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Despite the widely-spread consensus on the brain complexity, sprouts of thesingle neuron revolution emerged in neuroscience in the 1970s. They broughtmany unexpected discoveries, including grandmother or concept cells and sparsecoding of information in the brain.  In machine learning for a long time, the famous curse of dimensionalityseemed to be an unsolvable problem. Nevertheless, the idea of the blessing ofdimensionality becomes gradually more and more popular. Ensembles ofnon-interacting or weakly interacting simple units prove to be an effectivetool for solving essentially multidimensional problems. This approach isespecially useful for one-shot (non-iterative) correction of errors in largelegacy artificial intelligence systems.  These simplicity revolutions in the era of complexity have deep fundamentalreasons grounded in geometry of multidimensional data spaces. To explore andunderstand these reasons we revisit the background ideas of statisticalphysics. In the course of the 20th century they were developed into theconcentration of measure theory. New stochastic separation theorems reveal thefine structure of the data clouds.  We review and analyse biological, physical, and mathematical problems at thecore of the fundamental question: how can high-dimensional brain organisereliable and fast learning in high-dimensional world of data by simple tools?  Two critical applications are reviewed to exemplify the approach: one-shotcorrection of errors in intellectual systems and emergence of static andassociative memories in ensembles of single neurons.

The unreasonable effectiveness of small neural ensembles in high-dimensional brain