熱帶曲線(tropical curves) 是定義在熱帶半環(tropical semiring) 上的代數曲線。熱帶曲線是古典代數曲線經由某些賦值(valuation) 的映像，所以許多重要的代數曲線性質也同樣發生在熱帶曲線上。本篇論文我們試著將熱帶曲線圖形化。 首先，我們根據熱帶曲線的理論發展出幾個繪出熱帶曲線的演算法。再者，我們以電腦程式語言Python 去實現這些算演算法。我們發展的是跨平台的程式碼，可以在Linux, Mac OS X, Windows 等作業系統執行。 / Tropical curves are algebraic curves dened over the tropical semiring. They are the images of classical algebraic curves under some valuation maps, so reect many important properties of classical algebraic curves. In this thesis, we try to visualize tropical curves. We study the theory of tropical curves and develop several algorithms to draw the graphs of tropical curves. Furthermore, we implement these algorithms in Python programming language. These codes are cross-platform, running on Linux, Mac OS X, and Windows.

The cohomology of moduli spaces of curves has been extensively studied in classical algebraic geometry. The emergent field of tropical geometry gives new views and combinatorial tools for treating these classical problems. In particular, we study the cohomology of heavy/light Hassett spaces, moduli spaces of heavy/light weighted stable curves, denoted as $\calm_{g, w}$ for a particular genus $g$ and a weight vector $w \in (0, 1]^n$ using tropical geometry. We survey and build on the work of \citet{Cavalieri2014}, which proved that tropical compactification is a \textit{wonderful} compactification of the complement of hyperplane arrangement for these heavy/light Hassett spaces. For $g = 0$, we want to find the tropicalization of $\calm_{0, w}$, a polyhedral complex parametrizing leaf-labeled metric trees that can be thought of as Bergman fan, which furthermore creates a toric variety $X_{\Sigma}$. We use the presentation of $\overline{\calm}_{0,w}$ as a tropical compactification associated to an explicit Bergman fan, to give a concrete presentation of the cohomology.

A phylogenetic tree is a tree to represent an evolutionary history between species or other entities. Phylogenomics is a new field intersecting phylogenetics and genomics and it is well-known that we need statistical learning methods to handle and analyze a large amount of data which can be generated relatively cheaply with new technologies. Based on the existing Markov models, we introduce a new method, CURatio, to identify outliers in a given gene data set. This method, intrinsically an unsupervised method, can find outliers from thousands or even more genes. This ability to analyze large amounts of genes (even with missing information) makes it unique in many parametric methods. At the same time, the exploration of statistical analysis in high-dimensional space of phylogenetic trees has never stopped, many tree metrics are proposed to statistical methodology. Tropical metric is one of them. We implement a MCMC sampling method to estimate the principal components in a tree space with the tropical metric for achieving dimension reduction and visualizing the result in a 2-D tropical triangle.

If R is a list of positive integers with greatest common denominator equal to 1, calculating the Frobenius number of R is in general NP-hard. Dino Lorenzini defines the arithmetical graph, which naturally arises in arithmetic geometry, and a notion of genus, the g-number, that in specific cases coincides with the Frobenius number of R. A result of Dino Lorenzini's gives a method for quickly calculating upper bounds for the g-number of arithmetical graphs. We discuss the arithmetic geometry related to arithmetical graphs and present an example of an arithmetical graph that arises in this context. We also discuss the construction for Lorenzini's Riemann-Roch structure and how it relates to the Riemann-Roch theorem for finite graphs shown by Matthew Baker and Serguei Norine. We then focus on the connection between the Frobenius number and arithmetical graphs. Using the Laplacian of an arithmetical graph and a formulation of chip-firing on the vertices of an arithmetical graph, we show results that can be used to find arithmetical graphs whose g-numbers correspond to the Frobenius number of R. We describe how this can be used to quickly calculate upper bounds for the Frobenius number of R.

