多输入多输出GP？

Question

我试图用多维输入数据在GPFlow中实现多输出GP .

我从本期在GPflow中看到，多维输入可以通过‘定义多维基内核，然后在此基础上应用define’来实现。

我已经编写了以下代码，我知道对于同位素数据(所有的输出都是获得的)，可以使用类似于这个笔记本中描述的东西，但是在这里，由于我需要尝试ICM，所以让我们继续下面的代码。

但是，当我尝试运行以下代码时：

from gpflow.gpr import GPR
import gpflow
import numpy as np
from gpflow.kernels import Coregion


def f(x):
    def _y(_x):
        function_sum = 0
        for i in np.arange(0, len(_x) - 1):
            function_sum += (1 - _x[i]) ** 2 + 100 * ((_x[i + 1] - _x[i] ** 2) ** 2)
        return function_sum
    return np.atleast_2d([_y(_x) for _x in (np.atleast_2d(x))]).T


isotropic_X = np.random.rand(100, 2) * 4 - 2
Y1 = f(isotropic_X)
Y2 = f(isotropic_X) + np.random.normal(loc=2000, size=(100,1))
Y3 = f(isotropic_X) + np.random.normal(loc=-2000, size=(100,1))

# a Coregionalization kernel. The base kernel is Matern, and acts on the first ([0]) data dimension.
# the 'Coregion' kernel indexes the outputs, and actos on the second ([1]) data dimension
k1 = gpflow.kernels.Matern32(2)
coreg = Coregion(1, output_dim=3, rank=1, active_dims=[3]) # gpflow.kernels.Coregion(2, output_dim=2, rank=1)
coreg.W = np.random.rand(3, 1)
kern = k1 * coreg

# Augment the time data with ones or zeros to indicate the required output dimension
X_augmented = np.vstack((np.hstack((isotropic_X, np.zeros(shape=(isotropic_X.shape[0], 1)))),
                         np.hstack((isotropic_X, np.ones(shape=(isotropic_X.shape[0], 1)))),
                        np.hstack((isotropic_X, 2 * np.ones(shape=(isotropic_X.shape[0], 1))))))

# Augment the Y data to indicate which likeloihood we should use
Y_augmented = np.vstack((np.hstack((Y1, np.zeros(shape=(Y1.shape[0], 1)))),
                         np.hstack((Y2, np.ones(shape=(Y2.shape[0], 1)))),
                         np.hstack((Y3, 2 * np.ones(shape=(Y3.shape[0], 1))))))

# now buld the GP model as normal
m = GPR(X_augmented, Y_augmented, kern=kern)
m.optimize()

print(m.predict_f(np.array([[0.2, 0.2, 0], [0.4, 0.4, 0]])))

它给我的回报是：

  "Converting sparse IndexedSlices to a dense Tensor of unknown shape. "
Traceback (most recent call last):
  File "C:\Users\Administrator\Anaconda3\lib\site-packages\tensorflow\python\client\session.py", line 1356, in _do_call
    return fn(*args)
  File "C:\Users\Administrator\Anaconda3\lib\site-packages\tensorflow\python\client\session.py", line 1341, in _run_fn
    options, feed_dict, fetch_list, target_list, run_metadata)
  File "C:\Users\Administrator\Anaconda3\lib\site-packages\tensorflow\python\client\session.py", line 1429, in _call_tf_sessionrun
    run_metadata)
tensorflow.python.framework.errors_impl.InvalidArgumentError: indices[0] = 3 is not in [0, 3)
     [[{{node name.build_likelihood/name.kern.K/name.kern.coregion.K/GatherV2}}]]

所以我的问题是：

 - What is this problem and how to enable multi-output GP with multi-dimension input
 - I didn't quite get the workflow of gpflow with coregion, from [this multi-output gp slide](http://gpss.cc/gpss17/slides/multipleOutputGPs.pdf), The ICM returns output GP from a additive form of a latent process $u$ sampled from a GP parameterized by its weight $W$. But in the gpflow [notebook demo](https://nbviewer.jupyter.org/github/GPflow/GPflow/blob/develop/doc/source/notebooks/advanced/coregionalisation.ipynb) I can't see any latent process of that and the notebooks says 'The 'Coregion' kernel indexes the outputs, and acts on the last ([1]) data dimension (indices) of the augmented X values', which is quite different than the slides, I am really confused about these different descriptions, any hint on these?

Answer 1

问题仅仅在于您的偏移索引:共同区域化内核应该是

coreg = Coregion(input_dim=1, output_dim=3, rank=1, active_dims=[2])

因为active_dims=[2]索引第三列。

谢谢你提供了一个完全可复制的例子！我成功地运行了您的代码，并成功地使用AdamOptimizer和ScipyOptimizer的几个步骤对模型进行了优化，其日志似然值为-2023.4。