Rank deficient matricesCholesky factorizationLet A be a rank deficient square matrix. We characterize the unique full rank Cholesky factorization LALAT of A where the factor LA is a lower echelon matrix with positive leading entries. We compute an extended decomposition for the normal matrix BTB ...
1.The full rank factorization and Moore-Penrose inverse for generalized row(column) unitary symmetric matrix广义行(列)酉对称矩阵的满秩分解及其Moore-Penrose逆 2.The concept of row (column) transposed matrix and row (column) symmetric matrix is given,their basic property is studied,and the formula...
(A.152)rank(L⁎)=rank(D(I−VDW˜VDT)D†). First, we prove the sufficiency. According to the property of idempotent matrices, we have (A.153)rank(W˜)=tr(W˜)andrank(I−W˜)=tr(I−W˜). By substituting (VDW˜VDT,L⁎) into the objective function, the followi...
We use GCN to extract low-rank feature matrices of microbes and diseases, and construct low-rank feature vectors of microbe-disease pairs. 3. We concatenate the high-rank feature vectors and low-rank vectors of microbe-disease pairs and use Deep Forest for latent microbe-disease association pred...
offered the MSBMF method, where the matrix representing drug-disease associations is divided into separate feature matrices for drugs and diseases using multi-similarity bilinear matrix factorization, and these matrices subsequently help in predicting potential drug-disease associations [17]. These methods...
Factorization yielded 15 human tumor cell programs (referred to ash1-h15) and 90 mouse brain and TME programs (referred to asm1-m90), in line with the greater transcriptional diversity of the mouse brain [33,34]. To decipher if programs represented cell types, cell states, or anatomical ...
Parts-based representations, such as non-negative matrix factorization and topic modeling, have been used to identify structure from single-cell sequencing data sets, in particular structure that is not as well captured by clustering or other dimensionality reduction methods. However, interpreting the ...
pattern-based methods have sparsity problems in which hypernym-hyponym pairs that match the pattern are rare in the corpus [10,36]. Recently, matrix factorization techniques, such as Singular Value Decomposition, have been used to mitigate the sparsity problem of pattern-based methods and showed im...
The effect of microbes on their human host is often mediated through changes in metabolite concentrations. As such, multiple tools have been proposed to predict metabolite concentrations from microbial taxa frequencies. Such tools typically fail to captu
We calculated the CNV score for each cell using the method of Peng et al. [35]. To determine the cell state, a nonnegative matrix factorization (NMF) algorithm was first employed for dimension reduction of all tumor cells, and subsequently, cells were scored for state using the R package ...