I have had some time this weekend to take a deeper look at Sean's slides
from Berlin buzzwords, where he explains the math behind the distributed
item-based recommender. I think I found a way to extend it from using
only simple cooccurrence counts to using the standard computations of an
item-based recommender as defined in Sarwar et al "Item-Based
Collaborative Filtering Recommendation Algorithms"
(http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.144.9927&rep=rep1&type=pdf).


I'd be happy to see someone check and validate my thoughts!

If I understand the distributed recommender correctly, what it generally
does is that it computes the prediction values for all users towards all
items those users have not rated yet. And the computation is done in the
following way:

  u = a user
  i = an item not yet rated by u
  N = all items cooccurring with i

  Prediction(u,i) = sum(all n from N: cooccurrences(i,n) * rating(u,n))

The formula used in the paper which is used by
GenericItemBasedRecommender.doEstimatePreference(...) too, looks very
similar to the one above:

  u = a user
  i = an item not yet rated by u
  N = all items similar to i (where similarity is usually computed by
pairwisely comparing the item-vectors of the user-item matrix)

  Prediction(u,i) = sum(all n from N: similarity(i,n) * rating(u,n)) /
sum(all n from N: abs(similarity(i,n)))

There are only 2 differences:
 a) instead of the cooccurrence count, certain similarity measures like
pearson or cosine can be used
 b) the resulting value is normalized by the sum of the similarities

to overcome difference a) we would only need to replace the part that
computes the cooccurrence matrix with the code from ItemSimilarityJob or
the code introduced in MAHOUT-418, then we could compute arbitrary
similarity matrices and use them in the same way the cooccurrence matrix
is currently used

Regarding difference b) from a first look at the implementation I think
it should be possible to transfer the necessary similarity matrix
entries from the PartialMultiplyMapper to the
AggregateAndRecommendReducer to be able to compute the normalization
value in the denominator of the formula.

-sebastian

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