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There is an increasing need for algorithms that can accurately detect changepoints in long time-series, or equivalent, data. Many common approaches to detecting changepoints, for example based on penalised likelihood or minimum description…
In many applications, we need algorithms which can align partially overlapping point sets and are invariant to the corresponding transformations. In this work, a method possessing such properties is realized by minimizing the objective of…
We study the theoretical and practical aspects of computing braids described by approximate descriptions of paths in the plane. Exact algorithms rely on the lexicographic ordering of the points in the plane, which is unstable under…
New approaches to the theory of dynamic programming view dynamic programs as families of policy operators acting on partially ordered sets. In this paper, we extend these ideas by shifting from arbitrary partially ordered sets to ordered…
Let $P=(P_1, P_2, \ldots, P_n)$, $P_i \in \field{R}$ for all $i$, be a signal and let $C$ be a constant. In this work our goal is to find a function $F:[n]\rightarrow \field{R}$ which optimizes the following objective function: $$ \min_{F}…
We present a procedure to approximate a plane contour by piecewise polynomial functions, depending on various parameters, such as degree, number of local patches, selection of knots. This procedure aims to be adopted to study how…
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take…
Given values of a piecewise smooth function $f$ on a square grid within a domain $\Omega$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of…
We study the incremental knapsack problem, where one wishes to sequentially pack items into a knapsack whose capacity expands over a finite planning horizon, with the objective of maximizing time-averaged profits. While various…
The subdivision algorithm by Dellnitz and Hohmann for the computation of invariant sets of dynamical systems decomposes the relevant region of the state space into boxes and analyzes the induced box dynamics. Its convergence is proved in an…
The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time…
Our main interest in this paper is to study some approximation problems for classes of functions with mixed smoothness. We use technique, based on a combination of results from hyperbolic cross approximation, which were obtained in 1980s --…
Approximate linear programming (ALP) is an efficient approach to solving large factored Markov decision processes (MDPs). The main idea of the method is to approximate the optimal value function by a set of basis functions and optimize…
One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in…
We introduce a new approach aiming at computing approximate optimal designs for multivariate polynomial regressions on compact (semi-algebraic) design spaces. We use the moment-sum-of-squares hierarchy of semidefinite programming problems…
Neuro-dynamic programming is a class of powerful techniques for approximating the solution to dynamic programming equations. In their most computationally attractive formulations, these techniques provide the approximate solution only…
We consider the rate of piecewise constant approximation to a locally stationary process $X(t),t\in [0,1]$, having a variable smoothness index $\alpha(t)$. Assuming that $\alpha(\cdot)$ attains its unique minimum at zero and satisfies the…
A sequential piecewise linear programming method is presented where bounded domains of non-convex functions are successively contracted about the solution of a piecewise linear program at each iteration of the algorithm. Although…
Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By…
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…