Semantic Vectors for Recommenders

Semantic Vectors for Recommenders

 Start with a User->Item preference matrix. The values range from -0.5 to 0.5 in “linear” interpretation: 0 is neutral. -0.5 is twice as negative as -0.25. Blank values default to 0.

	Item1	Item2	Item3
User1	0.3	0.6	0.9
User2		0.1	0.4
User3			0.7

Now, let's do a semantic vectors projection. The User vector is random:

	Random
User1	0.2
User2	0.6
User3	0.9

The formula for the Item outputs is:

Item(i) =  ((sum(U)+ sum(pref(u,i)/#U))/2)/#U

Where #U = the number of users expressing a preference for Item(i)

I1	Not enough preferences
I2	(((U1 + U2) + ((pref( u1,i2)  + pref(u2,i2))/#U))/2)/#U
I3	(((U1 + U2 + U3) + ((pref( u1,i3)  + pref(u2,i3) + pref(u3,i3))/#U)/2//#U

I1	No vector
I2	(((0.2 + 0.6) + (0.6 + 0.1)/2)/2)/2
I3	(((0.2 + 0.6 + 0.9) + (0.9 + 0.4 + 0.7)/3)/2)/3

I1	No vector
I2	0.2875
I3	0.3944…..

The resulting semantic vectors projection:

	Item1	Item2	Item3		User Vector
User1	0.3	0.6	0.9		0.2
User2		0.1	0.4		0.6
User3			0.7		0.9
Item Vector		0.2875	0.3944…

Here is a very difficult-to-read graph of the relationships:

Recommendations

The recommendations for the users are their vector’s distance from each item vector:

·  
User1 would be most interested in Item3, and finally Item2.

·  
User2 has interests the same order, but would find the whole list less interesting.

·  
User3 would be most interested in Item2, then Item3.

Item-Item Similarity

The Item-Item similarity set is the distances between the
item vectors. Unfortunately, since item 1 has only one preference, it has no
vector projection.

       
Item2:Item3 = .11

User Vectors

The User-User distances are random, there is nothing to be learned.

Summary

The Semantic Vectors algorithm takes a matrix of Row->Column relationships and creates a set or Row-Column distances and a set of Column-Column relationships in a new, common numerical space.  In this example, we created two sets of recommenders, a User->Item recommender and an Item->Item recommender.