## 510 Mathematik

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In summary, this cumulative dissertation investigates the application of the conjugate gradient method CG for the optimization of artificial neural networks (NNs) and compares this method with common first-order optimization methods, especially the stochastic gradient descent (SGD).
The presented research results show that CG can effectively optimize both small and very large networks. However, the default machine precision of 32 bits can lead to problems. The best results are only achieved in 64-bits computations. The research also emphasizes the importance of the initialization of the NNs’ trainable parameters and shows that an initialization using singular value decomposition (SVD) leads to drastically lower error values. Surprisingly, shallow but wide NNs, both in Transformer and CNN architectures, often perform better than their deeper counterparts. Overall, the research results recommend a re-evaluation of the previous preference for extremely deep NNs and emphasize the potential of CG as an optimization method.

In the constrained planarity setting, we ask whether a graph admits a crossing-free drawing that additionally satisfies a given set of constraints. These constraints are often derived from very natural problems; prominent examples are Level Planarity, where vertices have to lie on given horizontal lines indicating a hierarchy, Partially Embedded Planarity, where we extend a given drawing without modifying already-drawn parts, and Clustered Planarity, where we additionally draw the boundaries of clusters which recursively group the vertices in a crossing-free manner. In the last years, the family of constrained planarity problems received a lot of attention in the field of graph drawing. Efficient algorithms were discovered for many of them, while a few others turned out to be NP-complete. In contrast to the extensive theoretical considerations and the direct motivation by applications, only very few of the found algorithms have been implemented and evaluated in practice.
The goal of this thesis is to advance the research on both theoretical as well as practical aspects of constrained planarity. On the theoretical side, we consider two types of constrained planarity problems. The first type are problems that individually constrain the rotations of vertices, that is they restrict the counter-clockwise cyclic orders of the edges incident to vertices. We give a simple linear-time algorithm for the problem Partially Embedded Planarity, which also generalizes to further constrained planarity variants of this type.
The second type of constrained planarity problem concerns more involved planarity variants that come down to the question whether there are embeddings of one or multiple graphs such that the rotations of certain vertices are in sync in a certain way. Clustered Planarity and a variant of the Simultaneous Embedding with Fixed Edges Problem (Connected SEFE-2) are well-known problems of this type. Both are generalized by our Synchronized Planarity problem, for which we give a quadratic algorithm. Through reductions from various other problems, we provide a unified modelling framework for almost all known efficiently solvable constrained planarity variants that also directly provides a quadratic-time solution to all of them.
For both our algorithms, a key ingredient for reaching an efficient solution is the usage of the right data structure for the problem at hand. In this case, these data structures are the SPQR-tree and the PC-tree, which describe planar embedding possibilities from a global and a local perspective, respectively. More specifically, PC-trees can be used to locally describe the possible cyclic orders of edges around vertices in all planar embeddings of a graph. This makes it a key component for our algorithms, as it allows us to test planarity while also respecting further constraints, and to communicate constraints arising from the surrounding graph structure between vertices with synchronized rotation.
Bridging over to the practical side, we present the first correct implementation of PC-trees. We also describe further improvements, which allow us to outperform all implementations of alternative data structures (out of which we only found very few to be fully correct) by at least a factor of 4. We show that this yields a simple and competitive planarity test that can also yield an embedding to certify planarity. We also use our PC-tree implementation to implement our quadratic algorithm for solving Synchronized Planarity. Here, we show that our algorithm greatly outperforms previous attempts at solving related problems like Clustered Planarity in practice. We also engineer its running time and show how degrees of freedom in the theoretical algorithm can be leveraged to yield an up to tenfold speed-up in practice.

