GPflow 2:带有MOK和多输入抛出ValueError的VGP模型

Question

我在跟踪GPflow 2.5.2文档的多输出内核笔记本。我试图用VGP或GPR模型代替SVGP模型，因为我只有很少的数据，不需要稀疏的方面。

我正在使用SharedIndependent多输出内核。

对于这两个模型，我得到了矩阵乘法中的维数不正确的ValueErrors。我想我需要改变输入数据的格式，但是我不知道怎么做，所以我只使用与SVGP模型相同的格式。

VGP模型的错误消息：

ValueError: Dimensions must be equal, but are 2 and 100 for '{{node MatMul}} = BatchMatMulV2[T=DT_DOUBLE, adj_x=false, adj_y=false](Cholesky, MatMul/identity_CONSTRUCTED_AT_top_level/forward/ReadVariableOp)' with input shapes: [100,2,2], [100,2].

探地雷达模型的错误信息：

ValueError: Dimensions must be equal, but are 2 and 100 for '{{node triangular_solve/MatrixTriangularSolve}} = MatrixTriangularSolve[T=DT_DOUBLE, adjoint=false, lower=true](Cholesky, sub)' with input shapes: [100,2,2], [100,2].

在像这样初始化VGP模型之后，我已经尝试将q_mu和q_sqrt值设置为建议的这里 (没有工作)：

m.q_mu = np.zeros((len(x_train)*len(y_train.T), 1), dtype=gpflow.config.default_float())
m.q_sqrt = np.expand_dims(np.identity(len(x_train)*len(y_train.T), dtype=gpflow.config.default_float()), axis=0)

代码如下：

import gpflow as gpf    

def generate_data(N=100):
    X1 = np.random.rand(N, 1)

    Y1 = np.sin(6 * X1) + np.random.randn(*X1.shape) * 0.03 + 2
    Y2 = np.sin(5 * X1 + 0.7) + np.random.randn(*X1.shape) * 0.1 + 0.5

    return X1, np.concatenate((Y1, Y2), axis=1)

N=100
M=15
P=2

data = (X, Y) = generate_data(N)

# create multi-output kernel
kernel = gpf.kernels.SharedIndependent(
    gpf.kernels.Matern52(active_dims=list(range(X.shape[1]))), output_dim=P
)

# initialization of inducing input locations (M random points from the training inputs)
Zinit = np.linspace(0, 1, M)[:, None]
Z = Zinit.copy()
# create multi-output inducing variables from Z
iv = gpf.inducing_variables.SharedIndependentInducingVariables(
    gpf.inducing_variables.InducingPoints(Z)
)

m = gpf.models.SVGP(kernel, gpf.likelihoods.Gaussian(), inducing_variable=iv, num_latent_gps=P)

optimizer = gpf.optimizers.Scipy()
optimizer.minimize(
    m.training_loss_closure(data),
    variables=m.trainable_variables,
    method="l-bfgs-b",
    options={"iprint": 0, "maxiter": ci_niter(2000)},
)

#  implementation of VGP

m = gpf.models.VGP(data, kernel, gpf.likelihoods.Gaussian(), num_latent_gps=P)
optimizer = gpf.optimizers.Scipy()
optimizer.minimize(
    m.training_loss,
    variables=m.trainable_variables,
    method="l-bfgs-b",
    options={"iprint": 0, "maxiter": ci_niter(2000)},
)

## implementation og gpflow.models.GPR

m = gpf.models.GPR(data, kernel)
optimizer = gpf.optimizers.Scipy()
optimizer.minimize(
    m.training_loss,
    variables=m.trainable_variables,
    method="l-bfgs-b",
    options={"iprint": 0, "maxiter": ci_niter(2000)},
)

Answer 1

不幸的是，目前只有SVGP支持多输出内核。

在更多的模型中支持它是一个共同要求，但是它是一个令人惊讶的大量工作，所以它从来没有完成。

好消息是，更简单的模型确实支持在多维上广播单个内核，这与SharedIndependent MOK完全相同。只要插入一个单输出内核，它就会广播。例如：

import gpflow as gpf
import numpy as np

D = 1
P = 2


def generate_data(N=100):
    X1 = np.random.rand(N, D)

    Y1 = np.sin(6 * X1) + np.random.randn(*X1.shape) * 0.03 + 2
    Y2 = np.sin(5 * X1 + 0.7) + np.random.randn(*X1.shape) * 0.1 + 0.5

    return X1, np.concatenate((Y1, Y2), axis=1)


N = 100
M = 15

data = generate_data(N)
train_data = (data[0][:70], data[1][:70])
test_data = (data[0][70:], data[1][70:])

# create multi-output kernel
kernel = gpf.kernels.SharedIndependent(gpf.kernels.Matern52(), output_dim=P)

# initialization of inducing input locations (M random points from the training inputs)
Zinit = np.linspace(0, 1, M)[:, None]
Z = Zinit.copy()
# create multi-output inducing variables from Z
iv = gpf.inducing_variables.SharedIndependentInducingVariables(
    gpf.inducing_variables.InducingPoints(Z)
)

m = gpf.models.SVGP(
    kernel, gpf.likelihoods.Gaussian(), inducing_variable=iv, num_latent_gps=P
)

optimizer = gpf.optimizers.Scipy()
optimizer.minimize(
    m.training_loss_closure(train_data),
    variables=m.trainable_variables,
    method="l-bfgs-b",
    options={"iprint": 0, "maxiter": 2000},
)
print("svgp", np.mean(m.predict_log_density(test_data)))

#  implementation of VGP

m = gpf.models.VGP(
    train_data, gpf.kernels.Matern52(), gpf.likelihoods.Gaussian(), num_latent_gps=P
)
optimizer = gpf.optimizers.Scipy()
optimizer.minimize(
    m.training_loss,
    variables=m.trainable_variables,
    method="l-bfgs-b",
    options={"iprint": 0, "maxiter": 2000},
)
print("vgp", np.mean(m.predict_log_density(test_data)))

## implementation og gpflow.models.GPR

m = gpf.models.GPR(
    train_data,
    gpf.kernels.Matern52(),
)
optimizer = gpf.optimizers.Scipy()
optimizer.minimize(
    m.training_loss,
    variables=m.trainable_variables,
    method="l-bfgs-b",
    options={"iprint": 0, "maxiter": 2000},
)
print("gpr", np.mean(m.predict_log_density(test_data)))