Bilinear Tensor Product in TensorFlow

Question

I'm working on re-implementing this paper and the key operation is a bilinear tensor product. I hardly know what that means, but the paper has a nice little graphic, which I understand.

The key operation is e_1 * W * e_2, and I want to know how to implement it in tensorflow, because the rest should be easy.

Basically, given 3D tensor W, slice it into matrices, and for the j'th slice (a matrix), multiply it on each side by e_1 and e_2, resulting in a scalar, which is the jth entry in the resulting vector (the output of this operation).

So I want to perform a product of e_1, a d-dimensional vector, W, the d x d x k tensor, and e_2, another d-dimensional vector. Could this product be expressed concisely in TensorFlow as it is now, or would I have to define my own op somehow?

EARLIER EDITS

Why doesn't multiplying these tensors work, and is there some way to define it more explicitly so that it works?

>>> import tensorflow as tf
>>> tf.InteractiveSession()
>>> a = tf.ones([3, 3, 3])
>>> a.eval()
array([[[ 1.,  1.,  1.],
        [ 1.,  1.,  1.],
        [ 1.,  1.,  1.]],

       [[ 1.,  1.,  1.],
        [ 1.,  1.,  1.],
        [ 1.,  1.,  1.]],

       [[ 1.,  1.,  1.],
        [ 1.,  1.,  1.],
        [ 1.,  1.,  1.]]], dtype=float32)
>>> b = tf.ones([3, 1, 1])
>>> b.eval()
array([[[ 1.]],

       [[ 1.]],

       [[ 1.]]], dtype=float32)
>>> 

The error message is

ValueError: Shapes TensorShape([Dimension(3), Dimension(3), Dimension(3)]) and TensorShape([Dimension(None), Dimension(None)]) must have the same rank

CURRENTLY

Turns out that multiplying two 3D tensors doesn't work either with tf.matmul, so but tf.batch_matmul does. tf.batch_matmul will also do 3D tensors and matrices. Then I tried 3D and a vector:

ValueError: Dimensions Dimension(3) and Dimension(1) are not compatible

Answer 1

You can do this with a simple reshape. For the first of the two matrix multiplies, you have k*d, length d vectors to dot product with.

This should be close:

temp = tf.matmul(E1,tf.reshape(Wddk,[d,d*k]))
result = tf.matmul(E2,tf.reshape(temp,[d,k]))