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From "Matthias Boehm (JIRA)" <j...@apache.org>
Subject [jira] [Created] (SYSTEMML-1761) Sparsity-exploiting weighted squared loss w/o weights
Date Wed, 12 Jul 2017 04:26:00 GMT
Matthias Boehm created SYSTEMML-1761:
----------------------------------------

             Summary: Sparsity-exploiting weighted squared loss w/o weights
                 Key: SYSTEMML-1761
                 URL: https://issues.apache.org/jira/browse/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:
{code}
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))
{code}

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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