PMC:4996402 / 30318-34071
Annnotations
{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/4996402","sourcedb":"PMC","sourceid":"4996402","source_url":"https://www.ncbi.nlm.nih.gov/pmc/4996402","text":"4.2.3. Positive NCA, Non-Negative NCA and Non-Iterative NCA\nPosNCA [33] modifies the original NCA algorithm in two regards. The first aspect pertains to evaluating matrix A via a convex optimization (instead of ALS, as in the original NCA). The second aspect refers to the addition of the positivity constraints on all of the nonzero elements in the connectivity matrix. This assumption has a biological support [35]. The positivity constraint is valid only in situations where all TFs play the same role (i.e., activating or deactivating) on their corresponding targeted genes. If all of the TFs regulate the genes in a negative way (deactivating), the positivity assumption is maintained by multiplying the activity value in the signal matrix by the value −1. This positivity assumption is a convex constraint, which perfectly integrates with the convex formulation of the problem.\nThe essence of the formulation of PosNCA as a convex optimization problem relies on the orthogonality between the range space and the left null space. However, the challenge is to find a basis for the left null space of A. Consider C to be a basis for the left null space of A; then, it follows that:(25) CTA=0.\nIn the ideal case (X=AS), the range space and left null space of A are the same as those of X. This is because A is a full column rank (first criterion of NCA) and S is full row rank (third criterion of NCA). Therefore, C is obtained directly from X. In contrast to the noiseless case, there is no direct access to C in the noisy case. Alternatively, SSP provides a robust approximation of C. Consider the SVD X=UΣVT, and let U be partitioned as U=[UL,UR], where UL is of dimensions N×M and UR is of dimensions N×(N−M). Then, based on the discussion in Section 4.2.1, UR represents an approximation of C (C^=UR). Therefore, A can be estimated by minimizing the Frobenius norm of ||C^TA||F, while maintaining both constraints, i.e., the structure of the connectivity matrix and the positivity of all nonzero elements in the connectivity matrix. Mathematically, this problem can be formulated as follows:(26) A^=argminA||C^TA||Fs.t. A(I)=0, A(J)≥c where J stands for the set of indices of the nonzero elements in A and c is small positive constant. The optimization problem in Equation (26) is a convex optimization problem, since both the objective function and constraints are convex. The authors of [33] used an interior point-based method [24] to solve Equation (26). After evaluating A, the signal matrix is estimated using the traditional ALS:(27) S=A†X\nThe authors of [34] pioneered nnNCA, which utilizes the separable nature of the estimation problem corresponding to the matrix A in Equation (26) to achieve a computationally-efficient version of their previously-reported algorithm PosNCA. In PosNCA, matrix A is estimated in one shot by solving the optimization problem Equation (26). On the other hand, nnNCA estimates the columns in A in parallel, since each column of the connectivity matrix can be estimated independently of the other columns [34].\nLater, NINCA [32] was proposed to further improve the computational efficiency and the estimation accuracy of the framework reported in PosNCA. Analogous to nnNCA, NINCA estimates matrix A on a column-by-column basis. In addition, NINCA does not assume a positive constraint on the non-zero elements of the connectivity matrix and further imposes the constraint 1T·ai=1 for each column of matrix A to avoid the trivial solution. In terms of the procedure to estimate the TF matrix S, instead of using the traditional least-squares error adopted in [33] and [34], NINCA employs a total least-squares (TLS) algorithm [36] that not only considers the error in S, but also weighs the error in A.","divisions":[{"label":"Title","span":{"begin":0,"end":59}}],"tracks":[{"project":"2_test","denotations":[{"id":"27600242-22641712-69476469","span":{"begin":3076,"end":3078},"obj":"22641712"}],"attributes":[{"subj":"27600242-22641712-69476469","pred":"source","obj":"2_test"}]}],"config":{"attribute types":[{"pred":"source","value type":"selection","values":[{"id":"2_test","color":"#eccf93","default":true}]}]}}