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We present an algorithm for approximately solving bounded convex vector optimization problems. The algorithm provides both an outer and an inner polyhedral approximation of the upper image. It is a modification of the primal algorithm…
Polynomial approximations of functions are widely used in scientific computing. In certain applications, it is often desired to require the polynomial approximation to be non-negative (resp. non-positive), or bounded within a given range,…
We consider offsets of a union of convex objects. We aim for a filtration, a sequence of nested cell complexes, that captures the topological evolution of the offsets for increasing radii. We describe methods to compute a filtration based…
We study the numerical integration problem for functions with infinitely many variables. The function spaces of integrands we consider are weighted reproducing kernel Hilbert spaces with norms related to the ANOVA decomposition of the…
Deep neural networks (DNNs) could be deceived by generating human-imperceptible perturbations of clean samples. Therefore, enhancing the robustness of DNNs against adversarial attacks is a crucial task. In this paper, we aim to train robust…
Many resource allocation problems in the cloud can be described as a basic Virtual Network Embedding Problem (VNEP): finding mappings of request graphs (describing the workloads) onto a substrate graph (describing the physical…
In this paper we consider the problem of constructing numerical algorithms for approximating of convex compact bodies in d-dimensional Euclidean space by polytopes with any given accuracy. It is well known that optimal with respect to the…
This study presents a novel algorithm for identifying the set of extreme points that constitute the exact convex hull of a point set in high-dimensional Euclidean space. The proposed method iteratively solves a sequence of dynamically…
Modeling of microlensing events poses computational challenges for the resolution of the lens equation and the high dimensionality of the parameter space. In particular, numerical noise represents a severe limitation to fast and efficient…
We focus on two central themes in this dissertation. The first one is on decomposing polytopes and polynomials in ways that allow us to perform nonlinear optimization. We start off by explaining important results on decomposing a polytope…
Let $\mathcal{T}$ be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles,…
In this paper, we present a deterministic algorithm for the closest vector problem for all l_p-norms, 1 < p < \infty, and all polyhedral norms, especially for the l_1-norm and the l_{\infty}-norm. We achieve our results by introducing a new…
We present polynomial-augmented neural networks (PANNs), a novel machine learning architecture that combines deep neural networks (DNNs) with a polynomial approximant. PANNs combine the strengths of DNNs (flexibility and efficiency in…
The polytope containment problem is deciding whether a polytope is a contained within another polytope. This problem is rooted in computational convexity, and arises in applications such as verification and control of dynamical systems. The…
Approximation problems involving a single convex body in $d$-dimensional space have received a great deal of attention in the computational geometry community. In contrast, works involving multiple convex bodies are generally limited to…
Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game…
We present a polynomial time algorithm to approximately scale tensors of any format to arbitrary prescribed marginals (whenever possible). This unifies and generalizes a sequence of past works on matrix, operator and tensor scaling. Our…
Dense subgraph extraction is a fundamental problem in graph analysis and data mining, aimed at identifying cohesive and densely connected substructures within a given graph. It plays a crucial role in various domains, including social…
First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general…
This paper considers the question of how to succinctly approximate a multidimensional convex body by a polytope. Given a convex body $K$ of unit diameter in Euclidean $d$-dimensional space (where $d$ is a constant) and an error parameter…