Related papers: Covering vectors by spaces: Regular matroids
We present a spatially efficient decomposition of matrices and arbitrary-order tensors as linear combinations of tensor products of $\{-1, 1\}$-valued vectors. For any matrix $A \in \mathbb{R}^{m \times n}$, $$A - R_w = S_w C_w T_w^\top =…
In the low-rank matrix completion (LRMC) problem, the low-rank assumption means that the columns (or rows) of the matrix to be completed are points on a low-dimensional linear algebraic variety. This paper extends this thinking to cases…
In this work we present a framework for studying the eigenvalues of a family of matrices with a particular displacement structure. The family admits a specific decomposition as the product of an upper and a lower triangular matrices having…
Subexponential parameterized algorithms are known for a wide range of natural problems on planar graphs, but the techniques are usually highly problem specific. The goal of this paper is to introduce a framework for obtaining…
We study the general norm optimization for combinatorial problems, initiated by Chakrabarty and Swamy (STOC 2019). We propose a general formulation that captures a large class of combinatorial structures: we are given a set $U$ of $n$…
We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct…
For a finite collection of graphs ${\cal F}$, the \textsc{${\cal F}$-TM-Deletion} problem has as input an $n$-vertex graph $G$ and an integer $k$ and asks whether there exists a set $S \subseteq V(G)$ with $|S| \leq k$ such that $G…
In this paper, we investigate the problem of decomposing 2D environments for robot coverage planning. Coverage path planning (CPP) involves computing a cost-minimizing path for a robot equipped with a coverage or sensing tool so that the…
We prove the decomposition of arbitrary diagonal operators into tensor and matrix products of smaller matrices, focusing on the analytic structure of the resulting formulas and their inherent symmetries. Diagrammatic representations are…
In this paper, we propose a new type of matroids, namely covering matroids, and investigate the connections with the second type of covering-based rough sets and some existing special matroids. Firstly, as an extension of partitions,…
Matrix factorization is an inference problem that has acquired importance due to its vast range of applications that go from dictionary learning to recommendation systems and machine learning with deep networks. The study of its fundamental…
In the classic Minimum Bisection problem we are given as input a graph $G$ and an integer $k$. The task is to determine whether there is a partition of $V(G)$ into two parts $A$ and $B$ such that $||A|-|B|| \leq 1$ and there are at most $k$…
As an extension of a classical tree-partition problem, we consider decompositions of graphs into edge-disjoint (rooted-)trees with an additional matroid constraint. Specifically, suppose we are given a graph $G=(V,E)$, a multiset…
We consider the problem of low-rank decomposition of incomplete multiway tensors. Since many real-world data lie on an intrinsically low dimensional subspace, tensor low-rank decomposition with missing entries has applications in many data…
Covering problems are classical computational problems concerning whether a certain combinatorial structure 'covers' another. For example, the minimum vertex covering problem aims to find the smallest set of vertices in a graph so that each…
We introduce a new data structure for answering connectivity queries in undirected graphs subject to batched vertex failures. Precisely, given any graph G and integer k, we can in fixed-parameter time construct a data structure that can…
In this work we present an algorithm to construct sparse-paving matroids over finite set $S$. From this algorithm we derive some useful bounds on the cardinality of the set of circuits of any Sparse-Paving matroids which allow us to prove…
Given a simple Eulerian binary matroid $M$, what is the minimum number of disjoint circuits necessary to decompose $M$? We prove that $|M| / (\operatorname{rank}(M) + 1)$ many circuits suffice if $M = \mathbb F_2^n \setminus \{0\}$ is the…
We present a geometric approach towards derandomizing the Isolation Lemma by Mulmuley, Vazirani, and Vazirani. In particular, our approach produces a quasi-polynomial family of weights, where each weight is an integer and quasi-polynomially…
Motivated by the Matrix Spencer conjecture, we study the problem of finding signed sums of matrices with a small matrix norm. A well-known strategy to obtain these signs is to prove, given matrices $A_1, \dots, A_n \in \mathbb{R}^{m \times…