In order to introduce more degrees of freedom but ensure that the additional unimodular matrices are still unitary, we allow matrices containing exactly one element from the set {±1, ±j} in each row and column and only zeros at all other positions.
Notice that, as earlier, each column of the transformed constraint matrices includes exactly one nonzero (unit) element so that ({mathbf{e}}_{N}^{prime } {mathbf{G}}_{N times NM} {mathbf{E}}_{J times NM}^{prime } = {mathbf{e}}_{M}^{prime } {mathbf{H}}_{M times NM} {mathbf{E}}_{J times NM}^{prime } = {mathbf{e}}_{J}^{prime }).
Here, for simpler computation, we exploit a more sparse random matrix which holds exactly one nonzero entry being ±1 equiprobably in each column.
Finally, if the jth column in the SNP matrix M contains exactly one value q from {0, 1}, then G M, F [ j]= q and the value f q, j is the degree of confidence that the jth column in the SNP matrix M and the jth column in the GenoSpectrum F are consistent.
We define a subset of n × n perfect phylogeny matrices, which we call n-clonal matrices that are in 1-1 correspondence with n-clonal trees T. D efinition 2 A matrix B ∈ { 0, 1 } n × n is an n-clonal matrix provided: There exists exactly one r ∈ [ n ] such that ∑ j = 1 n b r j = 1.
In the previous section we showed that each internal node in the generalized suffix tree, constructed for the set of strings corresponding to the rows in the discretized matrix after alphabet transformation, identifies exactly one CCC-Bicluster with at least two rows (maximal or not) (see Lemma 2).
Note that H can be viewed as a union of sub-matrices of the form H u, u ′, where each entry in H is covered by exactly one sub-matrix H u, u ′ for some u ∈ T, u′ ∈ S.
We make the assumption that there are correlated groups of variables in the data, and restrict the projection matrix V x to a clustering-type of matrix, where each variable comes from exactly one factor.
There is exactly one DV per distance matrix, but there are as many iDV as OTUs per distance matrix; for further details, see [ 48] and [ 49].
But since H + inv ( λ 0 ) is singular, it follows that the square summable solutions of (3.1) are generated by exactly one column of the matrix Φ ( λ 0 ) H + inv ( λ 0 ).