Sichtbarkeitsprobleme, wie das Folgende, gehören zu den grundlegenden Problemen der algorithmischen Geometrie: Berechne zu einem einfachen Polygon, dem sogenannten Kanal, und zu einem darin enthaltenen Punkt die von diesem Punkt aus sichtbare Punktmenge. Dabei ist ein Punkt von einem anderen Punkt aus sichtbar, wenn deren Verbindungsstrecke den Kanal nicht verlässt. Wir wollen uns in dieser Arbeit mit zirkulärer Sichtbarkeit beschäftigen. Zur Verbindung zweier Punkte sind dann nicht nur Strecken, sondern auch Kreisbögen zulässig. Außerdem betrachten wir als Ausgangspunkt dieser sogenannten Sichtbarkeitskreisbögen und -strecken eine Kante des Kanals anstatt eines einzelnen Punkts. Konkret liefert diese Arbeit einen Beitrag zur numerisch robusten Bestimmung der zirkulären Sichtbarkeitsmenge ausgehend von einer Kante des Kanals.
Hierfür wird in dieser Arbeit ein Algorithmus vorgestellt, mit dem für einen gegebenen Punkt festgestellt werden kann, ob dieser von der Startkante aus sichtbar ist. Im Fall eines sichtbaren Punkts wird ein Sichtbarkeitskreisbogen berechnet, der zwei Kanalberührungen besitzt. Damit kann der Algorithmus bei geeigneter Wahl des zu untersuchenden Punkts – der als dritte Kanalberührung fungiert – direkt zur Berechnung von sogenannten Grenzkreisbögen der Sichtbarkeitsmenge benutzt werden. Diese definieren den Rand der zirkulären Sichtbarkeitsmenge und zeichnen sich dadurch aus, dass sie vom Kanal dreimal abwechselnd von links und von rechts berührt werden.
Der beschriebene Algorithmus basiert auf der Untersuchung derjenigen Kreisbögen, die zwar nicht notwendigerweise vollständig im Kanal liegen, aber die Startkante mit dem Punkt verbinden, dessen Sichtbarkeit bestimmt werden soll. Insbesondere werden dabei die Bereiche untersucht, in denen der jeweilige Kreisbogen den Kanal
verlässt, die sogenannten Verletzungen. Da die „Schwere“ einer solchen Verletzung quantifizierbar ist, wird ein iteratives Vorgehen ermöglicht. Dabei wird der Kreisbogen iterativ so verändert, dass dieser bei gleichem Endpunkt den Kanal immer „weniger verlässt“. Ist der Endpunkt und damit der zu untersuchende Punkt nicht sichtbar, wird im Laufe des Algorithmus festgestellt, dass keine derartige Verbesserung möglich ist. Der vorgestellte Algorithmus ist numerisch robust, einfach umzusetzen und besitzt eine in der Anzahl der Kanalecken lineare Laufzeit.

This thesis investigates the quality of randomly collected data by employing a framework built on information-based complexity, a field related to the numerical analysis of abstract problems. The quality or power of gathered information is measured by its radius which is the uniform error obtainable by the best possible algorithm using it. The main aim is to present progress towards understanding the power of random information for approximation and integration problems.
In the first problem considered, information given by linear functionals is used to recover vectors, in particular from generalized ellipsoids. This is related to the approximation of diagonal operators which are important objects of study in the theory of function spaces. We obtain upper bounds on the radius of random information both in a convex and a quasi-normed setting, which extend and, in some cases, improve existing results. We conjecture and partially establish that the power of random information is subject to a dichotomy determined by the decay of the length of the semiaxes of the generalized ellipsoid.
Second, we study multivariate approximation and integration using information given by function values at sampling point sets. We obtain an asymptotic characterization of the radius of information in terms of a geometric measure of equidistribution, the distortion, which is well known in the theory of quantization of measures. This holds for isotropic Sobolev as well as Hölder and Triebel-Lizorkin spaces on bounded convex domains. We obtain that for these spaces, depending on the parameters involved, typical point sets are either asymptotically optimal or worse by a logarithmic factor, again extending and improving existing results.
Further, we study isotropic discrepancy which is related to numerical integration using linear algorithms with equal weights. In particular, we analyze the quality of lattice point sets with respect to this criterion and obtain that they are suboptimal compared to uniform random points. This is in contrast to the approximation of Sobolev functions and resolves an open question raised in the context of a possible low discrepancy construction on the two-dimensional sphere.

