Generalized Gapped-kmer Filters for Robust Frequency Estimation

Abstract

In this paper, we study the generalized gapped k-mer filters and derive a closed form solution for their coefficients. We consider nonnegative integers $\ell$ and $k$, with $k\le \ell$, and an $\ell$-tuple $B=(b_1,\dots ,b_{\ell })$ of integers $b_i\ge 2$, $i=1,\dots ,\ell$. We introduce and study an incidence matrix $A=A_{\ell ,k;B}$. We develop a Möbius-like function ${\nu }_B$ which helps us to obtain closed forms for a complete set of mutually orthogonal eigenvectors of $A^{\top }A$ as well as a complete set of mutually orthogonal eigenvectors of $A{A}^{\top }$ 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 ${\nu }_B$, to provide the entries of the Moore-Penrose pseudo-inverse matrix $A^{+}$ and the Gapped k-mer filter matrix $A^{+}A$.

arXiv admin note: substantial text overlap with arXiv:1605.06806