Related papers: Representative families for matroid intersections,…
In this thesis, we develop various techniques for working with sets in machine learning. Each input or output is not an image or a sequence, but a set: an unordered collection of multiple objects, each object described by a feature vector.…
This paper develops matroidal analogues of classical results on matchings in abelian groups. By embedding matroid ground sets in an abelian group, we introduce base matchings between matroid bases, recover the group-theoretic setting in the…
We present applications of rectangular matrix models to various combinatorial problems, among which the enumeration of face-bicolored graphs with prescribed vertex degrees, and vertex-tricolored triangulations. We also mention possible…
We study the representability of sets that admit extended formulations using mixed-integer bilevel programs. We show that feasible regions modeled by continuous bilevel constraints (with no integer variables), complementarity constraints,…
Cover-free families are set systems used as solutions for a large variety of problems, and in particular, problems where we deal with $n$ elements and want to identify $d$ invalid ones among them by performing only $t$ tests ($t \leq n$).…
This text investigates relations between two well-known family of algorithms, matrix factorisations and recursive linear filters, by describing a probabilistic model in which approximate inference corresponds to a matrix factorisation…
Embedding models trained separately on similar data often produce representations that encode stable information but are not directly interchangeable. This lack of interoperability raises challenges in several practical applications, such…
The problem of selecting a small, yet high quality subset of patterns from a larger collection of itemsets has recently attracted lot of research. Here we discuss an approach to this problem using the notion of decomposable families of…
A new fast algebraic method for obtaining an $\mathcal{H}^2$-approximation of a matrix from its entries is presented. The main idea behind the method is based on the nested representation and the maximum-volume principle to select…
Motivated by work in graph theory, we define the fixing number for a matroid. We give upper and lower bounds for fixing numbers for a general matroid in terms of the size and maximum orbit size (under the action of the matroid automorphism…
We study point-line configurations through the lens of projective geometry and matroid theory. Our focus is on their realisation spaces, where we introduce the concepts of liftable and quasi-liftable configurations, exploring cases in which…
We study the combinatorial properties of a tropical hyperplane arrangement. We define tropical oriented matroids, and prove that they share many of the properties of ordinary oriented matroids. We show that a tropical oriented matroid…
In this era of large-scale data, distributed systems built on top of clusters of commodity hardware provide cheap and reliable storage and scalable processing of massive data. Here, we review recent work on developing and implementing…
In this paper, we present and analyze a new set of low-rank recovery algorithms for linear inverse problems within the class of hard thresholding methods. We provide strategies on how to set up these algorithms via basic ingredients for…
This paper proposes a method to compute camera 6Dof poses to achieve a user defined coverage. The camera placement problem is modeled as a combinatorial optimization where given the maximum number of cameras, a camera set is selected from a…
We study rank-three matroids, known as point-line configurations, and their associated matroid varieties, defined as the Zariski closures of their realization spaces. Our focus is on determining finite generating sets of defining equations…
We give an overview of combinatorial methods to represent 3D data, such as graphs and meshes, from the viewpoint of their amenability to analysis using machine learning algorithms. We highlight pros and cons of various representations and…
We construct a family of independent sets for finite, atomic, and graded lattices, extending the well-known cryptomorphism between geometric lattices and matroids. This construction leads to an embedding theorem into geometric lattices that…
Suppose that there is a ground set which consists of a large number of vectors in a Hilbert space. Consider the problem of selecting a subset of the ground set such that the projection of a vector of interest onto the subspace spanned by…
Given an orientable surface with boundary and a free homotopy class, we present a purely combinatorial algorithm which produces a representative of that homotopy class with minimal self intersection.