General Lower Bounds based on Computer Generated Higher Order Expansions
نویسندگان
چکیده
In this article we show the rough outline of a computer algorithm to generate lower bounds on the exponential function of (in principle) arbitrary precision. We implemented this to generate all necessary analytic terms for the Boltzmann machine partition function thus leading to lower bounds of any order. It turns out that the extra variational parameters can be optimized analytically. We show that bounds upto nineth order are still reasonably calculable in practical situations. The gen erated terms can also be used as extra cor rection terms (beyond TAP) in mean field ex pansions.
منابع مشابه
Freezing in a Finite Slab Using Extensive Perturbation Expansions Method
In this paper Mathematica is used to solve the moving boundary problem of freezing in a finite slab for higher order perturbations. Mathematica is a new system which makes it possible to do algebra with computer. More specifically, it enables researchers to find the location of the ice at any time for as high order of perturbation as one whishes. Using of Mathematica and outer solution and an i...
متن کاملStochastic bounds for a single server queue with general retrial times
We propose to use a mathematical method based on stochastic comparisons of Markov chains in order to derive performance indice bounds. The main goal of this paper is to investigate various monotonicity properties of a single server retrial queue with first-come-first-served (FCFS) orbit and general retrial times using the stochastic ordering techniques.
متن کاملNonharmonic Gabor Expansions
We consider Gabor systems generated by a Gaussian function and prove certain classical results of Paley and Wiener on nonharmonic Fourier series of complex exponentials for the Gabor expansion. In particular, we prove a version of Plancherel-Po ́lya theorem for entire functions with finite order of growth and use the Hadamard factorization theorem to study regularity, exactness and deficienc...
متن کاملReal-Valued Special Functions: Upper and Lower Bounds
This development proves upper and lower bounds for several familiar realvalued functions. For sin, cos, exp and the square root function, it defines and verifies infinite families of upper and lower bounds, mostly based on Taylor series expansions. For tan−1, ln and exp, it verifies a finite collection of upper and lower bounds, originally obtained from the functions’ continued fraction expansi...
متن کاملOnline learning of positive and negative prototypes with explanations based on kernel expansion
The issue of classification is still a topic of discussion in many current articles. Most of the models presented in the articles suffer from a lack of explanation for a reason comprehensible to humans. One way to create explainability is to separate the weights of the network into positive and negative parts based on the prototype. The positive part represents the weights of the correct class ...
متن کامل