The Locality sensitive functions and its techniques are combined with the banding techniques in order to distinguish the pairs at low distance from the pairs of high distance. In this perspective, the steepness of S-curve gives a reflection on the process of avoiding the false positives as well as false negatives among the pair of candidates. It is required to explore the families of functions besides min-hash function. It serves to generate the pairs of candidate efficiently. The functions can be applied to space of the sets as well as Jaccard distance. There are three conditions, which are required to develop functions and distance measure. There are conditions of family functions that can be described as followed.
It is required to make the close pairs for the candidate compared to the distant pairs. It is required to make the notion précised. In addition, statistically independent in the sense is possible for estimating that there are more than two functions give a specific response by the products rule for independent events. In addition, it is important to detect the pairs of candidate in the time less than capability. It is combinable to develop such function, which are better for avoiding the false positives as well as negatives. The combined functions are required to take time less than the pairs. It is required to satisfy the technique for banding in single min-hash functions. On the other hand, locality sensitive functions need to consider that there are two items as well as rendering decision regarding the items that are required for pair of candidate. A collection of functions is called family of functions. For an instance, the family of min-hash functions are based on the possible permutations of rows of the characteristics matrix from the family. Hence, locality sensitive families for Jaccard Distance are required to be considered while amplifying locality sensitive family.
The process of constructing locality-sensitive families is important in the cosine distance as well as normal Euclidean distance. It is simple to develop locality sensitive family of the functions for Haming distance. Random hyperplanes as well as cosine distance are important to be considered. The cosine distance between the vectors is the angle between vectors. Hence, it is required to consider the vector as normal to hyperplane and then projection can be represented through dashed line. On contrary, randomly selected vector can be normal to the hyperplane such as dotted line. The probability for selecting the vector is r dashed in the dotted line. Therefore, the hyperline will be dotted dashed line. Instead of selecting the random vector from the possible vectors, it turns out sufficiently whether restricting the choice to the vectors and the elements are +1 and -1. In developing locality, it is important to consider the process for the functions. The techniques used in the family of has functions are required to develop and described as Euclidean spaces. There are points in the space for developing the locality sensitive families for any pair of the distances. On the other hand, it is required to derive the functions in bucket size in the partitions of life. The hash functions are required to protect the line. Hence, concluding the family F is explained as sensitive family of hash families and there are distances fall in the same bucket. Thus, amplifying the different examples of the locality-sensitive hash functions are achieved in this process. The techniques of amplification are adjusted with the probabilities in order to surround particular value. The probabilities of the hash functions need to use F function. Hence, it is required to formulate the functions properly to get appropriate result.
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