Related papers: Approximation by filter functions
Approximation theory is concerned with the ability to approximate functions by simpler and more easily calculated functions. The first question we ask in approximation theory concerns the {\it possibility of approximation}. Is the given…
Having a function $f$ and a set of functionals $\{\mathcal{C}_{n}\}$, $c_n^f \equiv \mathcal{C}_n \left(f\right)$, one can interpret function approximation very generally as a construction of some function $\mathcal{A}_{N}^{f}$ such that…
Natural Language Processing has moved rather quickly from modelling specific tasks to taking more general pre-trained models and fine-tuning them for specific tasks, to a point where we now have what appear to be inherently generalist…
Similarity functions measure how comparable pairs of elements are, and play a key role in a wide variety of applications, e.g., notions of Individual Fairness abiding by the seminal paradigm of Dwork et al., as well as Clustering problems.…
In this paper, we generalize the basic notions and results of Dempster-Shafer theory from predicates to formal concepts. Results include the representation of conceptual belief functions as inner measures of suitable probability functions,…
We introduce the notion of an approximation system as a generalization of Taylor approximation, and we give some first examples. Next we develop the general theory, including error bounds and a sufficient criterion for convergence. More…
Rough set theory is a new mathematical approach to imperfect knowledge. The notion of rough sets is generalized by using an arbitrary binary relation on attribute values in information systems, instead of the trivial equality relation. The…
A conceptual foundation for approximation of belief functions is proposed and investigated. It is based on the requirements of consistency and closeness. An optimal approximation is studied. Unfortunately, the computation of the optimal…
We define a generalized likelihood function based on uncertainty measures and show that maximizing such a likelihood function for different measures induces different types of classifiers. In the probabilistic framework, we obtain…
We introduce a notion of integration defined from filters over families of finite sets. This procedure corresponds to determining the average value of functions whose range lies in any algebraic structure in which finite averages make…
Approximation of entire functions by their pad\'e approximants has been examined in the past. It is true that generically such an approximation holds. However, examining this problem from another viewpoint, we obtain stronger generic…
The computational complexity of reasoning within the Dempster-Shafer theory of evidence is one of the main points of criticism this formalism has to face. To overcome this difficulty various approximation algorithms have been suggested that…
We study approximations of theories both in general context and with respect to some natural classes of theories. Some kinds of approximations are considered, connections with finitely axiomatizable theories and minimal generating sets of…
We consider approximation or recovery of functions based on a finite number of function evaluations. This is a well-studied problem in optimal recovery, machine learning, and numerical analysis in general, but many fundamental insights were…
We propose a new sampling-based approach for approximate inference in filtering problems. Instead of approximating conditional distributions with a finite set of states, as done in particle filters, our approach approximates the…
Usually, density functional models are considered approximations to density functional theory, However, there is no systematic connection between the two, and this can make us doubt about a linkage. This attitude can be further enforced by…
Similarity metric which is not positive definite, and present a general theorem which provides a large family of similarity metrics which are positive definite.
Fixpoints are ubiquitous in computer science and when dealing with quantitative semantics and verification one often considers least fixpoints of (higher-dimensional) functions over the non-negative reals. We show how to approximate the…
We introduce generalized filtration with which we can represent situations such as some agents forget information at some specific time. The filtration is defined as a functor to a category Prob whose objects are all probability spaces and…
This short article contains the construction of a construction that generalizes the concept of the derivative of a function of one variable, using the theory of filters. The paper presents a new concept, demonstrates that it really…