The cosine similarity between two vectors u = fu 1;u 2;:::;u Ngand v = fv 1;v 2;:::;v Ngis de ned as: sim(u;v) = uv jujjvj = P N r i=1 u iv i P N i=1 u 2 i P N i=1 v 2 i We cannot apply the formula directly to our semantic descriptors since we do not store the entries which are equal to zero. However, we can still compute the cosine similarity between vectors by only considering

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To work around this researchers have found that measuring the cosine similarity between two vectors is much better. To give an example, the red point and green point have a closer distance (Euclidean distance) with one another but in actuality, if you take the cosine similarity blue and red have a closer angular distance from one another. Cosine similarity is a measure of similarity between two non-zero vectors of an inner product space. It is defined to equal the cosine of the angle between them, which is also the same as the inner product of the same vectors normalized to both have length 1. The cosine of 0° is 1, and it is less than 1 for any angle in the interval (0, π ... Cosine similarity for users a and m are calculated using the following formula: $$s_{u}^{cos}(u_{k},u_{a}) = \frac{u_{k} \cdot u_{a}}{\left\| u_{k} \right\|\left\| u_{a} \right\|} = \frac{\sum x_{k,m}x_{a,m}}{\sqrt{\sum x_{k,m}^{2}\sum x_{a,m}^{2}}}$$ To calculate similarity between items m and b we used this formula:

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