Linear and Fisher Separability of Random Points in the d-dimensional Spherical Layer

Stochastic separation theorems play important role in high-dimensional data analysis and machine learning. It turns out that in high dimension any point of a random set of points can be separated from other points by a hyperplane with high probability even if the number of points is exponential in terms of dimension. This and similar facts can be used for constructing correctors for artificial intelligent systems, for determining an intrinsic dimension of data and for explaining various natural intelligence phenomena. In this paper, we refine the estimations for the number of points and for the probability in stochastic separation theorems, thereby strengthening some results obtained earlier. We propose the boundaries for linear and Fisher separability, when the points are drawn randomly, independently and uniformly from a $d$-dimensional spherical layer. These results allow us to better outline the applicability limits of the stochastic separation theorems in applications.Comment: 6 pages, 3 figures IJCNN 2020 Accepte

General stochastic separation theorems with optimal bounds

Phenomenon of stochastic separability was revealed and used in machine learning to correct errors of Artificial Intelligence (AI) systems and analyze AI instabilities. In high-dimensional datasets under broad assumptions each point can be separated from the rest of the set by simple and robust Fisher's discriminant (is Fisher separable). Errors or clusters of errors can be separated from the rest of the data. The ability to correct an AI system also opens up the possibility of an attack on it, and the high dimensionality induces vulnerabilities caused by the same stochastic separability that holds the keys to understanding the fundamentals of robustness and adaptivity in high-dimensional data-driven AI. To manage errors and analyze vulnerabilities, the stochastic separation theorems should evaluate the probability that the dataset will be Fisher separable in given dimensionality and for a given class of distributions. Explicit and optimal estimates of these separation probabilities are required, and this problem is solved in present work. The general stochastic separation theorems with optimal probability estimates are obtained for important classes of distributions: log-concave distribution, their convex combinations and product distributions. The standard i.i.d. assumption was significantly relaxed. These theorems and estimates can be used both for correction of high-dimensional data driven AI systems and for analysis of their vulnerabilities. The third area of application is the emergence of memories in ensembles of neurons, the phenomena of grandmother's cells and sparse coding in the brain, and explanation of unexpected effectiveness of small neural ensembles in high-dimensional brain.Comment: Numerical examples and illustrations are added, minor corrections extended discussion and the bibliograph

Bringing the Blessing of Dimensionality to the Edge

In this work we present a novel approach and algorithms for equipping Artificial Intelligence systems with capabilities to become better over time. A distinctive feature of the approach is that, in the supervised setting, the approaches' computational complexity is sub-linear in the number of training samples. This makes it particularly attractive in applications in which the computational power and memory are limited. The approach is based on the concentration of measure effects and stochastic separation theorems. The algorithms are illustrated with examples.N/

High--Dimensional Brain in a High-Dimensional World: Blessing of Dimensionality

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.Comment: 18 pages, 5 figure

Blessing of dimensionality at the edge

In this paper we present theory and algorithms enabling classes of Artificial Intelligence (AI) systems to continuously and incrementally improve with a-priori quantifiable guarantees - or more specifically remove classification errors - over time. This is distinct from state-of-the-art machine learning, AI, and software approaches. Another feature of this approach is that, in the supervised setting, the computational complexity of training is linear in the number of training samples. At the time of classification, the computational complexity is bounded by few inner product calculations. Moreover, the implementation is shown to be very scalable. This makes it viable for deployment in applications where computational power and memory are limited, such as embedded environments. It enables the possibility for fast on-line optimisation using improved training samples. The approach is based on the concentration of measure effects and stochastic separation theorems and is illustrated with an example on the identification faulty processes in Computer Numerical Control (CNC) milling and with a case study on adaptive removal of false positives in an industrial video surveillance and analytics system

“Brainland” vs. “flatland”: How many dimensions do we need in brain dynamics?: Comment on the paper “The unreasonable effectiveness of small neural ensembles in high-dimensional brain” by Alexander N. Gorban et al.

In their review article (this issue) [1], Gorban, Makarov and Tyukin develop a successful effort to show in biological, physical and mathematical problems the relevant question of how high-dimensional brain can organise reliable and fast learning in the high-dimensional world of data using reduction tools. In fact, this paper, and several recent studies, focuses on the crucial problem of how the brain manages the information it receives, how it is organized, and how mathematics can learn about this and use dimension related techniques in other fields. Moreover, the opposite problem is also relevant, that is, how we can recover high-dimensional information from low-dimensional ones, the relevant problem of embedding dimensions (the other side of reducing dimensions). The human brain is a real open problem and a great challenge in human knowledge. The way the memory is codified is a fundamental problem in Neuroscience. As mentioned by the authors, the idea of blessing the dimensionality (and the opposite curse of dimensionality), are becoming more and more relevant in machine learning..