How can we perform efficient inference and learning in directed probabilistic models, in the presence of continuous latent variables with intractable posterior distributions, and large datasets? We introduce a stochastic variational inference and learning algorithm that scales to large datasets and, under some mild differentiability conditions, even works in the intractable case. Our contributi...

In this paper we propose an approximation of piecewise linear membership functions with the help of sigmoid functions and certain arithmetic operations. The gradient-based tuning of piecewise linear membership functions can be achieved with the proposed efficient approximation because it has simple continuous derivatives. With this construction we can even approximate the Łukasiewicz operator f...

*Correspondence: [email protected] 1School of Mathematics, Anhui University, Hefei, 230039, China Full list of author information is available at the end of the article Abstract Discrete analogues of continuous-time neural models are of great importance in numerical simulations and practical implementations. In the current paper, a discrete model of continuous-time neural networks with variable c...

*Correspondence: [email protected] 1School of Mathematics, Anhui University, Hefei, 230039, China Full list of author information is available at the end of the article Abstract Discrete analogues of continuous-time neural models are of great importance in numerical simulations and practical implementations. In the current paper, a discrete model of continuous-time neural networks with variable c...

We consider a non-local Shrödinger problem driven by the fractional Orlicz g-Laplace operator as follows(P)(−△g)αu+g(u)=K(x)f(x,u),inRd, where d≥3,(−△g)α is operator, f:Rd×R→R measurable function and K positive continuous function. Employing Nehari manifold method without assuming well-known Ambrosetti-Rabinowitz differentiability conditions on non-linear term f, we prove that (P) has ground st...

In this article, a class of second order differential equations on [0, 1], driven by a γ-Hölder continuous function for any value of γ ∈ (0, 1) and with multiplicative noise, is considered. We first show how to solve this equation in a pathwise manner, thanks to Young integration techniques. We then study the differentiability of the solution with respect to the driving process and consider the...

The asymptotic distribution of sample quantiles in the classical definition is well-known to be normal for absolutely continuous distributions. However, this is no longer true for discrete distributions or sampleswith ties.We show that the definition of sample quantiles based on mid-distribution functions resolves this issue and provides a unified framework for asymptotic properties of sample q...

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Conditional expected values in Markov chains are solutions to a set of associated backward differential equations, which may be ordinary or partial depending on the number of relevant state variables. This paper investigates the validity of these differential equations by locating the points of non-smoothness of the state-wise conditional expected values, and it presents a numerical method for ...

It is frequently assumed in several domains of economics that demand functions are invertible prices. At the primitive level preferences, however, corresponding characterization has remained elusive. We identify necessary and sufficient conditions on a utility-maximizing consumer’s preferences for her function to be continuous invertible: strict convexity, monotonicity, differentiability sense ...