[jira] [Commented] (SPARK-2426) Quadratic Minimization for MLlib ALS

["Valeriy Avanesov (JIRA)" <ji...@apache.org>]

    [ https://issues.apache.org/jira/browse/SPARK-2426?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=14231375#comment-14231375 ] 

Valeriy Avanesov commented on SPARK-2426:
-----------------------------------------

I'm not sure if I understand your question...

As far as I can see, w_i stands for a row of the matrix w and h_j stands for a column of the matrix h.  

\sum_i \sum_j ( r_ij - w_i*h_j) -- is not a matrix norm. Probably, you either miss abs or square -- \sum_i \sum_j |r_ij - w_i*h_j| or \sum_i \sum_j ( r_ij - w_i*h_j)^2
It looks like l2 regularized stochastic matrix decomposition with respect to Frobenius (or l1) norm. But I don't understand why do you consider k optimization problems (do you? What does k \in {1 ... 25} stand for?). 

Anyway, l2 regularized stochastic matrix decomposition problem is defined as follows 

Minimize w.r.t. W and H : ||R - W*H|| + \lambda(||W|| + ||H||)
under non-negativeness and normalization constraints. 

||..|| stands for Frobenius norm (or l1). 

By the way: is the matrix of ranks r stochastic? Stochastic matrix decomposition doesn't seem reasonable if it's not. 

> Quadratic Minimization for MLlib ALS
> ------------------------------------
>
>                 Key: SPARK-2426
>                 URL: https://issues.apache.org/jira/browse/SPARK-2426
>             Project: Spark
>          Issue Type: New Feature
>          Components: MLlib
>    Affects Versions: 1.3.0
>            Reporter: Debasish Das
>            Assignee: Debasish Das
>   Original Estimate: 504h
>  Remaining Estimate: 504h
>
> Current ALS supports least squares and nonnegative least squares.
> I presented ADMM and IPM based Quadratic Minimization solvers to be used for the following ALS problems:
> 1. ALS with bounds
> 2. ALS with L1 regularization
> 3. ALS with Equality constraint and bounds
> Initial runtime comparisons are presented at Spark Summit. 
> http://spark-summit.org/2014/talk/quadratic-programing-solver-for-non-negative-matrix-factorization-with-spark
> Based on Xiangrui's feedback I am currently comparing the ADMM based Quadratic Minimization solvers with IPM based QpSolvers and the default ALS/NNLS. I will keep updating the runtime comparison results.
> For integration the detailed plan is as follows:
> 1. Add QuadraticMinimizer and Proximal algorithms in mllib.optimization
> 2. Integrate QuadraticMinimizer in mllib ALS



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