normal(shape=[input_dim, 1], dtype=tf.float64)
for i in range(1, n_features):
curr_omega = tf.random.normal(shape=[input_dim, 1], dtype=tf.float64)
omegas = tf.concat([omegas, curr_omega], axis=-1)
self.bias = tf.random.uniform(shape=[1, self.n_features], minval=np.float64(0), maxval=np.float64(2 * np.pi), dtype=tf.float64)
self.omegas = omegas
def compute_feature_vector(self, X: TensorType) -> TensorType:
return tf.math.sqrt(2 * self.variance) * tf.sqrt(tf.math.reciprocal(tf.convert_to_tensor(self.n_features, dtype=tf.float64))) \
* tf.math.cos(X @ (self.omegas / self.lengthscales) + self.bias)
@inherit_check_shapes
def K(self, X: TensorType, X2: Optional[TensorType] = None) -> tf.Tensor:
if X2 is None:
return self.compute_feature_vector(X) @ tf.transpose(self.compute_feature_vector(X))
else:
return self.compute_feature_vector(X) @ tf.transpose(self.compute_feature_vector(X2))
@inherit_check_shapes
def K_diag(self, X: TensorType) -> tf.Tensor:
return tf.linalg.diag_part(self.K(X, X))
class RFFGPR(gpflow.models.GPR):
@check_shapes(
"data[0]: [N, D]",
"data[1]: [N, P]",
"noise_variance: []",
)
def __init__(self, data: RegressionData,
kernel: SquaredExponentialRFF,
mean_function: Optional[MeanFunction] = None,
noise_variance: Optional[TensorData] = None,
likelihood: Optional[Gaussian] = None,):
super().__init__(data, kernel, mean_function, noise_variance, likelihood)
@check_shapes(
"return: []",
)
def log_marginal_likelihood(self) -> tf.Tensor:
r"""
Computes the log marginal likelihood.
.. math::
\log p(Y | \theta).
"""
X, Y = self.data
phi = self.kernel.compute_feature_vector(X)
sigma_2 = self.likelihood.variance_at(X)
sigma = tf.squeeze(tf.math.sqrt(sigma_2), axis=-1)
phi = tf.transpose(tf.transpose(phi) / sigma)
# regularized_inverse_id = tf.linalg.tensor_diag(tf.math.reciprocal(sigma_2))
inner_mat = tf.eye(phi.shape[1], dtype=tf.float64) + tf.transpose(phi) @ phi
L = tf.linalg.cholesky(inner_mat)
v = (Y - self.mean_function(X)) / sigma[..., None]
phiTv = tf.transpose(phi) @ v
L_invphiTv = tf.linalg.triangular_solve(L, phiTv, lower=True)
# mat_inv = regularized_inverse_id - tf.transpose(Lphi_inv) @ Lphi_inv # GP is homoscedastic
log_prob = - 0.5 * tf.reduce_sum(tf.square(v)) \
+ 0.5 * tf.reduce_sum(tf.square(L_invphiTv)) \
- tf.reduce_sum(tf.math.log(tf.linalg.diag_part(L))) \
- tf.reduce_sum(tf.math.log(sigma)) \
- 0.5 * Y.shape[0] * tf.math.log(2 * np.float64(np.pi))
return tf.reduce_sum(log_prob)
@inherit_check_shapes
def predict_f(
self, Xnew: InputData, full_cov: bool = False, full_output_cov: bool = False
) -> MeanAndVariance:
r"""
This method computes predictions at X \in R^{N \x D} input points
.. math::
p(F* | Y)
where F* are points on the GP at new data points, Y are noisy observations at training data
points.
"""
assert_params_false(self.predict_f, full_output_cov=full_output_cov)
X, Y = self.data
phi = self.kernel.compute_feature_vector(X)
phi_new = self.kernel.compute_feature_vector(Xnew)
sigma_2 = tf.squeeze(self.likelihood.variance_at(X), axis=-1)
sigma = tf.math.sqrt(sigma_2)
phi = tf.transpose(tf.transpose(phi) / sigma)
# regularized_inverse_id = tf.linalg.tensor_diag(tf.math.reciprocal(sigma_2))
inner_matrix = tf.eye(phi.shape[1], dtype=tf.float64) + tf.transpose(phi) @ phi
L = tf.linalg.cholesky(inner_matrix)
v = (Y - self.mean_function(X)) / sigma[..., None]
Linv_phi = tf.linalg.triangular_solve(L, tf.transpose(phi), lower=True)
# mat_inv = regularized_inverse_id - tf.transpose(Linv_phi) @ Linv_phi
f_mean = phi_new @ (tf.transpose(phi) @ v) \
- phi_new @ (tf.transpose(phi) @ (tf.transpose(Linv_phi) @ (Linv_phi @ v))) \
+ self.mean_function(Xnew)
# f_var = phi_new @ tf.transpose(phi_new) \
# - phi_new @ (tf.transpose(phi) @ (phi @ tf.transpose(phi_new)))\
# + phi_new @ (tf.transpose(phi) @ (tf.transpose(Linv_phi) @ (Linv_phi @ (phi @ tf.transpose(phi_new)))))
f_var = tf.reduce_sum(tf.math.square(phi_new), axis=-1) \
- tf.reduce_sum(tf.math.square(phi_new @ (tf.transpose(phi))), axis=-1) \
+ tf.reduce_sum(tf.math.square(phi_new @ (tf.transpose(phi) @ (tf.transpose(Linv_phi)))), axis=-1)
return f_mean, tf.expand_dims(f_var, axis=-1)
class SquaredExponentialNystrom(gpflow.kernels.Kernel):
@check_shapes(
"exact_kernel.variance: [broadcast n_active_dims]",
)
def __init__(
self, nystrom_centers: np.ndarray, exact_kernel: gpflow.kernels.RBF,
active_dims: Optional[ActiveDims] = None
) -> None:
"""
:param variance: the (initial) value for the variance parameter(s),
to induce ARD behaviour this must be initialised as an array the same
length as the number of active dimensions e.g. [1., 1., 1.]
