21 research outputs found
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
The center of a convex set and capital allocation
A capital allocation scheme for a company that has a random total profit Y and uses a coherent risk measure Ļ has been suggested. The scheme returns a unique real number ĪĻ*(X,Y), which determines the capital that should be allocated to companyās subsidiary with random profit X. The resulting capital allocation is linear and diversifying as defined by Kalkbrener (2005). The problem is reduced to selecting the ācenterā of a non-empty convex weakly compact subset of a Banach space, and the solution to the latter problem proposed by Lim (1981) has been used. Our scheme can also be applied to selecting the unique Pareto optimal allocation in a wide class of optimal risk sharing problems
Regression analysis: likelihood, error and entropy
In a regression with independent and identically distributed normal residuals, the log-likelihood function yields an empirical form of the L2L2-norm, whereas the normal distribution can be obtained as a solution of differential entropy maximization subject to a constraint on the L2L2-norm of a random variable. The L1L1-norm and the double exponential (Laplace) distribution are related in a similar way. These are examples of an āinter-regenerativeā relationship. In fact, L2L2-norm and L1L1-norm are just particular cases of general error measures introduced by Rockafellar et al. (Finance Stoch 10(1):51ā74, 2006) on a space of random variables. General error measures are not necessarily symmetric with respect to ups and downs of a random variable, which is a desired property in finance applications where gains and losses should be treated differently. This work identifies a set of all error measures, denoted by EE, and a set of all probability density functions (PDFs) that form āinter-regenerativeā relationships (through log-likelihood and entropy maximization). It also shows that M-estimators, which arise in robust regression but, in general, are not error measures, form āinter-regenerativeā relationships with all PDFs. In fact, the set of M-estimators, which are error measures, coincides with EE. On the other hand, M-estimators are a particular case of L-estimators that also arise in robust regression. A set of L-estimators which are error measures is identifiedāit contains EE and the so-called trimmed LpLp-norms
Direct data-based decision making under uncertainty
In a typical one-period decision making model under uncertainty, unknown consequences are modeled as random variables. However, accurately estimating probability distributions of the involved random variables from historical data is rarely possible. As a result, decisions made may be suboptimal or even unacceptable in the future. Also, an agent may not view data occurred at different time moments, e.g. yesterday and one year ago, as equally probable. The agent may apply a so-called ātimeā profile (weights) to historical data. To address these issues, an axiomatic framework for decision making based directly on historical time series is presented. It is used for constructing data-based analogues of mean-variance and maxmin utility approaches to optimal portfolio selection