Stochastic Separation Theorems Research Articles

Stochastic Separation Theorems: How Geometry May Help to Correct AI Errors

Stochastic Separation Theorems: How Geometry May Help to Correct AI Errors

High-Dimensional Separability for One- and Few-Shot Learning.

This work is driven by a practical question: corrections of Artificial Intelligence (AI) errors. These corrections should be quick and non-iterative. To solve this problem without modification of a legacy AI system, we propose special ‘external’ devices, correctors. Elementary correctors consist of two parts, a classifier that separates the situations with high risk of error from the situations in which the legacy AI system works well and a new decision that should be recommended for situations with potential errors. Input signals for the correctors can be the inputs of the legacy AI system, its internal signals, and outputs. If the intrinsic dimensionality of data is high enough then the classifiers for correction of small number of errors can be very simple. According to the blessing of dimensionality effects, even simple and robust Fisher’s discriminants can be used for one-shot learning of AI correctors. Stochastic separation theorems provide the mathematical basis for this one-short learning. However, as the number of correctors needed grows, the cluster structure of data becomes important and a new family of stochastic separation theorems is required. We refuse the classical hypothesis of the regularity of the data distribution and assume that the data can have a rich fine-grained structure with many clusters and corresponding peaks in the probability density. New stochastic separation theorems for data with fine-grained structure are formulated and proved. On the basis of these theorems, the multi-correctors for granular data are proposed. The advantages of the multi-corrector technology were demonstrated by examples of correcting errors and learning new classes of objects by a deep convolutional neural network on the CIFAR-10 dataset. The key problems of the non-classical high-dimensional data analysis are reviewed together with the basic preprocessing steps including the correlation transformation, supervised Principal Component Analysis (PCA), semi-supervised PCA, transfer component analysis, and new domain adaptation PCA.

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 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.

Blessing of dimensionality at the edge and geometry of few-shot learning

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. The theory enables building few-shot AI correction algorithms and provides conditions justifying their successful application. 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.

Linear and Fisher Separability of Random Points in the d-Dimensional Spherical Layer and Inside the d-Dimensional Cube.

Stochastic separation theorems play important roles in high-dimensional data analysis and machine learning. It turns out that in high dimensional space, 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 dimensions. This and similar facts can be used for constructing correctors for artificial intelligent systems, for determining the intrinsic dimensionality 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 and from the cube. These results allow us to better outline the applicability limits of the stochastic separation theorems in applications.

Motor intent recognition using deep learning ensemble to be used in prosthetic control

This paper presents a new approach for constructing simple and computationally efficient improvements of generic Artificial Intelligence (AI) systems, including Multilayer and Deep Learning neural networks. The improvements are small network ensembles added to the existing AI architectures. Theoretical foundations of the approach are based on stochastic separation theorems and the ideas of the concentration of measure. We show that, subject to mild technical assumptions on statistical properties of internal signals in the original AI system, the approach enables fast removal of the AI’s errors with probability close to one on the datasets which may be exponentially large in dimension. The approach is illustrated with numerical examples and a case study of digits recognition in American Sign Language.

Fast construction of correcting ensembles for legacy Artificial Intelligence systems: Algorithms and a case study

This paper presents a new approach for constructing simple and computationally efficient improvements of generic Artificial Intelligence (AI) systems, including Multilayer and Deep Learning neural networks. The improvements are small network ensembles added to the existing AI architectures. Theoretical foundations of the approach are based on stochastic separation theorems and the ideas of the concentration of measure. We show that, subject to mild technical assumptions on statistical properties of internal signals in the original AI system, the approach enables fast removal of the AI’s errors with probability close to one on the datasets which may be exponentially large in dimension. The approach is illustrated with numerical examples and a case study of digits recognition in American Sign Language.

One-trial correction of legacy AI systems and stochastic separation theorems

We consider the problem of efficient “on the fly” tuning of existing, or legacy, Artificial Intelligence (AI) systems. The legacy AI systems are allowed to be of arbitrary class, albeit the data they are using for computing interim or final decision responses should posses an underlying structure of a high-dimensional topological real vector space. The tuning method that we propose enables dealing with errors without the need to re-train the system. Instead of re-training a simple cascade of perceptron nodes is added to the legacy system. The added cascade modulates the AI legacy system’s decisions. If applied repeatedly, the process results in a network of modulating rules “dressing up” and improving performance of existing AI systems. Mathematical rationale behind the method is based on the fundamental property of measure concentration in high dimensional spaces. The method is illustrated with an example of fine-tuning a deep convolutional network that has been pre-trained to detect pedestrians in images.

