如何在 TensorFlow Probability 中创建不同内核对象的总和? [英] How to create sum of different kernel objects in TensorFlow Probability?

问题描述

我有一个关于在 Tensorflow-probability 中指定核函数的问题.

I have one question about specifying kernel function in Tensorflow-probability.

通常，如果我想创建一个内核对象，我会写

Usually, if I want to create a kernel object, I will write

import tensorflow as tf
import tensorflow_probability as tfp
tfp_kernels = tfp.positive_semidefinite_kernels

kernel_obj = tfp_kernels.ExponentiateQuadratic(*args, **karwgs)

我知道内核对象支持批量广播.但是如果我想构建一个内核对象，它是几个不同内核对象的总和，比如加性高斯过程?

I know that kernel object support batch broadcasting. But what if I want to build a kernel object that is the sum of several different kernel objects, like additive Gaussian processes?

我不确定如何在 Tensorflow 中总结"内核对象.我能做的就是创建几个单独的内核对象K1, ... KJ 网上好像没有类似的问题.

I am not sure how to "sum" up the kernel object in Tensorflow. What I am able to do is to create several separate kernel objects K1, ... KJ It seems that there is no similar question online.

提前感谢您的帮助.

更新:我试过直接+，但是协方差矩阵有一些奇怪的.

Updates: I tried direct +, but there is something strange with the covariance matrix.

我编写了以下示例:

feature1 = np.array([1, 2, 3, 5], dtype=np.float32)[:, np.newaxis]
feature2 = np.array([4.2, 6.5, 7.4, 8.3], dtype=np.float32)[:, np.newaxis]
features = np.concatenate([feature1, feature2], axis=1)

k1 = tfp_kernels.ExponentiatedQuadratic(amplitude=tf.cast(2.0, tf.float32),
                                        length_scale=tf.cast(2.0, tf.float32),
                                        feature_ndims=1,
                                        name='k1')

k2 = tfp_kernels.ExponentiatedQuadratic(amplitude=tf.cast(1.5, tf.float32),
                                        length_scale=tf.cast(1.5, tf.float32),
                                        feature_ndims=1,
                                        name='k2')

K = k1 + k2


gp_1 = tfd.GaussianProcess(kernel=k1,
                           index_points=feature1,
                           jitter=tf.cast(0, tf.float32),
                           name='gp_1')

gp_2 = tfd.GaussianProcess(kernel=k2,
                           index_points=feature2,
                           jitter=tf.cast(0, tf.float32),
                           name='gp_2')

gp_K1 = tfd.GaussianProcess(kernel=K,
                           index_points=feature1,
                           jitter=tf.cast(0, tf.float32),
                           name='gp_K')

gp_K2 = tfd.GaussianProcess(kernel=K,
                           index_points=feature2,
                           jitter=tf.cast(0, tf.float32),
                           name='gp_K')

gp_K = tfd.GaussianProcess(kernel=K,
                           index_points=features,
                           jitter=tf.cast(0, tf.float32),
                           name='gp_K')


gp_1_cov = gp_1.covariance()
gp_2_cov = gp_2.covariance()
gp_K1_cov = gp_K1.covariance()
gp_K2_cov = gp_K2.covariance()
gp_K_cov = gp_K.covariance()

with tf.Session() as my_sess:
    [gp_1_cov_, gp_2_cov_, gp_K1_cov_, gp_K2_cov_, gp_K_cov_] = my_sess.run([gp_1_cov, gp_2_cov, gp_K1_cov, gp_K2_cov, gp_K_cov])
my_sess.close()

print(gp_1_cov_)
print(gp_2_cov_)
print(gp_K1_cov_)
print(gp_K2_cov_)
print(gp_K_cov_)

前四个协方差矩阵很好，我通过比较 k(x_i, x_j) 元素来仔细检查它.

The first four covariance matrices are fine, and I double check it by comparing the k(x_i, x_j) element-wise.

但是，我不知道它是如何计算最后一个的.我试过

However, I don't know how it computes the last one. I tried

1. feature_1 与 kernel_1 和 feature_2 与 kernel_2
2. feature_1 与 kernel_2 和 feature_2 与 kernel_1

以下是最后三个矩阵的结果:

Below are the results of the last three matrices:

[[6.25       5.331647   3.3511252  0.60561347]
 [5.331647   6.25       5.331647   1.6031142 ]
 [3.3511252  5.331647   6.25       3.3511252 ]
 [0.60561347 1.6031142  3.3511252  6.25      ]]
[[6.25       2.7592793  1.3433135  0.54289836]
 [2.7592793  6.25       5.494186   3.7630994 ]
 [1.3433135  5.494186   6.25       5.494186  ]
 [0.54289836 3.7630994  5.494186   6.25      ]]
[[6.25       2.3782768  0.769587   0.06774138]
 [2.3782768  6.25       4.694947   1.0143608 ]
 [0.769587   4.694947   6.25       2.9651313 ]
 [0.06774138 1.0143608  2.9651313  6.25      ]]

它们与我的结果不匹配.有谁知道他们如何用不同的 index_points 计算最后一个矩阵?

They don't match with my result. Does anyone know how they compute the last matrix with different index_points?

或者一般来说，我如何指定内核，以便它们可以拟合模型，例如additive Gaussian processes，其中不同的index_points对应不同的内核函数，所以我可以在 TensorFlow Probability 框架下拟合模型 y_i = f_1(x_{1,i}) + f_2(x_{2,i}) + ...?

Or in general, how do I specify the kernel so that they can fit the model such as additive Gaussian processes, where different index_points correspond to different kernel functions, so that I can fit the model y_i = f_1(x_{1,i}) + f_2(x_{2,i}) + ... under TensorFlow Probability framework?

推荐答案

你可以直接写k_sum = k1 + k2！查看基类 PositiveSemidefineKernel，其中我们覆盖了加法和乘法运算符，您想了解它是如何工作的.

You can just write k_sum = k1 + k2! Check out the base class PositiveSemidefiniteKernel, where we've overridden the addition and multiplication operators, of you want to see how it works.

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