String Kernel

A string kernel is a function from any of various families of kernel functions (see kernel methods) that operate over strings and sequences. The most typical example is as follows. Suppose that we are dealing with strings over a finite alphabet Σ. Given a string a = a₁a₂… a _n ∈ Σ^∗, we say that a substring p = p₁p₂… p _k occurs in a on positions i₁i₂… i _k iff 1 ≤ i₁ < i₂ < … < i _k ≤ n and a _ij = p _j for all j = 1, …, k. We define the weight of this occurrence as $\lambda^{i_k-i_i-k+1}$, where λ ∈ [0, 1] is a constant chosen in advance; in other words, an occurrence weighs less if characters of p are separated by other characters. Let ϕ _p (a) be the sum of the weights of all occurrences of p in a, and let ϕ(a) = (ϕ _p (a))_{p ∈ ∑}^∗ be an infinite-dimensional feature vector consisting of ϕ _p (a) for all possible substrings p ∈ Σ*. It turns out that the dot product of two such infinite-length vectors, K(a, a^′) = ϕ(a)^Tϕ(a^′), can be computed in time polynomial in the length of a and a^′, e.g.,...