Basal Ganglia, a group of sub-cortical neuronal nuclei in the brain, are commonly described as the neuronal substratum to Reinforcement Learning. Since the seminal work by Schultz [1], a huge amount of work has been done to deepen that analogy, from functional and anatomic points of view. Nevertheless, a noteworthy architectural hint has been hardly explored: the outstanding reduction of dimensionality from the input to the output of the basal ganglia. Bar-Gad et al. [2] have suggested that this transformation could correspond to a Principal Component Analysis but did not explore the full functional consequences of this hypothesis. In this paper, we propose to study this mechanism within a model more realistic from a computational neuroscience point of view. Particularly, we show its feasibility when the loop is closed, in the framework of Action Selection.