Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK.
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK; Lobachevsky University, Nizhni Novgorod, Russia.
Neural Netw. 2021 Jun;138:33-56. doi: 10.1016/j.neunet.2021.01.034. Epub 2021 Feb 9.
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 the 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.
随机可分性现象在机器学习中被揭示并得到应用，以纠正人工智能 (AI) 系统的错误并分析 AI 的不稳定性。在广泛假设下的高维数据集中，每个点都可以通过简单而稳健的 Fisher 判别式（Fisher 可分）与集合的其余部分分开。错误或错误簇可以与其余数据分开。纠正 AI 系统的能力也为攻击它提供了可能性，而高维性则会因相同的随机可分性而产生漏洞，这种随机可分性是理解高维数据驱动 AI 中的稳健性和适应性的基础。为了管理错误和分析漏洞，随机分离定理应该评估在给定维度和给定分布类下数据集 Fisher 可分的概率。需要明确和最优的这些分离概率的估计，本工作解决了这个问题。对于重要的分布类：对数凹分布、它们的凸组合和乘积分布，获得了具有最优概率估计的一般随机分离定理。显著放宽了标准的独立同分布假设。这些定理和估计既可以用于纠正高维数据驱动的 AI 系统，也可以用于分析它们的漏洞。第三个应用领域是神经元集合中的记忆的出现、祖母细胞现象和大脑中的稀疏编码，以及对大脑中小神经元集合在高维中意外有效性的解释。
