High-dimensional data and high-dimensional representations of reality are inherent features of modern Artificial Intelligence systems and applications of machine learning. The well-known phenomenon of the "curse of dimensionality" states: many problems become exponentially difficult in high dimensions. Recently, the other side of the coin, the "blessing of dimensionality", has attracted much attention. It turns out that generic high-dimensional datasets exhibit fairly simple geometric properties. Thus, there is a fundamental tradeoff between complexity and simplicity in high dimensional spaces. Here we present a brief explanatory review of recent ideas, results and hypotheses about the blessing of dimensionality and related simplifying effects relevant to machine learning and neuroscience.

Complexity is an indisputable, well-known, and broadly accepted feature of the brain. Despite the apparently obvious and widely-spread consensus on the brain complexity, sprouts of the single neuron revolution emerged in neuroscience in the 1970s. They brought many unexpected discoveries, including grandmother or concept cells and sparse coding of information in the brain. In machine learning for a long time, the famous curse of dimensionality seemed to be an unsolvable problem. Nevertheless, the idea of the blessing of dimensionality becomes gradually more and more popular.

The journal Physics of Life Reviews, which has one of the highest impact factors in the categories "Biology" and "Biophysics", has published an article entitled "Symphony of high-dimensional brain". The authors of the article, world-famous scientists of Russian origin, including the chief researcher at the laboratory for advanced methods of multidimensional data analysis at Lobachevsky University, professor at the University of Leicester (UK) Alexander Gorban, the leading researcher at the laboratory for advanced methods of multidimensional data analysis at Lobachevsky University, professor at University of Leicester (UK) Ivan Tyukin, and the senior researcher at the laboratory of neural network technologies at Lobachevsky University, professor at the Complutense University (Madrid, Spain) Valery Makarov, sum up a broad discussion in the journal Physics of Life Reviews. A unified approach to these problems was proposed in the article by the same authors, "The unreasonable effectiveness of small neural ensembles in high-dimensional brain", which was published earlier in the same journal and triggered a wide discussion. According to the authors, the heart of the matter is in the geometry of multidimensional spaces. One of the discussion participants, the famous expert in neuroscience R. Quian Quiroga, proposed to call the new approach using the authors' initials: the GMT approach.

This paper is the final part of the scientific discussion organised by the Journal "Physics of Life Rviews" about the simplicity revolution in neuroscience and AI. This discussion was initiated by the review paper "The unreasonable effectiveness of small neural ensembles in high-dimensional brain". Phys Life Rev 2019, doi 10.1016/j.plrev.2018.09.005, arXiv:1809.07656. The topics of the discussion varied from the necessity to take into account the difference between the theoretical random distributions and "extremely non-random" real distributions and revise the common machine learning theory, to different forms of the curse of dimensionality and high-dimensional pitfalls in neuroscience. V. K{\r{u}}rkov{\'a}, A. Tozzi and J.F. Peters, R. Quian Quiroga, P. Varona, R. Barrio, G. Kreiman, L. Fortuna, C. van Leeuwen, R. Quian Quiroga, and V. Kreinovich, A.N. Gorban, V.A. Makarov, and I.Y. Tyukin participated in the discussion. In this paper we analyse the symphony of opinions and the possible outcomes of the simplicity revolution for machine learning and neuroscience.