Decomposition of discrete memoryless sources

In this paper, we consider a problem of decomposing two discrete memoryless sources (DMS´s) into three approximately independent streams, one common and two private parts, such that the original sources can be reconstructed from the streams. We assume a distributed source coding model that is similar to the Gray-Wyner network. In the Gray-Wyner network, two distributed decoders reconstruct the sources, each based on the common message and the corresponding private message. In our model, we assume a two-stage decoding, where the reconstruction of the sources is done symbolwise. It is not only practically simpler than the decoding steps in the Gray-Wyner network, but it can be also useful in revealing hidden structures in the sources. We show that decomposing two DMS´s into three approximately independent streams is possible when the rate of the common part of the message generated by the encoder is equal to the Wyner´s common information between the two sources. We also consider a related problem, where there are two separate encoders and a single reconstruction function and discuss its relationship with the coding for computing problem.