Additional File 1 for “A Binary Matrix Factorization Algorithm for Protein Complex Prediction”

This file is appended with the paper titled ”A Binary Matrix Factorization Algorithm for Protein Complex Prediction” as a supplementary document. In this file, all evaluation results on 42 percentage pairs of random additions and deletions are given. Also, a theoretical analysis on the computational efficiency and performance of the proposed BYY-BMF algorithm is presented. Some parts of theoretical analysis have been included in a working paper titled ”BYY Harmony Learning Algorithms for Binary Factor Analysis and Binary Matrix Factorization” to be submitted to the Neurocomputing journal. 1 The Evaluations Results of All 6 × 7 = 42 Percentage Pairs (add,del) As in [1], we build a test graph X from the MIPS complexes [2] by linking the protein nodes in the same complex. For a systematic evaluation, we alter the test graph X to be Xa,d, where a and d denote the percentages of randomly added or deleted edges with respect to the number of original edges in X. The set of percentage pairs (a, d) is PAD = {(a, d) | a ∈ {0, 0.05, 0.1, 0.2, 0.4, 0.8, 1.0}; d ∈ {0, 0.05, 0.1, 0.2, 0.4, 0.8} }. We evaluate the predictions on the 42 altered graphs by BYY-BMF(opt) that outputs the clustering result of the highest harmony measure under repeated random initializations, and MCL(opt) that uses the optimal value of the inflation parameter tuned by its prediction performance. All the results are given in Figure 1-10. 2 A Theoretical Analysis on the BYY-BMF algorithm 2.1 Algorithm Details of BYY-BMF The BYY-BMF algorithm is implemented to maximize the following harmony functional H(p‖q) = ∑ A,Y,X ∫ p(α,β|X)p(A, Y |X, α,β)p(X) ln[q(X|Y, A)q(Y |α)q(A|β)q(α|Ξ)q(β|Ξ)]dαdβ. (1) The architecture of the algorithm is sketched in the Section “Methods” of the paper. In “Yang-Step”, Y = [y1, . . . ,yN ] is estimated by a discrete optimization, which is simply decoupled into individual maximizations per yt, since the likelihood q(X|Y,A)q(Y |α) = ∏N t=1[q(xt|yt,A)q(yt|α)] is factorizable. It follows that ŷt = arg max yt∈Y1 ln[q(xt|yt,A)q(yt|α)] = arg max yt∈Y1 { n ∑ i=1 [xit ln(1− e−ai yt)− (1− xit)ai yt] + yt (lnα) } , (2) ? The correspondence should be addressed to Prof. Lei Xu, [email protected].