Related papers: Rigorous cubical approximation and persistent homo…
Computing the rate-distortion function for continuous sources is commonly regarded as a standard continuous optimization problem. When numerically addressing this problem, a typical approach involves discretizing the source space and…
The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of…
We consider the space of real-valued continuously differentiable functions on a compact subset of a euclidean space. We characterize the completeness of this space and prove that the space of restrictions of continuously differentiable…
We provide a general framework to construct finite dimensional approximations of the space of convex functions, which also applies to the space of c-convex functions and to the space of support functions of convex bodies. We give estimates…
Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…
We show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry. Using appropriate regularizations of the mass and of the first variation of a varifold we…
Any constructive continuous function must have a gradually varied approximation in compact space. However, the refinement of domain for $\sigma-$-net might be very small. Keeping the original discretization (square or triangulation), can we…
We present a fast algorithm for computing discrete cubical homology of graphs over finite fields with an appropriate characteristic. This algorithm improves on several computational steps compared to constructions in the existing…
Persistent homology is a method for computing the topological features present in a given data. Recently, there has been much interest in the integration of persistent homology as a computational step in neural networks or deep learning. In…
We consider several coding discretizations of continuous functions which reflect their variation at some given precision. We study certain statistical and combinatorial properties of the sequence of finite words obtained by coding a typical…
Continuous models used in physics and other areas of mathematics applications become discrete when they are computerized, e.g., utilized for computations. Besides, computers are controlling processes in discrete spaces, such as films and…
We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise…
Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently…
We solve the problem of minimizing the number of critical points among all functions on a surface within a prescribed distance {\delta} from a given input function. The result is achieved by establishing a connection between discrete Morse…
We consider the approximation of manifold-valued functions by embedding the manifold into a higher dimensional space, applying a vector-valued approximation operator and projecting the resulting vector back to the manifold. It is well known…
We study the problem of computing the value function from a discretely-observed trajectory of a continuous-time diffusion process. We develop a new class of algorithms based on easily implementable numerical schemes that are compatible with…
Regular convergence, together with various other types of convergence, has been studied since the 1970s for the discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of compact operators whose…
We comment on recent results in the field of information based complexity, which state (in a number of different settings), that approximation of infinitely differentiable functions is intractable and suffers from the curse of…
In this paper we present a new approach to computing homology (with field coefficients) and persistent homology. We use concepts from discrete Morse theory, to provide an algorithm which can be expressed solely in terms of simple graph…
We consider the problem of discretizing one-dimensional, real-valued functions as graphs. The goal is to find a small set of points, from which we can approximate the remaining function values. The method for approximating the unknown…