of future data mining. The effects of the blessing of dimensionality were discovered in many applications and found their foundation in the concentration of measure phenomena. One example of the blessing of dimensionality phenomenon is linear separability of a random point from a large finite random set with high probability even if this set is exponentially large: the number of elements in this random set can grow exponentially with dimension. Moreover, this linear functional can be selected in the form of the simplest linear Fisher discriminant. This separability theorem was proven for a wide class of probability distributions: general uniformly log-concave distributions, product distributions in a cube and many other families (reviewed recently in ). "The blessing of dimensionality and the curse of dimensionality are two sides of the same coin." For example, the typical property of essentially high-dimensional probability distributions in a high-dimensional space is: the squared distance of random points to a selected point is, with high probability, close to the average (or median) squared distance. This property significantly simplifies the expected geometry of data and indexing of high-dimensional data (blessing), but, at the same time, it makes the similarity search in high dimensions difficult and even useless (curse). Zimek et al. noted that while the typical formalizations of the curse of dimensionality affect i.i.d. data, having data that is separated in each attribute becomes easier even in high dimensions, and argued that the signal-to-noise ratio matters: data becomes easier with each attribute that adds signal, and harder with attributes that only add noise (irrelevant error) to the data. In particular for unsupervised data analysis this effect is known as swamping. See also Bellman equation Clustering high-dimensional data Concentration of measure Dimension reduction Dynamic programming Fourier-related transforms Linear least squares Model order reduction Multilinear PCA Multilinear subspace learning Principal component analysis Singular value decomposition References Numerical analysis Dynamic programming Machine learning Dimension