The generalization of univariate splines to higher dimensions is not straightforward. There are different approaches, each with its own advantages and drawbacks. A promising approach using Delaunay configurations and simplex splines is due to Neamtu.
After recalling fundamentals of univariate splines, simplex splines, and the wellknown, multivariate DMS-splines, we address Neamtu’s DCB-splines. He defined two variants that we refer to as the nonpooled and the pooled approach, respectively. Regarding these spline spaces, we contribute the following results.
We prove that, under suitable assumptions on the knot set, both variants exhibit the local finiteness property, i.e., these spline spaces are locally finite-dimensional and at each point only a finite number of basis candidate functions have a nonzero value. Additionally, we establish a criterion guaranteeing these properties within a compact region under mitigated assumptions.
Moreover, we show that the knot insertion process known from univariate splines does not work for DCB-splines and reason why this behavior is inherent to these spline spaces. Furthermore, we provide a necessary criterion for the knot insertion property to hold true for a specific inserted knot. This criterion is also sufficient for bivariate, nonpooled DCB-splines of degrees zero and one. Numerical experiments suggest that the sufficiency also holds true for arbitrary spline degrees.
Univariate functions can be approximated in terms of splines using the Schoenberg operator, where the approximation error decreases quadratically as the maximum distance between consecutive knots is reduced. We show that the Schoenberg operator can be defined analogously for both variants of DCB-splines with a similar error bound.
Additionally, we provide a counterexample showing that the basis candidate functions of nonpooled DCB-splines are not necessarily linearly independent, contrary to earlier statements in the literature. In particular, this implies that the corresponding functions are not a basis for the space of nonpooled DCB-splines.

Cryptography is the scientific study of techniques for securing information and communication against adversaries. It is about designing and analyzing encryption schemes and protocols that protect data from unauthorized reading. However, in our modern information-driven society with highly complex and interconnected information systems, encryption alone is no longer enough as it makes the data unintelligible, preventing any meaningful computation without decryption. On the one hand, data owners want to maintain control over their sensitive data. On the other hand, there is a high business incentive for collaborating with an untrusted external party.
Modern cryptography encompasses different techniques, such as secure multiparty computation, homomorphic encryption or order-preserving encryption, that enable cloud users to encrypt their data before outsourcing it to the cloud while still being able to process and search on the outsourced and encrypted data without decrypting it. In this thesis, we rely on these cryptographic techniques for computing on encrypted data to propose efficient multiparty protocols for order-preserving encryption, decision tree evaluation and kth-ranked element computation.
We start with Order-preserving encryption (OPE) which allows encrypting data, while still enabling efficient range queries on the encrypted data. However, OPE is symmetric limiting, the use case to one client and one server. Imagine a scenario where a Data Owner (DO) outsources encrypted data to the Cloud Service Provider (CSP) and a Data Analyst (DA) wants to execute private range queries on this data. Then either the DO must reveal its encryption key or the DA must reveal the private queries. We overcome this limitation by allowing the equivalent of a public-key OPE.
Decision trees are common and very popular classifiers because they are explainable. The problem of evaluating a private decision tree on private data consists of a server holding a private decision tree and a client holding a private attribute vector. The goal is to classify the client’s input using the server’s model such that the client learns only the result of the classification, and the server learns nothing. In a first approach, we represent the tree as an array and execute only d interactive comparisons (instead of 2 d as in existing solutions), where d denotes the depth of the tree. In a second approach, we delegate the complete tree evaluation to the server using somewhat or fully homomorphic encryption where the ciphertexts are encrypted under the client’s public key.
A generalization of a decision tree is a random forest that consists of many decision trees. A classification with a random forest evaluates each decision tree in the forest and outputs the classification label which occurs most often. Hence, the classification labels are ranked by their number of occurrences and the final result is the best ranked one. The best ranked element is a special case of the kth-ranked element. In this thesis, we consider the secure computation of the kth-ranked element in a distributed setting with applications in benchmarking and auctions. We propose different approaches for privately computing the kth-ranked element in a star network, using either garbled circuits or threshold homomorphic encryption.

