On the complexity of computing Gr\"obner bases for weighted homogeneous systems

Solving polynomial systems arising from applications is frequently made easier by the structure of the systems. Weighted homogeneity (or quasi-homogeneity) is one example of such a structure: given a system of weights $W=(w\_{1},\dots,w\_{n})$, $W$-homogeneous polynomials are polynomials which are homogeneous w.r.t the weighted degree $\deg\_{W}(X\_{1}^{\alpha\_{1}},\dots,X\_{n}^{\alpha\_{n}}) = \sum w\_{i}\alpha\_{i}$. Gr\"obner bases for weighted homogeneous systems can be computed by adapting existing algorithms for homogeneous systems to the weighted homogeneous case. We show that in this case, the complexity estimate for Algorithm~\F5 \left(\binom{n+\dmax-1}{\dmax}^{\omega}\right) can be divided by a factor $\left(\prod w\_{i} \right)^{\omega}$. For zero-dimensional systems, the complexity of Algorithm~\FGLM $nD^{\omega}$ (where $D$ is the number of solutions of the system) can be divided by the same factor $\left(\prod w\_{i} \right)^{\omega}$. Under genericity assumptions, for zero-dimensional weighted homogeneous systems of $W$-degree $(d\_{1},\dots,d\_{n})$, these complexity estimates are polynomial in the weighted B\'ezout bound $\prod\_{i=1}^{n}d\_{i} / \prod\_{i=1}^{n}w\_{i}$. Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by the weighted Macaulay bound $\sum (d\_{i}-w\_{i}) + w\_{n}$, and this bound is sharp if we can order the weights so that $w\_{n}=1$. For overdetermined semi-regular systems, estimates from the homogeneous case can be adapted to the weighted case. We provide some experimental results based on systems arising from a cryptography problem and from polynomial inversion problems. They show that taking advantage of the weighted homogeneous structure yields substantial speed-ups, and allows us to solve systems which were otherwise out of reach

Reducing the size and number of linear programs in a dynamic Gr\"obner basis algorithm

The dynamic algorithm to compute a Gr\"obner basis is nearly twenty years old, yet it seems to have arrived stillborn; aside from two initial publications, there have been no published followups. One reason for this may be that, at first glance, the added overhead seems to outweigh the benefit; the algorithm must solve many linear programs with many linear constraints. This paper describes two methods of reducing the cost substantially, answering the problem effectively.Comment: 11 figures, of which half are algorithms; submitted to journal for refereeing, December 201

ALGORITMA GROEBNER WALK LAMBAT?

Algoritma Groebner Walk dikembangkan karena Algoritma Buch‐berger memerlukan waktu dan memori yang sangat banyak untuk menghitung basis Groebner dari suatu ideal terhadap order leksikografis. Untuk menghitung basis Groebner terhadap order leksi‐kografis, Algoritma Groebner Walk harus menghitung terlebih dahulu beberapa basis Groebner terhadap order yang lain secara bertahap. Secara umum, Algoritma Groebner Walk jauh lebih efisien dari Algoritma Buchberger. Makalah ini menyajikan hasil eksperimen(pengimplemen‐tasian) dari Algoritma Groebner Walk di Sistem Aljabar Komputer Singular. Ternyata, proses penghitungan basis Groebner pada langkah terakhir, yaitu penghitungan basis Groebner dari ideal homogen terhadap order leksikografis sangat memakan waktu dan memori yang sangat lama. Akibatnya, Algoritma Groebner Walk menjadi tidak efisien. Kata Kunci: Basis Groebner, Algoritma Groebner Wal

Fast Reduction of Bivariate Polynomials with Respect to Sufficiently Regular Gröbner Bases

International audienc

A survey on signature-based Gr\"obner basis computations

This paper is a survey on the area of signature-based Gr\"obner basis algorithms that was initiated by Faug\`ere's F5 algorithm in 2002. We explain the general ideas behind the usage of signatures. We show how to classify the various known variants by 3 different orderings. For this we give translations between different notations and show that besides notations many approaches are just the same. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. This survey shall help to outline this field of active research.Comment: 53 pages, 8 figures, 11 table

Joint shape and motion estimation from echo-based sensor data

2018 Fall.Includes bibliographical references.Given a set of time-series data collected from echo-based ranging sensors, we study the problem of jointly estimating the shape and motion of the target under observation when the sensor positions are also unknown. Using an approach first described by Stuff et al., we model the target as a point configuration in Euclidean space and estimate geometric invariants of the configuration. The geometric invariants allow us to estimate the target shape, from which we can estimate the motion of the target relative to the sensor position. This work will unify the various geometric- invariant based shape and motion estimation literature under a common framework, and extend that framework to include results for passive, bistatic sensor systems

Combining Boolean Networks and Ordinary Differential Equations for Analysis and Comparison of Gene Regulatory Networks

