Fig. 3

A possible extension: getting E-GPS biomarkers by deep learning. a Considering many genes and multiple conditional measures in a bio-system (e.g., a pathway) that consists of far more than a few genes, we may consider a multiple-layer network by deep learning, such as stacked RBMs (Hinton and Salakhutdinov 2006) and LMSER (Xu 1991, 1993) featured by unsupervised learning for a hierarchical abstraction of biomarkers. b According to Turing–Church thesis, the class of partial recursive functions is precisely the functions that can be computed by Turing machines, which provide an interesting perspective for understanding deep learning. The basic functions are involved within each layer, and the operators of composition and primitive recursion correspond a forward processing across different layers (namely what is usually called ‘deep’), while the minimisation operator corresponds a recurrent process from upper layers back to lower layers. From this perspective, we speculate that the class of functions performed by deep neural networks is also the class of functions that can be computed by Turing machines, for which ‘deep’ and ‘recurrent’ are indispensable