For both item-item and user-user collaborative filtering the recommendation matrix $Γ_{m x n}$, which is an (m x n) matrix, can be defined as:

$$Γ(i,j)=r_{ij}$$ For user-user collaborative recommendation r_ij is defined as

$$r_{ij}= ∑_{(x∈ users)}[cosSim(x,i)*R_{xj} ]$$

For item-item collaborative recommendation $r_{ij}$ is defined as

$$r_{ij}= ∑_{(x ∈ items)}[R_{ix}*cosSim(x,j) ]$$

The matrix Γ can thus be defined as:

  • for user-user filtering: $Γ= S_U * R $
  • for user-user filtering $Γ= R * S_I$

In which scenarios does the equation Γ holds true. Assuming that we need to pick the k items with highest $r_{ij}$ in order to recommend the top-k items for a user i

  • $\begingroup$ Try looking into SVD. It should work for what you want. $\endgroup$
    – Anshul G.
    Feb 27 '17 at 12:29

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