:param active_dims: a slice or list specifying which columns of X are used
"""
super().__init__(active_dims)
self.exact_kernel = exact_kernel
self.variance = exact_kernel.variance
self.lengthscales = exact_kernel.lengthscales
self.nystrom_centers = nystrom_centers
self.n_centers = nystrom_centers.shape[0]
self._validate_ard_active_dims(self.variance)
self.Kmm_inv_sqrt = tf.linalg.sqrtm(tf.linalg.inv(self.exact_kernel.K(nystrom_centers)
+ 1e-4 * tf.eye(nystrom_centers.shape[0], dtype=tf.float64)))
def compute_feature_vector(self, X: TensorType) -> TensorType:
return self.exact_kernel.K(X, self.nystrom_centers) @ self.Kmm_inv_sqrt
@inherit_check_shapes
def K(self, X: TensorType, X2: Optional[TensorType] = None) -> tf.Tensor:
if X2 is None:
return self.exact_kernel.K(X, X)
else:
return self.exact_kernel.K(X, X2)
@inherit_check_shapes
def K_diag(self, X: TensorType) -> tf.Tensor:
return tf.linalg.diag_part(self.exact_kernel.K(X, X))
class NystromGPR(gpflow.models.GPR):
@check_shapes(
"data[0]: [N, D]",
"data[1]: [N, P]",
"noise_variance: []",
)
def __init__(self, data: RegressionData,
kernel: gpflow.kernels.SquaredExponential,
nystrom_centers = np.ndarray,
mean_function: Optional[MeanFunction] = None,
noise_variance: Optional[TensorData] = None,
likelihood: Optional[Gaussian] = None, ):
super().__init__(data, kernel, mean_function, noise_variance, likelihood)
self.nystrom_centers = nystrom_centers
self.Kmm = self.kernel.K(nystrom_centers)
self.Kmm_inv = tf.linalg.inv(self.Kmm + 1e-4 * tf.eye(self.Kmm.shape[0], dtype=tf.float64))
self.Kmn = self.kernel.K(nystrom_centers, data[0])
@check_shapes(
"return: []",
)
def log_marginal_likelihood(self) -> tf.Tensor:
r"""
Computes the log marginal likelihood.
.. math::
\log p(Y | \theta).
"""
X, Y = self.data
phi = self.kernel.compute_feature_vector(X)
sigma_2 = tf.squeeze(self.likelihood.variance_at(X))
regularized_inverse_id = tf.linalg.tensor_diag(tf.math.reciprocal(sigma_2))
inner_mat = tf.eye(phi.shape[1], dtype=tf.float64) + tf.transpose(phi) @ regularized_inverse_id @ phi
L = tf.linalg.cholesky(inner_mat)
Lphi_inv = tf.linalg.triangular_solve(L, tf.transpose(phi) @ regularized_inverse_id, lower=True)
mat_inv = regularized_inverse_id - tf.transpose(Lphi_inv) @ Lphi_inv # GP is homoscedastic
v = Y - self.mean_function(X)
log_prob = - 0.5 * tf.transpose(v) @ mat_inv @ v \
- 0.5 * (2 * tf.reduce_sum(tf.math.log(tf.linalg.diag_part(L))) + tf.reduce_sum(
tf.math.log(sigma_2))) \
- 0.5 * Y.shape[0] * tf.math.log(2 * np.float64(np.pi))
return tf.reduce_sum(log_prob)
@inherit_check_shapes
def predict_f(
self, Xnew: InputData, full_cov: bool = False, full_output_cov: bool = False
) -> MeanAndVariance:
r"""
This method computes predictions at X \in R^{N \x D} input points
.. math::
p(F* | Y)
where F* are points on the GP at new data points, Y are noisy observations at training data
points.
"""
assert_params_false(self.predict_f, full_output_cov=full_output_cov)
X, Y = self.data
k_new_m = self.kernel.K(Xnew, self.nystrom_centers)
sigma_2 = tf.squeeze(self.likelihood.variance_at(X))
regularized_inverse_id = tf.linalg.tensor_diag(tf.math.reciprocal(sigma_2))
inner_matrix = self.Kmm + self.Kmn @ regularized_inverse_id @ tf.transpose(self.Kmn)
L = tf.linalg.cholesky(inner_matrix + 1e-4 * tf.eye(tf.shape(inner_matrix)[0], dtype=tf.float64))
Linv_phi = tf.linalg.triangular_solve(L, self.Kmn @ regularized_inverse_id, lower=True)
mat_inv = regularized_inverse_id - tf.transpose(Linv_phi) @ Linv_phi
f_mean = k_new_m @ (self.Kmm_inv @ (self.Kmn @ (mat_inv @ Y))) + self.mean_function(Xnew)
f_var = k_new_m @ self.Kmm_inv @ tf.transpose(k_new_m) - k_new_m @ (
self.Kmm_inv @ self.Kmn @ (mat_inv @ tf.transpose(k_new_m @ self.Kmm_inv @ self.Kmn)))
return f_mean, tf.expand_dims(tf.linalg.diag_part(f_var), axis=-1)