This thesis is concerned with different groups of qualitative models of gene regulatory networks. Four types of models will be considered: interaction graphs, Boolean networks, models based on differential equations and discrete abstractions of differential equations. We will investigate the relations between these modeling frameworks and how they can be used in the analysis of individual models. The focus lies on the mathematical analysis of these models. This thesis makes several contributions in relating these different modeling frameworks. The first approach concerns individual Boolean models and parametrized families of ordinary differential equations (ODEs). To construct ODE models systematically from Boolean models several automatic conversion algorithms have been proposed. In Chapter 2 several such closely related algorithms will be considered. It will be proven that certain invariant sets are preserved during the conversion from a Boolean network to a model based on ODEs. In the second approach the idea of abstracting the dynamics of individual models to relate structure and dynamics will be introduced. This approach will be applied to Boolean models and models based on differential equations. This allows to compare groups of models in these modeling frameworks which have the same structure. We demonstrate that this constitutes an approach to link the interaction graph to the dynamics of certain sets of Boolean networks and models based on differential equations. The abstracted dynamics – or more precisely the restrictions on the abstracted behavior – of such sets of Boolean networks or models based on differential equations will be represented as Boolean state transitions graphs themselves. We will show that these state transition graphs can be considered as asynchronous Boolean networks. Despite the rather theoretical question this thesis tries to answer there are many potential applications of the results. The results in Chapter 2 can be applied to network reduction of ODE models based on Hill kinetics. The results of the second approach in Chapter 4 can be applied to network inference and analysis of Boolean model sets. Furthermore, in the last chapter of this thesis several ideas for applications with respect to experiment design will be considered. This leads to the question how different asynchronous Boolean networks or different behaviours of a single asynchronous Boolean network can be distinguishedDiese Arbeit beschäftigt sich mit unterschiedlichen Typen von qualitativen Modellen genregulatorischer Netzwerke. Vier Typen von Modellen werden betrachtet: Interaktionsgraphen, Boolesche Netzwerke, Modelle, die auf Differentialgleichungen basieren und diskrete Abstraktionen von Differentialgleichungen. Wir werden mehr über die Beziehungen zwischen diesen Modellgruppen lernen und wie diese Beziehungen genutzt werden können, um einzelne Modelle zu analysieren. Der Schwerpunkt liegt hierbei auf der mathematischen Analyse dieser Modellgruppen. In dieser Hinsicht leistet diese Arbeit mehrere Beiträge. Zunächst betrachten wir Boolesche Netzwerke und parametrisierte Familien von gewöhnlichen Differentialgleichungen (ODEs). Um solche ODE-Modelle systematisch aus Booleschen Modellen abzuleiten, wurden in der Vergangenheit verschiedene automatische Konvertierungsalgorithmen vorgeschlagen. In Kapitel 2 werden einige dieser Algorithmen näher untersucht. Wir werden beweisen, dass bestimmte invariante Mengen bei der Konvertierung eines Booleschen Modells in ein ODE-Modell erhalten bleiben. Der zweite Ansatz, der in dieser Arbeit verfolgt wird, beschäftigt sich mit diskreten Abstraktionen der Dynamik von Modellen. Mit Hilfe dieser Abstraktionen ist es möglich, die Struktur – den Interaktionsgraphen – und die Dynamik der zugehörigen Modelle in Bezug zu setzen. Diese Methode wird sowohl auf Boolesche Modelle als auch auf ODE-Modelle angewandt. Gleichzeitig erlaubt dieser Ansatz Mengen von Modellen in unterschiedlichen Modellgruppen zu vergleichen, die dieselbe Struktur haben. Die abstrahierten Dynamiken (genauer die Einschränkungen der abstrahierten Dynamiken) der Booleschen Modellmengen oder ODE-Modellmengen können als Boolesche Zustandsübergangsgraphen repräsentiert werden. Wir werden zeigen, dass diese Zustandsübergangsgraphen wiederum selber als (asynchrone) Boolesche Netzwerke aufgefasst werden können. Trotz der theoretischen Ausgangsfrage werden in dieser Arbeit zahlreiche Anwendungen aufgezeigt. Die Ergebnisse aus Kapitel 2 können zur Modellreduktion benutzt werden, indem die Dynamik der ODE-Modelle auf den zu den Booleschen Netzwerken gehörigen “trap spaces” betrachtet wird. Die Resultate aus Kapitel 4 können zur Netzwerkinferenz oder zur Analyse von Modellmengen genutzt werden. Weiterhin werden im letzten Kapitel dieser Arbeit einige Anwendungsideen im Bezug auf Experimentdesign eingeführt. Dies führt zu der Fragestellung, wie verschiedene asynchrone Boolesche Netzwerke oder unterschiedliche Dynamiken, die mit einem einzelnen Modell vereinbar sind, unterschieden werden können

Computational Methods for Computer Vision : Minimal Solvers and Convex Relaxations

Robust fitting of geometric models is a core problem in computer vision. The most common approach is to use a hypothesize-and-test framework, such as RANSAC. In these frameworks the model is estimated from as few measurements as possible, which minimizes the risk of selecting corrupted measurements. These estimation problems are called minimal problems, and they can often be formulated as systems of polynomial equations. In this thesis we present new methods for building so-called minimal solvers or polynomial solvers, which are specialized code for solving such systems. On several minimal problems we improve on the state-of-the-art both with respect to numerical stability and execution time.In many computer vision problems low rank matrices naturally occur. The rank can serve as a measure of model complexity and typically a low rank is desired. Optimization problems containing rank penalties or constraints are in general difficult. Recently convex relaxations, such as the nuclear norm, have been used to make these problems tractable. In this thesis we present new convex relaxations for rank-based optimization which avoid drawbacks of previous approaches and provide tighter relaxations. We evaluate our methods on a number of real and synthetic datasets and show state-of-the-art results