The Coupled Sum Function is a lower priority as it is not likely to be needed for the Coupled VAE development. Nevertheless, for completeness, this would be nice to have in the library.
There are two ways the function could be developed.
1) follow the approach used in the mathematical code, which is a straightforward implementation of the basic function;
2) A better alternative is to utilize the exp_k and log_k functions. This is sketched below.
The Coupled Sum arises from the product of coupled exponential functions which in turn results in the coupled sum of the exponents of a resultant. exp_k x * exp_k y = exp_k (x +_k y). The coupled sum is defined as x +_k y = x + y + k x y. However, a more complete version that accounts for the parameters alpha and dimension like the coupled product function does. Thus a better implementation would be as follows.
Coupled_Sum (X) = log_k ( Product_Total (exp_k (x_i) )), where X is an array and x_i are the elements. I'm not showing the alpha and dimension terms but these would be included to complete the expression.
The Coupled Sum Function is a lower priority as it is not likely to be needed for the Coupled VAE development. Nevertheless, for completeness, this would be nice to have in the library.
There are two ways the function could be developed. 1) follow the approach used in the mathematical code, which is a straightforward implementation of the basic function; 2) A better alternative is to utilize the exp_k and log_k functions. This is sketched below.
The Coupled Sum arises from the product of coupled exponential functions which in turn results in the coupled sum of the exponents of a resultant. exp_k x * exp_k y = exp_k (x +_k y). The coupled sum is defined as x +_k y = x + y + k x y. However, a more complete version that accounts for the parameters alpha and dimension like the coupled product function does. Thus a better implementation would be as follows.
Coupled_Sum (X) = log_k ( Product_Total (exp_k (x_i) )), where X is an array and x_i are the elements. I'm not showing the alpha and dimension terms but these would be included to complete the expression.