On Boundaries of Statistical Models

In the thesis "On Boundaries of Statistical Models" problems related to a description of probability distributions with zeros, lying in the boundary of a statistical model, are treated. The distributions considered are joint distributions of finite collections of finite discrete random variables. Owing to this restriction, statistical models are subsets of finite dimensional real vector spaces. The support set problem for exponential families, the main class of models considered in the thesis, is to characterize the possible supports of distributions in the boundaries of these statistical models. It is shown that this problem is equivalent to a characterization of the face lattice of a convex polytope, called the convex support. The main tool for treating questions related to the boundary are implicit representations. Exponential families are shown to be sets of solutions of binomial equations, connected to an underlying combinatorial structure, called oriented matroid. Under an additional assumption these equations are polynomial and one is placed in the setting of commutative algebra and algebraic geometry. In this case one recovers results from algebraic statistics. The combinatorial theory of exponential families using oriented matroids makes the established connection between an exponential family and its convex support completely natural: Both are derived from the same oriented matroid. The second part of the thesis deals with hierarchical models, which are a special class of exponential families constructed from simplicial complexes. The main technical tool for their treatment in this thesis are so called elementary circuits. After their introduction, they are used to derive properties of the implicit representations of hierarchical models. Each elementary circuit gives an equation holding on the hierarchical model, and these equations are shown to be the "simplest", in the sense that the smallest degree among the equations corresponding to elementary circuits gives a lower bound on the degree of all equations characterizing the model. Translating this result back to polyhedral geometry yields a neighborliness property of marginal polytopes, the convex supports of hierarchical models. Elementary circuits of small support are related to independence statements holding between the random variables whose joint distributions the hierarchical model describes. Models for which the complete set of circuits consists of elementary circuits are shown to be described by totally unimodular matrices. The thesis also contains an analysis of the case of binary random variables. In this special situation, marginal polytopes can be represented as the convex hulls of linear codes. Among the results here is a classification of full-dimensional linear code polytopes in terms of their subgroups. If represented by polynomial equations, exponential families are the varieties of binomial prime ideals. The third part of the thesis describes tools to treat models defined by not necessarily prime binomial ideals. It follows from Eisenbud and Sturmfels'' results on binomial ideals that these models are unions of exponential families, and apart from solving the support set problem for each of these, one is faced with finding the decomposition. The thesis discusses algorithms for specialized treatment of binomial ideals, exploiting their combinatorial nature. The provided software package Binomials.m2 is shown to be able to compute very large primary decompositions, yielding a counterexample to a recent conjecture in algebraic statistics

Algebraic Systems Biology: A Case Study for the Wnt Pathway

Steady state analysis of dynamical systems for biological networks give rise to algebraic varieties in high-dimensional spaces whose study is of interest in their own right. We demonstrate this for the shuttle model of the Wnt signaling pathway. Here the variety is described by a polynomial system in 19 unknowns and 36 parameters. Current methods from computational algebraic geometry and combinatorics are applied to analyze this model.Comment: 24 pages, 2 figure

