我想使用tensorflow(版本2(来使用高斯过程回归 为了适应一些数据，我在这里在线找到了谷歌 colab 示例 [1]。 我已经把这个笔记本的一些变成了下面一个最小的例子。
有时，在使用 MCMC 边缘化超参数时，代码会失败并出现以下错误：我想知道是否有人以前见过这个或知道如何解决这个问题？
tensorflow.python.framework.errors_impl.InvalidArgumentError: Input matrix is not invertible.
[[{{node mcmc_sample_chain/trace_scan/while/body/_168/smart_for_loop/while/body/_842/dual_averaging_step_size_adaptation___init__/_one_step/transformed_kernel_one_step/mh_one_step/hmc_kernel_one_step/leapfrog_integrate/while/body/_1244/leapfrog_integrate_one_step/maybe_call_fn_and_grads/value_and_gradients/value_and_gradient/gradients/leapfrog_integrate_one_step/maybe_call_fn_and_grads/value_and_gradients/value_and_gradient/PartitionedCall_grad/PartitionedCall/gradients/JointDistributionNamed/log_prob/JointDistributionNamed_log_prob_GaussianProcess/log_prob/JointDistributionNamed_log_prob_GaussianProcess/get_marginal_distribution/Cholesky_grad/MatrixTriangularSolve}}]] [Op:__inference_do_sampling_113645]
Function call stack:
do_sampling
[1] https://colab.research.google.com/github/tensorflow/probability/blob/master/tensorflow_probability/examples/jupyter_notebooks/Gaussian_Process_Regression_In_TFP.ipynb#scrollTo=jw-_1yC50xaM
请注意，下面的一些代码有点多余，但它应该 在某些部分中，但它应该能够重现错误。
谢谢！
import time
import numpy as np
import tensorflow.compat.v2 as tf
import tensorflow_probability as tfp
tfb = tfp.bijectors
tfd = tfp.distributions
tfk = tfp.math.psd_kernels
tf.enable_v2_behavior()
import matplotlib
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
#%pylab inline
# Configure plot defaults
plt.rcParams['axes.facecolor'] = 'white'
plt.rcParams['grid.color'] = '#666666'
#%config InlineBackend.figure_format = 'png'
def sinusoid(x):
return np.sin(3 * np.pi * x[..., 0])
def generate_1d_data(num_training_points, observation_noise_variance):
"""Generate noisy sinusoidal observations at a random set of points.
Returns:
observation_index_points, observations
"""
index_points_ = np.random.uniform(-1., 1., (num_training_points, 1))
index_points_ = index_points_.astype(np.float64)
# y = f(x) + noise
observations_ = (sinusoid(index_points_) +
np.random.normal(loc=0,
scale=np.sqrt(observation_noise_variance),
size=(num_training_points)))
return index_points_, observations_
# Generate training data with a known noise level (we'll later try to recover
# this value from the data).
NUM_TRAINING_POINTS = 100
observation_index_points_, observations_ = generate_1d_data(
num_training_points=NUM_TRAINING_POINTS,
observation_noise_variance=.1)
def build_gp(amplitude, length_scale, observation_noise_variance):
"""Defines the conditional dist. of GP outputs, given kernel parameters."""
# Create the covariance kernel, which will be shared between the prior (which we
# use for maximum likelihood training) and the posterior (which we use for
# posterior predictive sampling)
kernel = tfk.ExponentiatedQuadratic(amplitude, length_scale)
# Create the GP prior distribution, which we will use to train the model
# parameters.
return tfd.GaussianProcess(
kernel=kernel,
index_points=observation_index_points_,
observation_noise_variance=observation_noise_variance)
gp_joint_model = tfd.JointDistributionNamed({
'amplitude': tfd.LogNormal(loc=0., scale=np.float64(1.)),
'length_scale': tfd.LogNormal(loc=0., scale=np.float64(1.)),
'observation_noise_variance': tfd.LogNormal(loc=0., scale=np.float64(1.)),
'observations': build_gp,
})
x = gp_joint_model.sample()
lp = gp_joint_model.log_prob(x)
print("sampled {}".format(x))
print("log_prob of sample: {}".format(lp))
# Create the trainable model parameters, which we'll subsequently optimize.
# Note that we constrain them to be strictly positive.
constrain_positive = tfb.Shift(np.finfo(np.float64).tiny)(tfb.Exp())
amplitude_var = tfp.util.TransformedVariable(
initial_value=1.,
bijector=constrain_positive,
name='amplitude',
dtype=np.float64)
length_scale_var = tfp.util.TransformedVariable(
initial_value=1.,
bijector=constrain_positive,
name='length_scale',
dtype=np.float64)
observation_noise_variance_var = tfp.util.TransformedVariable(
initial_value=1.,
bijector=constrain_positive,
name='observation_noise_variance_var',
dtype=np.float64)
trainable_variables = [v.trainable_variables[0] for v in
[amplitude_var,
length_scale_var,
observation_noise_variance_var]]
# Use `tf.function` to trace the loss for more efficient evaluation.
@tf.function(autograph=False, experimental_compile=False)
def target_log_prob(amplitude, length_scale, observation_noise_variance):
return gp_joint_model.log_prob({
'amplitude': amplitude,
'length_scale': length_scale,
'observation_noise_variance': observation_noise_variance,
'observations': observations_
})
# Now we optimize the model parameters.
num_iters = 1000
optimizer = tf.optimizers.Adam(learning_rate=.01)
# Store the likelihood values during training, so we can plot the progress
lls_ = np.zeros(num_iters, np.float64)
for i in range(num_iters):
with tf.GradientTape() as tape:
loss = -target_log_prob(amplitude_var, length_scale_var,
observation_noise_variance_var)
grads = tape.gradient(loss, trainable_variables)
optimizer.apply_gradients(zip(grads, trainable_variables))
lls_[i] = loss
print('Trained parameters:')
print('amplitude: {}'.format(amplitude_var._value().numpy()))
print('length_scale: {}'.format(length_scale_var._value().numpy()))
print('observation_noise_variance: {}'.format(observation_noise_variance_var._value().numpy()))
num_results = 100
num_burnin_steps = 50
sampler = tfp.mcmc.TransformedTransitionKernel(
tfp.mcmc.HamiltonianMonteCarlo(
target_log_prob_fn=target_log_prob,
step_size=tf.cast(0.1, tf.float64),
num_leapfrog_steps=8),
bijector=[constrain_positive, constrain_positive, constrain_positive])
adaptive_sampler = tfp.mcmc.DualAveragingStepSizeAdaptation(
inner_kernel=sampler,
num_adaptation_steps=int(0.8 * num_burnin_steps),
target_accept_prob=tf.cast(0.75, tf.float64))
initial_state = [tf.cast(x, tf.float64) for x in [1., 1., 1.]]
# Speed up sampling by tracing with `tf.function`.
@tf.function(autograph=False, experimental_compile=False)
def do_sampling():
return tfp.mcmc.sample_chain(
kernel=adaptive_sampler,
current_state=initial_state,
num_results=num_results,
num_burnin_steps=num_burnin_steps,
trace_fn=lambda current_state, kernel_results: kernel_results)
t0 = time.time()
samples, kernel_results = do_sampling()
t1 = time.time()
print("Inference ran in {:.2f}s.".format(t1-t0))
如果您有多个非常接近的索引点，则可能会发生这种情况，因此您可以考虑使用np.linspace或只是对随机抽奖进行一些后期过滤。我还建议更大的epsilon，也许是1e-6。