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From "Agnonchik (JIRA)" <j...@apache.org>
Subject [jira] [Comment Edited] (MAHOUT-1106) SVD++
Date Fri, 16 Nov 2012 11:38:12 GMT

    [ https://issues.apache.org/jira/browse/MAHOUT-1106?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13498742#comment-13498742
] 

Agnonchik edited comment on MAHOUT-1106 at 11/16/12 11:37 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 my point is that applying SVD++ doesn't make much sense in comparison to biased
SVD which ignores implicit feedback (Y parameter).
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 the point is that applying SVD++ 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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