TensorFlow vector times vector multiplication

Question

I have a vector of volatilities and a matrix of correlations

volatilities = tf.constant([0.2, 0.4, 0.6], dtype=tf.float32)
correlations = tf.constant([[1, 0.25, 0.5], [0.25, 1, 0.3], [0.5, 0.3, 1]], dtype=tf.float32)

I want to create a covariance matrix using these values

covariance = tf.tensordot(volatilities, volatilities, axes=0) * correlations

So far, so good. Now I'm using this inside a batched training process, and the shapes of my volatilities and correlations are (batch_size, 3) and (batch_size, 3, 3), respectively. I spent a lot of time trying to figure out how to get the covariance matrix. I tried first to obtain a volatility matrix of shape (batch_size, 3, 3), and then scalar multiplying with the correlations, but I did not succeed in getting the volatility matrix. The closest I get is a (3, 3) tensor by doing this:

volatility_matrix = tf.tensordot(volatilities, volatilities, axes=[[0], [0]])

How do I computationally efficiently obtain my correlation matrix with the batch_size dimension?

Answer 1

As the post referred to by @Lescurel mentioned, this problem can be solved by using tf.einsum:

volatility_matrix = tf.einsum("ij,ik->ijk", volas, volas)

solves the problem of obtaining the volatility matrix. From there, it's just scalar multiplication along the second axis. Specifically, to then get the covariance matrix, I did

covariance_matrix = tf.linalg.matmul(volatility_matrix, correlation_matrix)

Answer 2

These are the application matrix products of the covalence matrix and volatility present when some information is not represented helps determine the target relation when reading from batches of information.

[ Sample ]:

import tensorflow as tf

Cell_A = 0.20
Cell_B = 0.20
Cell_C = 0.20

volatilities = tf.constant([[ Cell_A, Cell_B, Cell_C ]], dtype=tf.float32)
correlations = tf.constant([[ 33405, 33405, 33405 ], [ 33405, 40050, 33405 ], [ 33405, 33405, 37025 ]], dtype=tf.float32)

volatility_matrix = tf.einsum('ij,jk->ik', volatilities, correlations)
correlation_matrix = correlations

print( ' volatilities : ' )
print( volatilities )
print( ' correlations : ' )
print( correlations )
print( ' volatility_matrix : ' )
print( volatility_matrix )

print( ' Shortest: ' + str( tf.math.argmin( volatility_matrix[0] ).numpy() ) )
print( ' Hghest: ' + str( tf.math.argmax( volatility_matrix[0] ).numpy() ) )

[ Output ]:

 volatilities :
tf.Tensor([[0.2 0.2 0.2]], shape=(1, 3), dtype=float32)
 correlations :
tf.Tensor(
[[33405. 33405. 33405.]
 [33405. 40050. 33405.]
 [33405. 33405. 37025.]], shape=(3, 3), dtype=float32)
 volatility_matrix :
tf.Tensor([[20043. 21372. 20767.]], shape=(1, 3), dtype=float32)
 Shortest: 0
 Hghest: 1