Related papers: On the Petras algorithm for verified integration o…
We present an implementation of arbitrary-precision numerical integration with rigorous error bounds in the Arb library. Rapid convergence is ensured for piecewise complex analytic integrals by use of the Petras algorithm, which combines…
The complexity of continuous piecewise affine (CPA) functions can be measured by the number of pieces $p$ or the number of distinct affine functions $n$. For CPA functions on $\mathbb{R}^d$, this paper shows an upper bound of $p=O(n^{d+1})$…
In this paper, we extend an available neural network verification technique to support a wider class of piece-wise linear activation functions. Furthermore, we extend the algorithms, which provide in their original form exact respectively…
A complex-analytic structure within the unit disk of the complex plane is presented. It can be used to represent and analyze a large class of real functions. It is shown that any integrable real function can be obtained by means of the…
In a previous paper, an implementable algorithm was introduced to compute discrete solutions of sweeping processes (i.e. specific first order differential inclusions). The convergence of this numerical scheme was proved thanks to…
We refine the bit complexity analysis of an algorithm for the computation of at least one point per connected component of a smooth real algebraic set, yielding exponential speedup (with respect to the number of variables) compared to prior…
The paper provides results regarding the computational complexity of hybrid system identification. More precisely, we focus on the estimation of piecewise affine (PWA) maps from input-output data and analyze the complexity of computing a…
We present an algorithm which uses Fujiwara's inequality to bound algebraic functions over ellipses of a certain type, allowing us to concretely implement a rigorous Gauss-Legendre integration method for algebraic functions over a line…
In process mining, alignments quantify the degree of deviation between an observed event trace and a business process model and constitute the most important conformance checking technique. We study the algorithmic complexity of computing…
In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision…
In probabilistic program analysis, quantitative analysis aims at deriving tight numerical bounds for probabilistic properties such as expectation and assertion probability. Most previous works consider numerical bounds over the whole…
The study of some parametric integrals is presented with a combined approach of analytical development, the usage of a Computed Algebra System (CAS) and of the Online Encyclopedia of Integer Sequences. The methodology for the solution…
In many high-impact applications, it is important to ensure the quality of output of a machine learning algorithm as well as its reliability in comparison with the complexity of the algorithm used. In this paper, we have initiated a…
The efficient evaluation of high-dimensional integrals is of importance in both theoretical and practical fields of science, such as data science, statistical physics, and machine learning. However, exact computation methods suffer from the…
An analytical-numeric calculation method of extremely complicated integrals is presented. These integrals appear often in magnet soliton theory. The appropriate analytical continuation and a corresponding integration contour allow to reduce…
We show that the decision problem for the basic system of interpretability logic IL is PSPACE-complete. For this purpose we present an algorithm which uses polynomial space with respect to the complexity of a given formula. The existence of…
We investigate the problem of fitting piecewise affine functions (PWA) to data. Our algorithm divides the input domain into finitely many polyhedral regions whose shapes are specified using a user-defined template such that the data points…
A growing trend in the database and system communities is to augment conventional index structures, such as B+-trees, with machine learning (ML) models. Among these, error-bounded Piecewise Linear Approximation ($\epsilon$-PLA) has emerged…
We distinguish two kinds of piecewise linear functions and provide an interesting representation for a piecewise linear function between two normed spaces. Based on such a representation, we study a fully piecewise linear vector…
In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise…