Convolution, Separation and Concurrency

A notion of convolution is presented in the context of formal power series together with lifting constructions characterising algebras of such series, which usually are quantales. A number of examples underpin the universality of these constructions, the most prominent ones being separation logics, where convolution is separating conjunction in an assertion quantale; interval logics, where convolution is the chop operation; and stream interval functions, where convolution is used for analysing the trajectories of dynamical or real-time systems. A Hoare logic is constructed in a generic fashion on the power series quantale, which applies to each of these examples. In many cases, commutative notions of convolution have natural interpretations as concurrency operations.Comment: 39 page

Matrix factorization over dioids and its applications in data mining

Matrix factorizations are an important tool in data mining, and they have been used extensively for finding latent patterns in the data. They often allow to separate structure from noise, as well as to considerably reduce the dimensionality of the input matrix. While classical matrix decomposition methods, such as nonnegative matrix factorization (NMF) and singular value decomposition (SVD), proved to be very useful in data analysis, they are limited by the underlying algebraic structure. NMF, in particular, tends to break patterns into smaller bits, often mixing them with each other. This happens because overlapping patterns interfere with each other, making it harder to tell them apart. In this thesis we study matrix factorization over algebraic structures known as dioids, which are characterized by the lack of additive inverse (“negative numbers”) and the idempotency of addition (a + a = a). Using dioids makes it easier to separate overlapping features, and, in particular, it allows to better deal with the above mentioned pattern breaking problem. We consider different types of dioids, that range from continuous (subtropical and tropical algebras) to discrete (Boolean algebra). Among these, the Boolean algebra is perhaps the most well known, and there exist methods that allow one to obtain high quality Boolean matrix factorizations in terms of the reconstruction error. In this work, however, a different objective function is used – the description length of the data, which enables us to obtain compact and highly interpretable results. The tropical and subtropical algebras, on the other hand, are much less known in the data mining field. While they find applications in areas such as job scheduling and discrete event systems, they are virtually unknown in the context of data analysis. We will use them to obtain idempotent nonnegative factorizations that are similar to NMF, but are better at separating the most prominent features of the data.Matrix-Faktorisierungen sind ein wichtiges Werkzeug in Data-Mining und wurden umfangreich zum Auffinden latenter Muster in den Daten verwendet. Oft erlauben sie, die Struktur vom Rauschen zu trennen, sowie Dimensionalität von der Eingabematrix wesentlich zu reduzieren. Obwohl klassische Methoden für die Matrix-Zerlegung, wie z.B. nicht negative Matrixfaktorisierung (NMF) und Singulärwertzerlegung (SVD), in der Datenanalyse sich als sehr nützlich erwiesen haben, sind sie durch die zugrunde liegende algebraische Struktur eingeschränkt. Insbesondere neigt NMF dazu, Muster in kleinere Bits zu brechen, und vermischt sie oft miteinander. Das passiert, weil überschneidende Muster sich gegenseitig stören, sodass es schwieriger ist, sie auseinander zu halten. In dieser Dissertation werden Matrix-Faktorisierungen über algebraische Strukturen, sogenannte Dioiden, untersucht, die sich durch die fehlende additive Inverse (“negative Zahlen”) und Idempotenz der Addition (a + a = a) auszeichnen. Mit Dioiden ist es einfacher überschneidende Merkmale zu trennen. Insbesondere erlauben sie besser mit dem erwähnten Musterbrechenproblem umzugehen. Es werden unterschiedliche Dioiden untersucht, die von kontinuierlichen (subtropische und tropische Algebren) bis zu diskreter (Boolesche Algebra) reichen. Unter diesen, die Boolesche Algebra ist wahrscheinlich die bekannteste, und es gibt Methoden, die ermöglichen hochwertiger Matrix-Faktorisierungen in Bezug auf den Rekonstruktionsfehler zu erzielen. In dieser Arbeit aber wird eine andere Zielfunktion verwendet: Die Länge der Beschreibung von den Daten. Die Zielfunktion ermöglicht uns kompakte und hochinterpretierbare Ergebnisse zu erzielen. Andererseits sind die tropische und subtropische Algebren viel weniger im Bereich Data-Mining bekannt. Sie finden zwar Anwendungen in Bereichen wie Job-Scheduling und diskrete Ereignissysteme, jedoch sind sie im Kontext von Datenanalyse nahezu unbekannt. Hier werden sie verwendet, um idempotente, nicht negative Faktorisierungen zu erhalten, die NMF ähneln, aber die wichtigsten Merkmale der Daten besser voneinander trennen

Detecting Deterministic Chaotic Inter-arrival Times in Material Flow Systems

Automated, modular, asynchronous and locally controlled material flow systems promise high routing flexibility in production lines because their conveying modules can be reconfigured without reprogramming PLCs. However, if such material flow systems comprise cycles and different routes, they may exhibit undesirable deterministic chaotic inter-arrival times, which can lead to conveying bottlenecks when approaching maximum capacity. Since existing analytical models have not been practically adopted for planning material flow systems, an approach for detecting deterministic chaotic inter-arrival times during production is proposed. It employs the Hough transform to identify trajectories in inter-arrival time phase space. The approach is tested with a laboratory double belt conveyor system, in which non-deterministic behavior is minimized. Results are compared with a previously published analytical model. It is shown that the proposed approach is able to detect deterministic chaotic inter-arrival times for the test cases. Phase trajectories are only partly identified. Future research should test and compare different line detection algorithms for their influence on the approach’s robustness in practical production environments

Characterizing matrices with XX-simple image eigenspace in max-min semiring

A matrix $A$ is said to have $X$-simple image eigenspace if any eigenvector $x$ belonging to the interval $X=\{x\colon \underline{x}\leq x\leq\overline{x}\}$ is the unique solution of the system $A\otimes y=x$ in $X$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.Comment: 23 page

Discrete events: Perspectives from system theory

Systems Theory;differentiaal/ integraal-vergelijkingen

Algorithms for Approximate Subtropical Matrix Factorization

Matrix factorization methods are important tools in data mining and analysis. They can be used for many tasks, ranging from dimensionality reduction to visualization. In this paper we concentrate on the use of matrix factorizations for finding patterns from the data. Rather than using the standard algebra -- and the summation of the rank-1 components to build the approximation of the original matrix -- we use the subtropical algebra, which is an algebra over the nonnegative real values with the summation replaced by the maximum operator. Subtropical matrix factorizations allow "winner-takes-it-all" interpretations of the rank-1 components, revealing different structure than the normal (nonnegative) factorizations. We study the complexity and sparsity of the factorizations, and present a framework for finding low-rank subtropical factorizations. We present two specific algorithms, called Capricorn and Cancer, that are part of our framework. They can be used with data that has been corrupted with different types of noise, and with different error metrics, including the sum-of-absolute differences, Frobenius norm, and Jensen--Shannon divergence. Our experiments show that the algorithms perform well on data that has subtropical structure, and that they can find factorizations that are both sparse and easy to interpret.Comment: 40 pages, 9 figures. For the associated source code, see http://people.mpi-inf.mpg.de/~pmiettin/tropical

Convolution as a unifying concept: Applications in separation logic, interval calculi and concurrency

The research reported here was supported in part by Australian Research Council Grant No. DP130102901 and EPSRC Grant No. EP/J003727/1