Master of Science / Department of Mathematics / Ilia Zharkov / When considering an unramified double cover π: C’→ C of nonsingular algebraic curves, the Prym variety (P; θ) of the cover arises from the sheet exchange involution of C’ via extension to the Jacobian J(C’). The Prym is defined to be the anti-invariant (odd) part of this induced map on J(C’), and it carries twice a principal polarization of J(C’). The pair (P; θ), where θ is a representative of a theta divisor of J(C’) on P, makes the Prym a candidate for the Jacobian of another curve. In 1974, David Mumford proved that for an unramified double cover π : C’η →C of a plane quintic curve, where η is a point of order two in J(C), then the Prym (P; θ) is not a Jacobian if the theta characteristic L(η) is odd, L the hyperplane section. We sought to find an analog of Mumford's result in the tropical geometry setting. We consider the Prym variety of certain unramified double covers of three types of tropical plane quintics. Applying the theory of lattice dicings, which give affine invariants of the Prym lattice, we found that when the parity α(H3) is even, H3 the cycle associated to the hyperplane section and the analog to η in the classical setting, then the Prym is not a Jacobian, and is a Jacobian when the parity is odd.

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This thesis tackles the problem of understanding deep neural network with piece- wise linear activation functions. We leverage tropical geometry, a relatively new field in algebraic geometry to characterize the decision boundaries of a single hidden layer neural network. This characterization is leveraged to understand, and reformulate three interesting applications related to deep neural network. First, we give a geo- metrical demonstration of the behaviour of the lottery ticket hypothesis. Moreover, we deploy the geometrical characterization of the decision boundaries to reformulate the network pruning problem. This new formulation aims to prune network pa- rameters that are not contributing to the geometrical representation of the decision boundaries. In addition, we propose a dual view of adversarial attack that tackles both designing perturbations to the input image, and the equivalent perturbation to the decision boundaries.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22976号 / 理博第4653号 / 新制||理||1669(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)准教授 伊藤 哲史, 教授 入谷 寛, 教授 池田 保 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

This dissertation aims at theoretically studying a block of layers that is common in al- most all deep learning models. The block of layers of interest is the composition of an affine layer followed by a nonlinear activation that is followed by another affine layer. We study this block from three perspectives. (i) An Optimization Perspective. Is it possible that the output of the forward pass through this block is an optimal solution to a certain convex optimization problem? We show an equivalency between the forward pass through this block and a single iteration of deterministic and stochastic algorithms solving a ten- sor formulated convex optimization problem. As consequence, we derive for the first time a formula for computing the singular values of convolutional layers surpassing the need for the prohibitive construction of the underlying linear operator. Thereafter, we show that several deep networks can have this block replaced with the corresponding optimiza- tion algorithm predicted by our theory resulting in networks with improved generalization performance. (ii) A Probabilistic Perspective. Is it possible to analytically analyze the output of a deep network upon subjecting the input to Gaussian noise? To that regard, we derive analytical formulas for the first and second moments of this block under Gaussian input noise. We demonstrate that the derived expressions can be used to efficiently analyze the output of an arbitrary deep network in addition to constructing Gaussian adversarial attacks surpassing any need for prohibitive data augmentation procedures. (iii) A Tropi- cal Geometry Perspective. Is it possible to characterize the decision boundaries of this block as a geometric structure representing a solution set to a certain class of polynomials (tropical polynomials)? If so, then, is it possible to utilize this geometric representation of the decision boundaries for novel reformulations to classical computer vision and machine learning tasks on arbitrary deep networks? We show that the decision boundaries of this block are a subset of a tropical hypersurface, which is intimately related to a the polytope that is the convex hull of two zonotopes. We utilize this geometric characterization to shed lights on new perspectives of network pruning.

The field of algebraic tropical geometry establishes a deep connection between algebraic geometry and combinatorics by associating to certain classical algebraic varieties so called tropical varieties, which are polyhedral complexes in some real vectorspaces. Tropical varieties are closely related to the Groebner complexes of the ideal defining the classical variety. In this thesis the tropical variety of an ideal is studied under a generic change of coodinates. Analogously to the existence of generic initial ideals the existence of generic Groebner complexes and generic tropical varieties is proved. Moreover, it is shown that in the constant coefficient case information on the invariants dimension, Hilbert-Samuel multiplicity and depth of the corresponding coordinate rings can be obtained from generic tropical varieties.