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https://issues.apache.org/jira/browse/MAHOUT-1106?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13498742#comment-13498742
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Agnonchik edited comment on MAHOUT-1106 at 11/16/12 11:30 AM:
--------------------------------------------------------------
May I ask here some abstract question regarding the SVD++ algorithm? It has
nothing to do with the code. Please excuse me if I'm posting it in the wrong
place.
I wonder if the optimization problem solved by the SVD++ algorithm has a unique
solution?
Seems that in some cases, for example, when the regularization parameter lambda
is equal to zero, the problem permits multiple solutions.
We can write the SVD++ model as
ratingPrediction(user, item) = mu + bu(user) + bi(item) + (p(user) +
|N(user)|^(-0.5) * sum_{implItem from N(user)} y(implItem)) * q(item)^T
and the learning algorithm try to optimize the following cost function
sum_{(user, item) from R} (ratingPrediction - observedRating)^2 + lambda *
(||bu||_2^2 + ||bi||_2^2 + ||P||_F^2 + ||Q||_F^2 + ||Y||_F^2)
where P = [p(1); ... ;p(m)], Q = [q(1); ... ;q(n)], Y = [y(1); ... ;y(n)].
Lets introduce the matrix Z such that
[Z * Y](user) = |N(user)|^(-0.5) * sum_{implItem from N(user)} y(implItem)
Then for any solution P and Y of the optimization problem and an arbitrary
vector Y2, P2 = P + Z * (Y - Y2) and Y2 is also a solution.
Am I right?
If yes, then the point is that applying SVD++ doesn't make much sense in
comparison to biased SVD.
Thanks!
was (Author: agnonchik):
May I ask here some abstract question regarding the SVD++ algorithm? It has
nothing to do with the code. Please excuse me if I'm posting it in the wrong
place.
I wonder if the optimization problem solved by the SVD++ algorithm has a unique
solution?
Seems that in some cases, for example, when the regularization parameter lambda
is equal to zero, the problem permits multiple solutions.
We can write the SVD++ model as
ratingPrediction(user, item) = mu + bu(user) + bi(item) + (p(user) +
|N(user)|^(-0.5) * sum_{implItem from N(user)} y(implItem)) * q(item)^T
and the learning algorithm try to optimize the following cost function
sum_{(user, item) from R} (ratingPrediction - observedRating)^2 + lambda *
(||bu||_2^2 + ||bi||_2^2 + ||P||_F^2 + ||Q||_F^2 + ||Y||_F^2)
where P = [p(1); ... ;p(m)], Q = [q(1); ... ;q(n)], Y = [y(1); ... ;y(n)].
Lets introduce the matrix Z such that
[Z * Y](user) = |N(user)|^(-0.5) * sum_{implItem from N(user)} y(implItem)
Then for any solution P and Y of the optimization problem and an arbitrary
vector Y2, P2 = P + Z * (Y - Y2) and Y2 is also a solution.
Am I right?
If yes, then applying SVD++ to rating data doesn't make much sense in
comparison to biased SVD.
Thanks!
> SVD++
> -----
>
> Key: MAHOUT-1106
> URL: https://issues.apache.org/jira/browse/MAHOUT-1106
> Project: Mahout
> Issue Type: New Feature
> Components: Collaborative Filtering
> Reporter: Zeno Gantner
> Assignee: Sebastian Schelter
> Attachments: SVDPlusPlusFactorizer.java
>
>
> Initial shot at SVD++.
> Relies on the RatingsSGDFactorizer class introduced in MAHOUT-1089.
> One could also think about several enhancements, e.g. having separate
> regularization constants for user and item factors.
> I am also the author of the SVDPlusPlus class in MyMediaLite, so if there are
> any similarities, no need to worry -- I am okay with relicensing this to the
> Apache 2.0 license.
> https://github.com/zenogantner/MyMediaLite/blob/master/src/MyMediaLite/RatingPrediction/SVDPlusPlus.cs
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