  Hierarchical Sparse Coding (HSC) is a powerful model to efficiently represent
multi-dimensional, structured data such as images. The simplest solution to
solve this computationally hard problem is to decompose it into independent
layer-wise subproblems. However, neuroscientific evidence would suggest
inter-connecting these subproblems as in the Predictive Coding (PC) theory,
which adds top-down connections between consecutive layers. In this study, a
new model called 2-Layers Sparse Predictive Coding (2L-SPC) is introduced to
assess the impact of this inter-layer feedback connection. In particular, the
2L-SPC is compared with a Hierarchical Lasso (Hi-La) network made out of a
sequence of independent Lasso layers. The 2L-SPC and the 2-layers Hi-La
networks are trained on 4 different databases and with different sparsity
parameters on each layer. First, we show that the overall prediction error
generated by 2L-SPC is lower thanks to the feedback mechanism as it transfers
prediction error between layers. Second, we demonstrate that the inference
stage of the 2L-SPC is faster to converge than for the Hi-La model. Third, we
show that the 2L-SPC also accelerates the learning process. Finally, the
qualitative analysis of both models dictionaries, supported by their activation
probability, show that the 2L-SPC features are more generic and informative.