Our subject of study is strong approximation of stochastic differential equations (SDEs) with respect to the supremum and the L_p error criteria, and we seek approximations that are strongly asymptotically optimal in specific classes of approximations. For the supremum error, we prove strong asymptotic optimality for specific tamed Euler schemes relating to certain adaptive and to equidistant time discretizations. For the L_p error, we prove strong asymptotic optimality for specific tamed Milstein schemes relating to certain adaptive and to equidistant time discretizations. To illustrate our findings, we numerically analyze the SDE associated with the Heston–3/2–model originating from mathematical finance.

Algebraic solving of polynomial systems and satisfiability of propositional logic formulas are not two completely separate research areas, as it may appear at first sight. In fact, many problems coming from cryptanalysis, such as algebraic fault attacks, can be rephrased as solving a set of Boolean polynomials or as deciding the satisfiability of a propositional logic formula. Thus one can analyze the security of cryptosystems by applying standard solving methods from computer algebra and SAT solving. This doctoral thesis is dedicated to studying solvers that are based on logic and algebra separately as well as integrating them into one such that the combined solvers become more powerful tools for cryptanalysis.
This disseration is divided into three parts. In this first part, we recall some theory and basic techniques for algebraic and logic solving. We focus mainly on DPLL-based SAT solving and techniques that are related to border bases and Gröbner bases. In particular, we describe in detail the Border Basis Algorithm and discuss its specialized version for Boolean polynomials called the Boolean Border Basis Algorithm.
In the second part of the thesis, we deal with connecting solvers based on algebra and logic. The ultimate goal is to combine the strength of different solvers into one. Namely, we fuse the XOR reasoning from algebraic solvers with the light, efficient design of SAT solvers. As a first step in this direction, we design various conversions from sets of clauses to sets of Boolean polynomials, and vice versa, such that solutions and models are preserved via the conversions. In particular, based on a block-building mechanism, we design a new blockwise algorithm for the CNF to ANF conversion which is geared towards producing fewer and lower degree polynomials. The above conversions allow usto integrate both solvers via a communication interface.
To reach an even tighter integration, we consider proof systems that combine resolution and polynomial calculus, i.e. the two most used proof systems in logic and algebraic solving. Based on such a proof system, which we call SRES, we introduce new types of solving algorithms that demostrate the synergy between Gröbner-like and DPLL-like solving. At the end of the second part of the dissertation, we provide some experiments based on a new benchmark which illustrate that the our new method based on DPLL has the potential to outperform CDCL SAT solvers.
In the third part of the thesis, we focus on practical attacks on various cryptograhic primitives. For instance, we apply SAT solvers in the case of algebraic fault attacks on the symmetric ciphers LED and derivatives of the block cipher AES. The main goal there is to derive so-called fault equations automatically from the hardware description of the cryptosystem and thus automatizate the attack. To give some extra power to a SAT solver that inverts the hash functions SHA-1 and SHA-2, we describe how to tweak the SAT solver using a programmatic interface such that the propagation of the solver and thus the attack itself is improved.

In various fields of image analysis, determining the precise geometry of occurrent edges, e.g. the contour of an object, is a crucial task. Especially the curvature of an edge is of great practical relevance. In this thesis, we develop different methods to detect a variety of edge features, among them the curvature.
We first examine the properties of the parabolic Radon transform and show that it can be used to detect the edge curvature, as the smoothness of the parabolic Radon transform changes when the parabola is tangential to an edge and also, when additionally the curvature of the parabola coincides with the edge curvature. By subsequently introducing a parabolic Fourier transform and establishing a precise relation between the smoothness of a certain class of functions and the decay of the Fourier transform, we show that the smoothness result for the parabolic Radon transform can be translated into a change of the decay rate of the parabolic Fourier transform.
Furthermore, we introduce an extension of the continuous shearlet transform which additionally utilizes shears of higher order. This extension, called the Taylorlet transform, allows for a detection of the position and orientation, as well as the curvature and other higher order geometric information of edges. We introduce novel vanishing moment conditions which enable a more robust detection of the geometric edge features and examine two different constructions for Taylorlets. Lastly, we translate the results of the Taylorlet transform in R^2 into R^3 and thereby allow for the analysis of the geometry of object surfaces.