Linear and Rational Factorization of Tropical Polynomials

Lin, Bo; Tran, Ngoc Mai

doi:10.48550/arxiv.1707.03332

Cited by 6 publications

(9 citation statements)

References 12 publications

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“…Study the factorization of tropical polynomials in more than one variable. Work in this direction has been done in [37], who provide efficient algorithms for certain classes of polynomials, even though in general this is an NP-complete problem.…”

Section: Tropical Polynomials and Tropical Varietiesmentioning

confidence: 99%

Tropical Geometry

Morrison

2020

Foundations for Undergraduate Research in Mathematics

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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 with these new operations. These equations define piecewise-linear geometric objects called tropical varieties. We explore these tropical varieties in two and three dimensions, building up discrete tools for studying them and determining their geometric properties. We then discuss the relationship between tropical geometry and algebraic geometry, which considers shapes defined by usual polynomial equations.Suggested prerequisites. We use standard set theory notation (unions, functions, etc.) throughout this chapter. Section 1 draws on terminology and motivation from abstract algebra and linear algebra, but can be understood without them. Section 2 draws on topics from discrete geometry, although it is mostly self-contained. Section 3 includes geometry in three dimensions, which uses some notation from a standard course in multivariable calculus. Section 4 uses ring theory terminology from an abstract algebra course.

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Section: Tropical Polynomials and Tropical Varietiesmentioning

confidence: 99%

Tropical Geometry

Morrison

2020

Foundations for Undergraduate Research in Mathematics

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“…Balanced Coarsenings. Signed Minkowsi sums are intricate, for instance, they do not commute [Pos09,LT17]. Importantly, on the level of polyhedral fans, extended weight functions can be added and subtracted commutatively while respecting the non-negativity constraints imposed by Proposition 4.2.…”

Section: Minkowski Factorization Of Polytopesmentioning

confidence: 99%

“…While polytope factorization bases are not unique, their cardinality is, and so is any expansion with respect to a fixed basis. For a different result in this direction we emphasize [LT17,Cor. 23].…”

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confidence: 99%

The Tropical Division Problem and the Minkowski Factorization of Generalized Permutahedra

Crowell¹

2019

Preprint

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Given two tropical polynomials f, g on R n , we provide a characterization for the existence of a factorization f = h ⊙ g and the construction of h. As a ramification of this result we obtain a parallel result for the Minkowski factorization of polytopes. Using our construction we show that for any given polytopal fan there is a polytope factorization basis, i.e. a finite set of polytopes with respect to which any polytope whose normal fan is refined by the original fan can be uniquely written as a signed Minkowski sum. We explicitly study the factorization of polymatroids and their generalizations, Coxeter matroid polytopes, and give a hyperplane description of the cone of deformations for this class of polytopes.

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“…Besides, in some cases, it is conceivable that an additional neuron may lead to no change of the model's expressivity at all. On the other hand, counting the exact number of linear regions was shown to be very hard ( [12,16]). As a result, many recent studies focus on the methods for estimating the tight upper bounds of numbers of linear regions in the entire input space, as reviewed in a recent article ([16]).…”

Section: Introductionmentioning

confidence: 99%

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

Zhu¹,

Lin²,

Tang³

2020

Preprint

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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 property reflected the expressivity of a neural network family ([14]); as a result, it can be used to characterize how the structural complexity of a neural network model affects the function it aims to compute. Nonetheless, it is challenging to directly compute the number of linear regions; therefore, many researchers focus on estimating the bounds (in particular the upper bound) of the number of linear regions for deep neural networks using ReLU. These methods, however, attempted to estimate the upper bound in the entire input space. The theoretical methods are still lacking to estimate the number of linear regions within a specific area of the input space, e.g., a sphere centered at a training data point such as an adversarial example or a backdoor trigger. In this paper, we present the first method to estimate the upper bound of the number of linear regions in any sphere in the input space of a given ReLU neural network. We implemented the method, and computed the bounds in deep neural networks using the piece-wise linear active function. Our experiments showed that, while training a neural network, the boundaries of the linear regions tend to move away from the training data points. In addition, we observe that the spheres centered at the training data points tend to contain more linear regions than any arbitrary points in the input space. To the best of our knowledge, this is the first study of bounding linear regions around a specific data point. We consider our research, with both theoretical proof and software implementation, as a first step toward the investigation of the structural complexity of deep neural networks in a specific input area.

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Linear and Rational Factorization of Tropical Polynomials

Cited by 6 publications

References 12 publications

Tropical Geometry

Tropical Geometry

The Tropical Division Problem and the Minkowski Factorization of Generalized Permutahedra

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

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