Correction of AI systems by linear discriminants: Probabilistic foundations

Artificial Intelligence (AI) systems sometimes make errors and will make errors in the future, from time to time. These errors are usually unexpected, and can lead to dramatic consequences. Intensive development of AI and its practical applications makes the problem of errors more important. Total re-engineering of the systems can create new errors and is not always possible due to the resources involved. The important challenge is to develop fast methods to correct errors without damaging existing skills. We formulated the technical requirements to the ‘ideal’ correctors. Such correctors include binary classifiers, which separate the situations with high risk of errors from the situations where the AI systems work properly. Surprisingly, for essentially high-dimensional data such methods are possible: simple linear Fisher discriminant can separate the situations with errors from correctly solved tasks even for exponentially large samples. The paper presents the probabilistic basis for fast non-destructive correction of AI systems. A series of new stochastic separation theorems is proven. These theorems provide new instruments for fast non-iterative correction of errors of legacy AI systems. The new approaches become efficient in high-dimensions, for correction of high-dimensional systems in high-dimensional world (i.e. for processing of essentially high-dimensional data by large systems).We prove that this separability property holds for a wide class of distributions including log-concave distributions and distributions with a special ‘SMeared Absolute Continuity’ (SmAC) property defined through relations between the volume and probability of sets of vanishing volume. These classes are much wider than the Gaussian distributions. The requirement of independence and identical distribution of data is significantly relaxed. The results are supported by computational analysis of empirical data sets.

High-Dimensional Brain: A Tool for Encoding and Rapid Learning of Memories by Single Neurons

Codifying memories is one of the fundamental problems of modern Neuroscience. The functional mechanisms behind this phenomenon remain largely unknown. Experimental evidence suggests that some of the memory functions are performed by stratified brain structures such as the hippocampus. In this particular case, single neurons in the CA1 region receive a highly multidimensional input from the CA3 area, which is a hub for information processing. We thus assess the implication of the abundance of neuronal signalling routes converging onto single cells on the information processing. We show that single neurons can selectively detect and learn arbitrary information items, given that they operate in high dimensions. The argument is based on stochastic separation theorems and the concentration of measure phenomena. We demonstrate that a simple enough functional neuronal model is capable of explaining: (i) the extreme selectivity of single neurons to the information content, (ii) simultaneous separation of several uncorrelated stimuli or informational items from a large set, and (iii) dynamic learning of new items by associating them with already “known” ones. These results constitute a basis for organization of complex memories in ensembles of single neurons. Moreover, they show that no a priori assumptions on the structural organization of neuronal ensembles are necessary for explaining basic concepts of static and dynamic memories.

Blessing of dimensionality: mathematical foundations of the statistical physics of data.

The concentrations of measure phenomena were discovered as the mathematical background to statistical mechanics at the end of the nineteenth/beginning of the twentieth century and have been explored in mathematics ever since. At the beginning of the twenty-first century, it became clear that the proper utilization of these phenomena in machine learning might transform the curse of dimensionality into the blessing of dimensionality This paper summarizes 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 with such classifiers and determine a non-iterative (one-shot) procedure for their construction.This article is part of the theme issue 'Hilbert's sixth problem'.

Stochastic separation theorems

The problem of non-iterative one-shot and non-destructive correction of unavoidable mistakes arises in all Artificial Intelligence applications in the real world. Its solution requires robust separation of samples with errors from samples where the system works properly. We demonstrate that in (moderately) high dimension this separation could be achieved with probability close to one by linear discriminants. Based on fundamental properties of measure concentration, we show that for M<aexp(bn) random M-element sets in Rn are linearly separable with probability p, p>1−ϑ, where 1>ϑ>0 is a given small constant. Exact values of a,b>0 depend on the probability distribution that determines how the random M-element sets are drawn, and on the constant ϑ. These stochastic separation theorems provide a new instrument for the development, analysis, and assessment of machine learning methods and algorithms in high dimension. Theoretical statements are illustrated with numerical examples.

The Blessing of Dimensionality: Separation Theorems in the Thermodynamic Limit

Abstract: We consider and analyze properties of large sets of randomly selected (i.i.d.) points in high dimensional spaces. In particular, we consider the problem of whether a single data point that is randomly chosen from a finite set of points can be separated from the rest of the data set by a linear hyperplane. We formulate and prove stochastic separation theorems, including: 1) with probability close to one a random point may be separated from a finite random set by a linear functional; 2) with probability close to one for every point in a finite random set there is a linear functional separating this point from the rest of the data. The total number of points in the random sets are allowed to be exponentially large with respect to dimension. Various laws governing distributions of points are considered, and explicit formulae for the probability of separation are provided. These theorems reveal an interesting implication for machine learning and data mining applications that deal with large data sets (big data) and high-dimensional data (many attributes): simple linear decision rules and learning machines are surprisingly efficient tools for separating and filtering out arbitrarily assigned points in large dimensions.