Tamper-Resistant Arithmetic for Public-Key Cryptography

Cryptographic hardware has found many uses in many ubiquitous and pervasive security devices with a small form factor, e.g. SIM cards, smart cards, electronic security tokens, and soon even RFIDs. With applications in banking, telecommunication, healthcare, e-commerce and entertainment, these devices use cryptography to provide security services like authentication, identification and confidentiality to the user. However, the widespread adoption of these devices into the mass market, and the lack of a physical security perimeter have increased the risk of theft, reverse engineering, and cloning. Despite the use of strong cryptographic algorithms, these devices often succumb to powerful side-channel attacks. These attacks provide a motivated third party with access to the inner workings of the device and therefore the opportunity to circumvent the protection of the cryptographic envelope. Apart from passive side-channel analysis, which has been the subject of intense research for over a decade, active tampering attacks like fault analysis have recently gained increased attention from the academic and industrial research community. In this dissertation we address the question of how to protect cryptographic devices against this kind of attacks. More specifically, we focus our attention on public key algorithms like elliptic curve cryptography and their underlying arithmetic structure. In our research we address challenges such as the cost of implementation, the level of protection, and the error model in an adversarial situation. The approaches that we investigated all apply concepts from coding theory, in particular the theory of cyclic codes. This seems intuitive, since both public key cryptography and cyclic codes share finite field arithmetic as a common foundation. The major contributions of our research are (a) a generalization of cyclic codes that allow embedding of finite fields into redundant rings under a ring homomorphism, (b) a new family of non-linear arithmetic residue codes with very high error detection probability, (c) a set of new low-cost arithmetic primitives for optimal extension field arithmetic based on robust codes, and (d) design techniques for tamper resilient finite state machines

The Design and Implementation of a High-Performance Polynomial System Solver

This thesis examines the algorithmic and practical challenges of solving systems of polynomial equations. We discuss the design and implementation of triangular decomposition to solve polynomials systems exactly by means of symbolic computation. Incremental triangular decomposition solves one equation from the input list of polynomials at a time. Each step may produce several different components (points, curves, surfaces, etc.) of the solution set. Independent components imply that the solving process may proceed on each component concurrently. This so-called component-level parallelism is a theoretical and practical challenge characterized by irregular parallelism. Parallelism is not an algorithmic property but rather a geometrical property of the particular input system’s solution set. Despite these challenges, we have effectively applied parallel computing to triangular decomposition through the layering and cooperation of many parallel code regions. This parallel computing is supported by our generic object-oriented framework based on the dynamic multithreading paradigm. Meanwhile, the required polynomial algebra is sup- ported by an object-oriented framework for algebraic types which allows type safety and mathematical correctness to be determined at compile-time. Our software is implemented in C/C++ and have extensively tested the implementation for correctness and performance on over 3000 polynomial systems that have arisen in practice. The parallel framework has been re-used in the implementation of Hensel factorization as a parallel pipeline to compute roots of a polynomial with multivariate power series coefficients. Hensel factorization is one step toward computing the non-trivial limit points of quasi-components

Equations of parametric surfaces with base points via syzygies

Suppose $S$ is a parametrized surface in complex projective 3-space $mathbf{P}^3$ given as the image of $phi: mathbf{P}^1 imes mathbf{P}^1 o mathbf{P}^3$. The implicitization problem is to compute an implicit equation $F=0$ of $S$ using the parametrization $phi$. An algorithm using syzygies exists for computing $F$ if $phi$ has no base points, i.e. $phi$ is everywhere defined. This work is an extension of this algorithm to the case of a surface with multiple base points of total multiplicity k. We accomplish this in three chapters. In Chapter 2, we develop the concept and properties of Castelnuovo-Mumford regularity in biprojective spaces. In Chapter 3, we give a criterion for regularity in biprojective spaces. These results are applied to the implicitization problem in Chapter 4

