Euclidean Distance. For any√︃𝑣, 𝑣′ ∈ R𝐷 , the eu- If the parties in set 𝑆𝑑 are honest, they cannot distinguish be- tween the following scenarios: Scenario d.i, where 𝒊 ≠ 𝒅: The 𝑡𝑠 parties in 𝑆𝑖 are corrupted, and execute protocol Π correctly with input 𝑒𝑖 . Then, since Π achieves 𝑡𝑠 -Validity and 𝑡𝑠 -Liveness, the honest parties in 𝑆 obtain outputs where 𝑣𝑑 is the projection of 𝑣 on coordinate 1 ≤ 𝑑 ≤ 𝐷. clidean distance between 𝑣 and 𝑣′ is 𝛿 (𝑣, 𝑣′) = 𝐷 . (𝑣 𝑑=1 𝑑 − 𝑣 ′𝑑 )2, in convex({xx : 𝑖 ≠ 𝑑 Then, the output of any honest party in 𝑆𝑑 is in 𝑖≠𝑑 convex({xx : 𝑗 }), as shown in Figure 1. We use 𝛿max (𝑉 ) = max{𝛿 (𝑣, 𝑣′) : 𝑣, 𝑣′ ∈ 𝑉 } to denote the diam- 𝑖 ≠ 𝑗 }) = {𝑒𝑑 }, meaning that each honest party outputs its own □ input in Π. The diameter of the output set is 𝛿max ({𝑒𝑑 : 0 ≤ 𝑑 ≤ ⊆ eter of 𝑉 R𝐷 . ⊆ ∈ Below we provide the definitions of a convex set and the convex hull of a set. Roughly, 𝑉 R𝐷 is convex if, for any 𝑣, 𝑣′ 𝑉 , the segment between 𝑣 and 𝑣′ is also included in 𝑉 . for any 𝑣1, 𝑣2, . . . , 𝑣𝑘 ∈ 𝑉 and 𝜆1, 𝜆2, . . . , 𝜆𝑘 ≥ 0 such that 𝑘 𝜆𝑖 =
Euclidean Distance d u , v =∥u−v∥ (6.1) Although this is a very basic type of measure, Euclidean distance – which measures the direct distance from one point in graph space to another in a uniform manner – is widely used and has been built into many classification systems. Its relative simplicity lends it speed and enables easy implementation.
Euclidean Distance. Euclidean distance is the most common method used for measuring the distance between two objects. Sometimes known also as the Pythagorean metric, this method finds the distance by calculating the root of square differences between coordinates of a pair of objects. The Euclidean formula is: = √∑( − )2 =1
Euclidean Distance. We already stated, in the above sections, that the Euclidean distance will not be such a good measure for grouping the users. The reason behind this was that, in the case of Boolean data, when comparing two users, the distance represents nothing more than the number of groups in which they differ. Therefore, users will be grouped more according to the number of groups they belong to, than according to the groups themselves. Nevertheless, we performed k-means with Euclidean Distance as input as well. In almost all the cases, what we see is that one of the clusters is very good (has a silhouette value of between 0.4 and 0.5), while the other clusters have negative silhouette values (meaning that they were “not well” assigned). This is due to two reasons:
