Let X be a space and F a family of 0, 1-valued functions on X. Vapnik and Chervonenkis showed that if F is “simple” (finite VC dimension), then for every probability measure μ on X and ε > 0 there is a finite set S such that for all f ∈ F , ∑ x∈S f(x)/|S| = [ ∫ f(x)dμ(x)]± ε. Think of S as a “universal ε-approximator” for integration in F . S can actually be obtained w.h.p. just by sampling a f...

Uncertain inference is a process of deriving consequences from uncertain knowledge or evidences via the tool of conditional uncertain set. Based on uncertain inference, uncertain system is a function from its inputs to outputs. This paper proves that uncertain systems are universal approximators, which means that uncertain systems are capable of approximating any continuous function on a compac...

The tradeoff between accuracy and interpretability in fuzzy modeling has shifted into focus in last few years. This paper aims at improving accuracy of linguistic models while maintaining a good interpretability. A new gradient-based method, extended version of Jager approach, is proposed for the optimization of transparent Mamdani systems. The advantage of Mamdani systems if compared to 0 orde...

We show that deep narrow Boltzmann machines are universal approximators of probability distributions on the activities of their visible units, provided they have sufficiently many hidden layers, each containing the same number of units as the visible layer. Besides from this existence statement, we provide upper and lower bounds on the sufficient number of layers and parameters. These bounds sh...

Due to its significance in the morphodynamic processes of natural streams and associated impacts on riverine aquatic habitats, sediment transport has been studied extensively for decades. Hydraulic researchers have endeavored to describe the above phenomenon using physical reasoning, dimensional analysis, empirical or pra ctical approaches. As a result a plethora of mathematical formulas have b...

Diierent predictors and their approximators in nonlinear prediction regression models are studied. The minimal value of the mean squared error (MSE) is derived. Some approximate formulae for the MSE of ordinary and weighted least squares predictors are given.

This dissertation shows that careful injection of noise into sample data can substantially speed up Expectation-Maximization algorithms. Expectation-Maximization algorithms are a class of iterative algorithms for extracting maximum likelihood estimates from corrupted or incomplete data. The convergence speed-up is an example of a noise benefit or"stochastic resonance"in statistical signal proce...

In this paper we present an approach to designing a novel type of fuzzy controller B spline basis functions are used for input variables and fuzzy singletons for output variables to specify linguistic terms Product is chosen as the fuzzy conjunction and centroid as the defuzzi ca tion method By appropriately designing the rule base a fuzzy controller can be interpreted as a B spline interpolato...

The employment of proper codings, such as the base-2 coding, has allowed to establish the universal approximation property of Boolean functions: if a sufficient number b of inputs (bits) is taken, they are able to approximate arbitrarily well any real Borel measurable mapping. However, if the reduced set of monotone Boolean functions, whose expression involves only and and or operators, is cons...

Deep Belief Networks (DBN) are generative models with many layers of hidden causal variables, recently introduced by Hinton et al. (2006), along with a greedy layer-wise unsupervised learning algorithm. Building on Le Roux and Bengio (2008) and Sutskever and Hinton (2008), we show that deep but narrow generative networks do not require more parameters than shallow ones to achieve universal appr...