LAUSR

The title of this article by Gorban et al. refers to Wigner’s famous lecture, “The unreasonable effectiveness of mathematics in the natural sciences” [1], delivered 60 years ago in 1959. In the lecture, Wigner emphasized the crucial role of mathematics in developing consistent theories in physics. Similarly, Gorban et al. focus on the role of mathematics in understanding nature, namely the functioning and structure of brains. They utilize mathematics of high-dimensional spaces to explain “how can high-dimensional brain organize reliable and fast learning in high-dimensional world of data by simple tools?”. A number of neurobiological studies have observed the energy efficiency of the brain which seems to exhibit both sparse activity (only a small fraction of neurons have a high rate of firing at any time) and sparse connectivity (each neuron is connected to only a limited number of other neurons) [2]. Gorban et al. suggest that sparse coding of information in the brain can be explained using high-dimensional geometry. They approach the investigation of biological neural networks by exploring a much simpler case of artificial ones. In recent years, neurocomputing achieved impressive successes [3]. In particular, randomized models and algorithms for neural networks have turned out to be quite efficient for performing high-dimensional tasks. Theoretical analysis complements the experimental evidence of almost deterministic behavior of stochastic algorithms on large networks and/or large data sets. With increase in data dimension and network size, outputs tend to be sharply concentrated around precalculated values. This behavior can be explained by the geometry of high-dimensional spaces, which have many counter-intuitive properties difficult to visualize for us who live in three-dimensional space. Mathematics alone guides us in these higher dimensions, where senses cannot reach. Using classical calculus (integration in spherical polar coordinates), one can compute the relative area of the d-dimensional sphere, which is occupied by the polar cap. More precisely, let Sd−1 denote the unit sphere (the set of vectors of length 1) in the d-dimensional Euclidean space and C(g, ε) = {f ∈ Sd−1 | 〈u, v〉 ≥ ε} the polar cap centered at a fixed vector g, which contains all vectors f which have the angular distance from g at most α = arccos ε (the inner product 〈f, g〉 is at least ε), see Fig. 1. For a fixed angle α, with increasing dimension d the normalized surface area μ of such cap decreases exponentially fast to zero as μ(C(g, ε)) ≤ e− dε2 2 (see, e.g., [4]). It is quite surprising to