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https://issues.apache.org/jira/browse/SYSTEMML-1761?page=com.atlassian.jira.plugin.system.issuetabpanels:all-tabpanel
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Matthias Boehm updated SYSTEMML-1761:
-------------------------------------
Description:
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 to huge (unnecessary) computation overhead for the outer-product-like
multiply of factors. 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.
was:
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.
> 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 to huge (unnecessary) computation overhead for the outer-product-like
> multiply of factors. 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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