21,928 research outputs found
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 -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
Stochastic kinetic models: Dynamic independence, modularity and graphs
The dynamic properties and independence structure of stochastic kinetic
models (SKMs) are analyzed. An SKM is a highly multivariate jump process used
to model chemical reaction networks, particularly those in biochemical and
cellular systems. We identify SKM subprocesses with the corresponding counting
processes and propose a directed, cyclic graph (the kinetic independence graph
or KIG) that encodes the local independence structure of their conditional
intensities. Given a partition of the vertices, the graphical
separation in the undirected KIG has an intuitive chemical
interpretation and implies that is locally independent of given . It is proved that this separation also results in global independence of
the internal histories of and conditional on a history of the jumps in
which, under conditions we derive, corresponds to the internal history of
. The results enable mathematical definition of a modularization of an SKM
using its implied dynamics. Graphical decomposition methods are developed for
the identification and efficient computation of nested modularizations.
Application to an SKM of the red blood cell advances understanding of this
biochemical system.Comment: Published in at http://dx.doi.org/10.1214/09-AOS779 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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