Machine Learning 3. Nearest Neighbor and Kernel Methods - ISMLL

Machine Learning / 1. Distance Measures
Distances for Strings / Sequences
edit distance / Levenshtein distance:
d(x, y) := minimal number of deletions, insertions or substitions to transform x in y
Examples:
d(man, men) =1
d(house, spouse) =2
d(order, express order) =8
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of Hildesheim
Course on Machine Learning, winter term 2007 11/48
Machine Learning / 1. Distance Measures
Distances for Strings / Sequences
The edit distance is computed recursively. With
x 1:i := (x i ′) i ′ =1,...,i = (x 1 , x 2 , . . . , x i ),
i ∈ N
we compute the number of operations to transform x 1:i into y 1:j as
c(x 1:i , y 1:j ) := min{ c(x 1:i−1 , y 1:j ) + 1, // delete x i , x 1:i−1 y 1:j
c(x 1:i , y 1:j−1 ) + 1,
// x 1:i y 1:j−1 , insert y j
c(x 1:i−1 , y 1:j−1 ) + I(x i ≠ y j )} // x 1:i−1 y 1:j−1 , substitute y j for x i
starting from
c(x 1:0 , y 1:j ) = c(∅, y 1:j ) := j // insert y 1 , . . . , y j
c(x 1:i , y 1:0 ) = c(x 1:i , ∅) := i // delete x 1 , . . . , x i
Such a recursive computing scheme is called dynamic
programming.
Lars Schmidt-Thieme, Information Systems and Machine Learning Lab (ISMLL), Institute BW/WI & Institute for Computer Science, University of Hildesheim
Course on Machine Learning, winter term 2007 12/48