  The concentration of measure phenomena were discovered as the mathematical
background of statistical mechanics at the end of the XIX - beginning of the XX
century and were then explored in mathematics of the XX-XXI centuries. At the
beginning of the XXI century, it became clear that the proper utilisation of
these phenomena in machine learning might transform the curse of dimensionality
into the blessing of dimensionality.
  This paper summarises recently discovered phenomena of measure concentration
which drastically simplify some machine learning problems in high dimension,
and allow us to correct legacy artificial intelligence systems. The classical
concentration of measure theorems state that i.i.d. random points are
concentrated in a thin layer near a surface (a sphere or equators of a sphere,
an average or median level set of energy or another Lipschitz function, etc.).
  The new stochastic separation theorems describe the thin structure of these
thin layers: the random points are not only concentrated in a thin layer but
are all linearly separable from the rest of the set, even for exponentially
large random sets. The linear functionals for separation of points can be
selected in the form of the linear Fisher's discriminant.
  All artificial intelligence systems make errors. Non-destructive correction
requires separation of the situations (samples) with errors from the samples
corresponding to correct behaviour by a simple and robust classifier. The
stochastic separation theorems provide us by such classifiers and a
non-iterative (one-shot) procedure for learning.
