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https://issues.apache.org/jira/browse/SPARK-2426?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=14231375#comment-14231375
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Valeriy Avanesov edited comment on SPARK-2426 at 12/2/14 11:47 AM:
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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.
was (Author: acopich):
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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