Generalized Gapped k-mer Filters for Robust Frequency Estimation - Mohammad-Noori, Morteza; Ghareghani, Narges; Ghandi, Mahmoud (Bulletin of the Iranian Mathematical Society, 2024)
In this paper, we study the generalized gapped k -mer filters and derive a closed form solution for their coefficients. We consider nonnegative integers ℓ and k , with k ≤ ℓ , and an ℓ -tuple B = ( b 1 , … , b ℓ ) of integers b i ≥ 2 , i = 1 , … , ℓ . We introduce and study an incidence matrix A = A ℓ , k ; B . We develop a Möbius-like function ν B which helps us to obtain closed forms for a complete set of mutually orthogonal eigenvectors of A ⊤ A as well as a complete set of mutually orthogonal eigenvectors of A A ⊤ corresponding to nonzero eigenvalues. The reduced singular value decomposition of A and combinatorial interpretations for the nullity and rank of A , are among the consequences of this approach. We then combine the obtained formulas, some results from linear algebra, and combinatorial identities of elementary symmetric functions and ν B , to provide the entries of the Moore–Penrose pseudo-inverse matrix A + and the Gapped k -mer filter matrix A + A .