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We analyse the axioms of Euclidean geometry according to standard object-oriented software development methodology. We find a perfect match: the main undefined concepts of the axioms translate to object classes. The result is a suite of C++…
Several problems in modeling and control of stochastically-driven dynamical systems can be cast as regularized semi-definite programs. We examine two such representative problems and show that they can be formulated in a similar manner. The…
We describe a method for approximating a single-variable function $f$ using persistence diagrams of sublevel sets of $f$ from height functions in different directions. We provide algorithms for the piecewise linear case and for the smooth…
This paper investigates connections between discrete and continuous approaches for decomposable submodular function minimization. We provide improved running time estimates for the state-of-the-art continuous algorithms for the problem…
A broad range of inverse problems can be abstracted into the problem of minimizing the sum of several convex functions in a Hilbert space. We propose a proximal decomposition algorithm for solving this problem with an arbitrary number of…
The aim of this study is to analyze the properties of harmonic fields in the vicinity of rough boundaries where either a constant potential or a zero flux is imposed, while a constant field is prescribed at an infinite distance from this…
Persistent homology is a popular tool in Topological Data Analysis. It provides numerical characteristics of data sets which reflect global geometric properties. In order to be useful in practice, for example for feature generation in…
For a continuous function $f$ defined on a closed and bounded domain, there is at least one maximum and one minimum. First, we introduce some preliminaries which are necessary through the paper. We then present an algorithm, which is…
Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…
We present a general technique for the analysis of first-order methods. The technique relies on the construction of a duality gap for an appropriate approximation of the objective function, where the function approximation improves as the…
We present robust algorithms for set operations and Euclidean transformations of curved shapes in the plane using approximate geometric primitives. We use a refinement algorithm to ensure consistency. Its computational complexity is…
We study the problem of detecting zeros of continuous functions that are known only up to an error bound, extending the earlier theoretical work with explicit algorithms and experiments with an implementation. More formally, the robustness…
We prove that every nonnegative continuous real-valued function on a given compact metric space is the uniform limit of some increasing sequence of nonnegative simple functions being linear combinations of indicators of open sets; here the…
We propose novel smooth approximations to the classical rounding function, suitable for differentiable optimization and machine learning applications. Our constructions are based on two approaches: (1) localized sigmoid window functions…
Topological data analysis is becoming increasingly relevant to support the analysis of unstructured data sets. A common assumption in data analysis is that the data set is a sample---not necessarily a uniform one---of some high-dimensional…
This survey addresses sampling discretization and its connections with other areas of mathematics. The survey concentrates on sampling discretization of norms of elements of finite-dimensional subspaces. We present here known results on…
We develop a method to compute $H^2$-conforming finite element approximations in both two and three space dimensions using readily available finite element spaces. This is accomplished by deriving a novel, equivalent mixed variational…
We consider the problem of sketching set valuation functions, defined as the expectation of a valuation function applied to independent random item values. For valuation functions that are monotone and either subadditive or submodular, and…
A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first…
We study variational problems for curves approximated by B-spline curves. We show that, one can obtain discrete Euler-Lagrange equations, for the data describing the approximated curves. Our main application is to the curve completion…