Source Themes

Bounding The Number of Linear Regions in Local Area for Neural Networks with ReLU Activations

The number of linear regions is one of the distinct properties of the neural networks using piecewise linear activation functions such as ReLU, comparing with those conventional ones using other activation functions. Previous studies showed this …

Tropical Optimal Transport and Wasserstein Distances in Phylogenetic Tree Space

We study the problem of optimal transport on phylogenetic tree space from the perspective of tropical geometry, and thus define the Wasserstein-p distances for probability measures in this continuous metric measure space setting. With respect to the …

Two-player incentive compatible outcome functions are affine maximizers

In mechanism design, for a given type space, there may be incentive compatible outcome functions which are not affine maximizers. We prove that for two-player games on a discrete type space, any given outcome function can be turned into an affine …

Tropical Fermat-Weber Points

In a metric space, the Fermat-Weber points of a sample are statistics to measure the central tendency of the sample and it is well known that the Fermat--Weber point of a sample is not necessarily unique in the metric space. We investigate the …

Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective

Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. As data objects, they are characterized by the challenges associated with "big data," as well as the complication that their discrete geometric …

Towards a Tropical Hodge Bundle

The moduli space $M_g^{trop}$ of tropical curves of genus $g$ is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus $g$. For each such graph $\Gamma$, the associated canonical linear system $\vert K_\Gamma\vert$ has …

Convexity in Tree Spaces

We study the geometry of metrics and convexity structures on the space of phylogenetic trees, which is here realized as the tropical linear space of all ultrametrics. The $CAT-0$ metric of Billera-Holmes-Vogtman arises from the theory of orthant …

Linear and Rational Factorization of Tropical Polynomials

Already for bivariate tropical polynomials, factorization is an NP-Complete problem. In this paper, we give an efficient algorithm for factorization and rational factorization of a rich class of tropical polynomials in $n$ variables. Special families …

Computing Linear Systems on Metric Graphs

The linear system of a divisor $D$ on a metric graph has the structure of a cell complex. We introduce the anchor divisors and anchor cells in it – they serve as the landmarks for us to compute the $f$-vector of the complex and find all cells in the …

Almost-toric Hypersurfaces

An almost-toric hypersurface is parameterized by monomials multiplied by polynomials in one extra variable. We determine the Newton polytope of such a hypersurface, and apply this to give an algorithm for computing the implicit polynomial equation.