Algorithms for Mori Dream Spaces

Mori dream spaces are algebraic varieties with finitely generated Cox ring; basic examples are toric varieties or rational varieties with a torus action of complexity one. Due to the finite generation of the Cox ring, Mori dream spaces allow an explicit approach in terms of commutative algebra and polyhedral combinatorics. Based on this approach we develop a series of algorithms to explore the geometry of Mori dream spaces. We first present a toolkit for basic computations with Mori dream spaces, e.g., determining the Picard group, cones of divisor classes, the canonical toric ambient variety, singularities or a test for being factorial. Specialized algorithms are presented for the case of complete intersection Cox rings or varieties with a torus action of complexity one, e.g., the computation of intersection numbers, the test for the Fano or Gorenstein properties, roots of the automorphism group, resolution of singularities. We apply these algorithms to explore and classify certain (combinatorially) minimal singular del Pezzo k*-surfaces of Picard rank two. As a first advanced algorithm, we show how to compute the Mori chamber decomposition and the GIT-fan of a torus action on an affine variety. A second series of advanced algorithms concerns the problem of the behavior of the Cox ring under modifications, e.g., of blow ups. We develop algorithms to verify finite generation, verify a guess of generators, systematically produce generators and to determine the ideal of relations of the Cox ring of the modified variety. This includes an algorithm to compute Cox rings of blow ups of Mori dream spaces that terminates if and only if the new variety is a Mori dream space. As applications, we determine the Cox rings of certain blow ups of the three-dimensional projective space and of the Gorenstein log-terminal del Pezzo surfaces of Picard number one without a non-trival k*-action. As further application, we determine the Cox rings of the smooth rational surfaces of Picard number at most six; for Picard number six, we restrict ourselves to the cases without a non-trival k*-action whereas the classification is complete for Picard number up to five.Mori-Dream-Spaces sind algebraische Varietäten mit endlich erzeugtem Coxring, etwa torische Varietäten oder rationale Varietäten mit einer Toruswirkung der Komplexität eins. Aufgrund des endlichen Erzeugendensystems für den Coxring können Mori-Dream-Spaces durch Methoden der kommutativen Algebra und polyedrischen Kombinatorik behandelt werden. Aufbauend auf dieser Beschreibung entwickeln wir eine Reihe an Algorithmen zur Untersuchung der Geometrie von Mori-Dream-Spaces. Zunächst präsentieren wir Werkzeuge, um grundlegende Berechnungen mit Mori-Dream-Spaces durchführen zu können, z.B. das Bestimmen der Picardgruppe, Kegel von Divisorenklassen, kanonische torische ambiente Varietät, Singularitäten oder der Test auf Faktorialität. Spezialisierte Algorithmen werden für Varietäten, deren Coxring ein vollständigen Durchschnitt ist, oder für Varietäten mit einer Toruswirkung der Komplexität eins vorgestellt, etwa das Berechnen von Schnittzahlen, das Testen auf die Fano- oder Gorenstein-Eigenschaft, das Berechnen von Wurzeln der Automorphismengruppe oder das Auflösen von Singularitäten. Als Anwendung unserer Algorithmen untersuchen und klassifizieren wir gewisse (kombinatorisch) minimale del-Pezzo k*-Flächen mit Picardzahl zwei. Als ersten fortgeschrittenen Algorithmus zeigen wir, wie man die Mori-Kammerzerlegung eines Mori-Dream-Spaces und den GIT-Fächer von Toruswirkungen auf affinen Varietäten berechnet. Eine zweite Reihe an fortgeschritten Algorithmen beschäftigt sich mit der Auswirkung von Modifikationen, etwa von Aufblasungen, auf Coxringe. Wir entwickeln Algorithmen, um zu verifizieren, dass der Coxring der modifizierten Varietät endlich erzeugt ist, um geratene Erzeuger zu verifizieren, um systematisch Erzeuger zu produzieren und um Relationen zu berechnen. Dies beinhaltet einen Algorithmus zur Berechnung von Coxringen von Aufblasungen eines Mori-Dream-Spaces, der genau dann terminiert, wenn der Coxring der Aufblasung endlich erzeugt ist. Als Anwendungen bestimmen wir die Coxringe gewisser Aufblasungen des drei-dimensionalen projektiven Raumes und von Gorenstein log-terminalen del-Pezzo-Flächen mit Picardzahl eins ohne nicht-triviale k*-Wirkung. Als weitere Anwendung bestimmen wir die Coxringe der glatten, rationalen Flächen, deren Picardzahl höchstens sechs ist; für Picardzahl sechs beschränken wir uns auf die Fälle ohne nicht-triviale k*-Wirkung, bis zu Picardzahl fünf ist die Klassifikation vollständig

Q(sqrt(-3))-Integral Points on a Mordell Curve

We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4