Generalized Gapped k-mer Filters for Robust Frequency Estimation
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...
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| Published in: | Bulletin of the Iranian Mathematical Society Vol. 50; no. 5 |
|---|---|
| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Singapore
Springer Nature Singapore
01.10.2024
|
| Subjects: | |
| ISSN: | 1017-060X, 1735-8515 |
| Online Access: | Get full text |
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| Summary: | 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
. |
|---|---|
| ISSN: | 1017-060X 1735-8515 |
| DOI: | 10.1007/s41980-024-00901-z |