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The smoothed analysis of algorithms is concerned with the expected running time of an algorithm under slight random perturbations of arbitrary inputs. Spielman and Teng proved that the shadow-vertex simplex method has polynomial smoothed…
There are (at least) two reasons to study random polytopes. The first is to understand the combinatorics and geometry of random polytopes especially as compared to other classes of polytopes, and the second is to analyze average-case…
In this paper we provide some exact formulas for projective dimension and the regularity of powers of edge ideals of vertex-weighted rooted forests. These formulas are functions of the weight of the vertices and the number of edges. We also…
We consider the bipartite boolean quadric polytope (BQP) with multiple-choice constraints and analyse its combinatorial properties. The well-studied BQP is defined as the convex hull of all quadric incidence vectors over a bipartite graph.…
Archetypal analysis is an unsupervised learning method that uses a convex polytope to summarize multivariate data. For fixed $k$, the method finds a convex polytope with $k$ vertices, called archetype points, such that the polytope is…
We present a polyhedral algorithm to manipulate positive dimensional solution sets. Using facet normals to Newton polytopes as pretropisms, we focus on the first two terms of a Puiseux series expansion. The leading powers of the series are…
Every regular polytope has the remarkable property that it inherits all symmetries of each of its facets. This property distinguishes a natural class of polytopes which are called hereditary. Regular polytopes are by definition hereditary,…
The main purpose of this paper is twofold. We first want to analyze in details the meaningful geometric aspect of the method introduced in the previous paper [12], concerning regularity of families of irreducible, nodal "curves" on a…
We investigate symmetric edge polytopes generated by Erd\H{o}s--R\'enyi random graphs in a high-dimensional regime. These objects provide a natural and largely unexplored model of random lattice polytopes, in which geometric properties are…
The story of positive geometry of massless scalar theories was pioneered in [1] in the context of bi-adjoint $\phi^3$ theories. Further study proposed that the positive geometry for a generic massless scalar theory with polynomial…
Shapley values are a standard tool for explaining predictions of tree ensembles, with Path-Dependent SHAP being the most widely used variant. Despite substantial progress, existing methods still exhibit trade-offs between depth-dependent…
Recursive max-linear vectors provide models for causal dependence between large values of random variables that are supported on directed acyclic graphs, but the standard assumption that all nodes of such a graph are observed can be…
Our study focuses on determining the best weight windows for a weighted moving average smoother under squared loss. We show that there exists an optimal weight window that is symmetrical around its center. We study the class of tapered…
Decision trees are well-known due to their ease of interpretability. To improve accuracy, we need to grow deep trees or ensembles of trees. These are hard to interpret, offsetting their original benefits. Shapley values have recently become…
Deep learning has received much attention lately due to the impressive empirical performance achieved by training algorithms. Consequently, a need for a better theoretical understanding of these problems has become more evident in recent…
We develop tools for characterizing vertices of fiber products of polytopes and apply them to simplicial distribution polytopes, a class of probability polytopes arising in quantum foundations and quantum information. In the theory of…
Uniform bounds on sketched inner products of vectors or matrices underpin several important computational and statistical results in machine learning and randomized algorithms, including the Johnson-Lindenstrauss (J-L) lemma, the Restricted…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals of vertex-weighted oriented unicyclic graphs. These formulas are in function of the weight of the vertices, the numbers of edges. We…
The modelling of interconnection networks by graphs motivated the study of several extremal problems that involve well known parameters of a graph (degree, diameter, girth and order) and ask for the optimal value of one of them while…