tri-factorization. For example, consider three data objects, mutual relations of
which are given in three different relation matrices,R 12 ,R13,andR 23 , while
relations between objects of the same type (intratype) are given by the constraint
matrices,L 1 ,L 2 , andL 3. In this setup, the penalized NMTF takes as an input the
following block matrices:
R¼
0R 12 R 13
R 12 T 0R 23
R 13 T R 23 T 0
2
4
3
(^5) ,L¼
L 1 00
0L 2 0
00L 3
2
4
3
(^5) ð 13 Þ
and randomly initialized matrices:
S¼
0S 12 S 13
S 12 T 0S 23
S 13 T S 23 T 0
2
4
3
(^5) ,G¼
G 1 00
0G 2 0
00G 3
2
4
3
(^5) ð 14 Þ
After minimizing the following objective function:
min
G 0
J¼ RGSGT
2
FþtrGTLG ð 15 Þit outputs block representation of matricesSandG. The blocks of matrixGare
further used to cluster each data set. Solving this optimization problem (Eq. 15 )
involves in themultiplicative update rules, which start with some randomly initial-
ized matricesGandSand then iteratively improve matrix values until stopping
criterion is achieved. The stopping criterion is achieved if the difference between
values of the objective function in two consecutive iterations is smaller than same
threshold valueE:|Jn–Jn- 1 |<E. Here, we did not provide equations for this
multiplicative update rules due to their complex derivation, but we refer a reader
to (see Zˇitniket al.( 2013 )) where these rules are provided, along with their detailed
description.
Following this formalism, we can integrate any number of data sources. The
basic idea of matrix factorization-based data fusion lies in the fact that matrix
factors,Gi, which represent the model of biological data of a particular type, are
shared across all data types. This data integration approach belongs to the class of
intermediateintegration approaches, which have been proven to be the most
accurate. For example, in the previous example, matrixG 2 is shared between the
first and the second data types. Namely,G 2 participates in the decomposition ofR 12
and at the same time in the decomposition ofR 23 (see Fig. 5 for a simple
illustration).^124 The clustering assignment obtained fromG 2 matrix is therefore
influenced by the two data sources, represented by matricesR 12 andR 23.
(^124) Zˇitnik et al. ( 2013 ).
168 V. Gligorijevic ́and N. Pržulj