Adjusted Cosine Similarity

One fundamental difference between the similarity computation in user-based CF and item-based CF is that in case of user-based CF the similarity is computed along the rows of the matrix but in case of the item-based CF the similarity is computed along the columns i.e., each pair in the co-rated set corresponds to a different user (Figure 2). Computing similarity using basic cosine measure in item-based case has one important drawback-the difference in rating scale between different users are not taken into account. The adjusted cosine similarity offsets this drawback by subtracting the corresponding user average from each co-rated pair. Formally, the similarity between items i and j using this scheme is given by

$$sim(i,j) = \frac{\sum_{u \in U} ( R_{u,i} - \bar{R_{u}})( R_{... ...})^{2}} {\sqrt{\sum_{u \in U}(R_{u,j}-\bar{R_{u}})^{2} } } }.$$

Here $\bar{R_{u}}$ is the average of the u-th user's ratings.