Related papers: Combinatorial and inductive methods for the tropic…
This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its…
In this paper, we consider the Perron theorem over the real Puiseux field. We introduce a recursive method for calculating Perron roots and Perron vectors of positive Puiseux matrices (which satisfy some condition of genericness) by means…
Building on recent work of the authors, we use degenerations to chains of elliptic curves to prove two cases of the Aprodu-Farkas strong maximal rank conjecture, in genus $22$ and $23$. This constitutes a major step forward in Farkas'…
The orthogonal decomposition factorizes a tensor into a sum of an orthogonal list of rankone tensors. We present several properties of orthogonal rank. We find that a subtensor may have a larger orthogonal rank than the whole tensor and…
Tropical polyhedra have been recently used to represent disjunctive invariants in static analysis. To handle larger instances, tropical analogues of classical linear programming results need to be developed. This motivation leads us to…
The major challenge in designing a discriminative learning algorithm for predicting structured data is to address the computational issues arising from the exponential size of the output space. Existing algorithms make different assumptions…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
The tensor rank decomposition, also known as canonical polyadic(CP) or simply tensor decomposition, has a long history in multilinear algebra. However, computing a rank decomposition becomes particularly challenging when the rank lies…
We give a simple combinatorial proof of the $\lambda_g$ conjectue in genus 2. We use a description of the class $\lambda_2$ as a linear combination of boundary strata, and show the conjecture follows inductively from applications of the…
In this work we introduce and study various generalizations of the notion of increasingly labelled trees, where the label of a child node is always larger than the label of its parent node, to multilabelled tree families, where the nodes in…
We give a complete description of Green's D relation for the multiplicative semigroup of all n-by-n tropical matrices. Our main tool is a new variant on the duality between the row and column space of a tropical matrix (studied by Cohen,…
A multidimensional optimization problem is formulated in the tropical mathematics setting as to maximize a nonlinear objective function, which is defined through a multiplicative conjugate transposition operator on vectors in a…
We consider systems of recursively defined combinatorial structures. We give algorithms checking that these systems are well founded, computing generating series and providing numerical values. Our framework is an articulation of the…
The tropical semifield, i.e., the real numbers enhanced by the operations of addition and maximum, serves as a base of tropical mathematics. Addition is an abelian group operation, whereas the maximum defines an idempotent semigroup…
We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related…
We present a novel preconditioning technique for proximal optimization methods that relies on graph algorithms to construct effective preconditioners. Such combinatorial preconditioners arise from partitioning the graph into forests. We…
We generalize methods to compute various kinds of rank to the case of a toric variety $X$ embedded into projective space using a very ample line bundle $\mathcal{L}$. We find an upper bound on the cactus rank. We use this to compute rank,…
This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then…
We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in…
Ryser's max term rank formula with graph theoretic terminology is equivalent to a characterization of degree sequences of simple bipartite graphs with matching number at least $\ell$. In a previous paper by the authors, a generalization was…