Matthias Boehm created SYSTEMML-1761:

             Summary: Sparsity-exploiting weighted squared loss w/o weights
                 Key: SYSTEMML-1761
             Project: SystemML
          Issue Type: Task
            Reporter: Matthias Boehm

There are existing rewrites and fused operators for weighted squared loss 
(wsloss). However, for the wsloss type {{NONE}}, i.e., without weights 
{{sum((X-(U%*%t(V)))^2)}}, the implementation is not sparsity-exploiting 
leading huge (unnecessary) computation overhead. As it turns out this 
expression can be rewritten into a sparsity-exploiting form as follows:
sum ((X - U %*% t(V)) ^ 2)
-> sum(X^2) - sum(2 * (X * (U%*%t(V)))) + sum((U%*%t(V))^2)
-> sum(X^2) - sum(2 * (X * (U%*%t(V)))) + sum ((t(U) %*% U) * (t(V) %*% V))

This task aims to change the block-level wsloss NONE implementation to exploit 
this logical rewrite by computing {{sum(X^2) - sum(2 * (X * (U%*%t(V))))}} in a 
sparsity-exploiting pass over non-zeros in X and a subsequent correct for {{+ 
sum ((t(U) %*% U) * (t(V) %*% V))}} via two tsmm operations.

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