Minimizing and maximizing a linear objective function under a fuzzy max⁡−∗\max -\ast relational equation and an inequality constraint

summary:This paper provides an extension of results connected with the problem of the optimization of a linear objective function subject to $\max-\ast$ fuzzy relational equations and an inequality constraint, where $\ast$ is an operation. This research is important because the knowledge and the algorithms presented in the paper can be used in various optimization processes. Previous articles describe an important problem of minimizing a linear objective function under a fuzzy $\max-\ast$ relational equation and an inequality constraint, where $\ast$ is the $t$-norm or mean. The authors present results that generalize this outcome, so the linear optimization problem can be used with any continuous increasing operation with a zero element where $\ast$ includes in particular the previously studied operations. Moreover, operation $\ast$ does not need to be a t-norm nor a pseudo-$t$-norm. Due to the fact that optimal solutions are constructed from the greatest and minimal solutions of a $\max-\ast$ relational equation or inequalities, this article presents a method to compute them. We note that the linear optimization problem is valid for both minimization and maximization problems. Therefore, for the optimization problem, we present results to find the largest and the smallest value of the objective function. To illustrate this problem a numerical example is provided

Unsupervised and semi-supervised fuzzy clustering with multiple kernels.

For real-world clustering tasks, the input data is typically not easily separable due to the highly complex data structure or when clusters vary in size, density and shape. Recently, kernel-based clustering has been proposed to perform clustering in a higher-dimensional feature space spanned by embedding maps and corresponding kernel functions. Although good results were obtained using the Gaussian kernel function, its performance depends on the selection of the scaling parameter among an extensive range of possibilities. This step is often heavily influenced by prior knowledge about the data and by the patterns we expect to discover. Unfortunately, it is often unclear which kernels are more suitable for a particular task. The problem is aggravated for many real-world clustering applications, in which the distributions of the different clusters in the feature space exhibit large variations. Thus, in the absence of a priori knowledge, a single kernel selected from a predefined group is sometimes insufficient to represent the data. One way to learn optimal scaling parameters is through an exhaustive search of one optimal scaling parameter for each cluster. However, this approach is not practical since it is computationally expensive, especially when the data includes a large number of clusters and when the dynamic range of possible values of the scaling parameters is large. Moreover, the evaluation of the resulting partition in order to select the optimal parameters is not an easy task. To overcome the above drawbacks, we introduce two novel fuzzy clustering techniques that use Multiple Kernel Learning to provide an elegant solution for parameter selection. The Fuzzy C-Means with Multiple Kernels algorithm (FCMK) simultaneously finds the optimal partition and the cluster-dependent kernel combination weights that reflect the intrinsic structure of the data. The Relational Fuzzy Clustering with Multiple Kernels (RFCMK) learns the kernel combination weights by optimizing the relational dissimilarities. Consequently, the learned kernel combination weights reflect the relative density, size, and position of each cluster with respect to the other clusters. We also extended FCMK and RFCMK to the semi-supervised paradigms. We show that the incorporation of prior knowledge in the unsupervised clustering task in the form of a small set of constraints on which instances should or should not reside in the same cluster, guides the unsupervised approaches to a better partitioning of the data and avoid local minima, especially for high dimensional real world data. All of the proposed algorithms are optimized iteratively by dynamically updating the partition and the kernel combination weights in each iteration. This makes these algorithms simple and fast. Moreover, our algorithms are formulated to work on both vector and relational data. This makes them applicable to data where objects cannot be represented by vectors or when clusters of similar objects cannot be represented efficiently by a single prototype. We also introduced two relational fuzzy clustering with multiple kernel algorithms for large data to deal with the scalability issue of RFCMK. The random sample and extend RFCMK (rseRFCMK) computes cluster prototypes from a smaller sample of randomly selected objects, and then extends the partition to the remainder of the data. The single pass RFCMK (spRFCMK) sequentially loads manageable sized chunks, clustering the chunks in a single pass, and then combining the results from each chunk. Our extensive experiments show that RFCMK and SS-RFCMK outperform existing algorithms. In particular, we show that when data include clusters with various intrinsic structures and densities, learning kernel weights that vary over clusters is crucial in obtaining a good partition

Methods in Industrial Biotechnology for Chemical Engineers

In keeping with the definition that biotechnology is really no more than a name given to a set of techniques and processes, the authors apply some set of fuzzy techniques to chemical industry problems such as finding the proper proportion of raw mix to control pollution, to study flow rates, to find out the better quality of products. We use fuzzy control theory, fuzzy neural networks, fuzzy relational equations, genetic algorithms to these problems for solutions. When the solution to the problem can have certain concepts or attributes as indeterminate, the only model that can tackle such a situation is the neutrosophic model. The authors have also used these models in this book to study the use of biotechnology in chemical industries. This book has six chapters. First chapter gives a brief description of biotechnology. Second chapter deals will proper proportion of mix of raw materials in cement industries to minimize pollution using fuzzy control theory. Chapter three gives the method of determination of temperature set point for crude oil in oil refineries. Chapter four studies the flow rates in chemical industries using fuzzy neutral networks. Chapter five gives the method of minimization of waste gas flow in chemical industries using fuzzy linear programming. The final chapter suggests when in these studies indeterminancy is an attribute or concept involved, the notion of neutrosophic methods can be adopted.Comment: 125 pages, 20 figure

Unsupervised multiple kernel learning approaches for integrating molecular cancer patient data

Cancer is the second leading cause of death worldwide. A characteristic of this disease is its complexity leading to a wide variety of genetic and molecular aberrations in the tumors. This heterogeneity necessitates personalized therapies for the patients. However, currently defined cancer subtypes used in clinical practice for treatment decision-making are based on relatively few selected markers and thus provide only a coarse classifcation of tumors. The increased availability in multi-omics data measured for cancer patients now offers the possibility of defining more informed cancer subtypes. Such a more fine-grained characterization of cancer subtypes harbors the potential of substantially expanding treatment options in personalized cancer therapy. In this thesis, we identify comprehensive cancer subtypes using multidimensional data. For this purpose, we apply and extend unsupervised multiple kernel learning methods. Three challenges of unsupervised multiple kernel learning are addressed: robustness, applicability, and interpretability. First, we show that regularization of the multiple kernel graph embedding framework, which enables the implementation of dimensionality reduction techniques, can increase the stability of the resulting patient subgroups. This improvement is especially beneficial for data sets with a small number of samples. Second, we adapt the objective function of kernel principal component analysis to enable the application of multiple kernel learning in combination with this widely used dimensionality reduction technique. Third, we improve the interpretability of kernel learning procedures by performing feature clustering prior to integrating the data via multiple kernel learning. On the basis of these clusters, we derive a score indicating the impact of a feature cluster on a patient cluster, thereby facilitating further analysis of the cluster-specific biological properties. All three procedures are successfully tested on real-world cancer data. Comparing our newly derived methodologies to established methods provides evidence that our work offers novel and beneficial ways of identifying patient subgroups and gaining insights into medically relevant characteristics of cancer subtypes.Krebs ist eine der häufigsten Todesursachen weltweit. Krebs ist gekennzeichnet durch seine Komplexität, die zu vielen verschiedenen genetischen und molekularen Aberrationen im Tumor führt. Die Unterschiede zwischen Tumoren erfordern personalisierte Therapien für die einzelnen Patienten. Die Krebssubtypen, die derzeit zur Behandlungsplanung in der klinischen Praxis verwendet werden, basieren auf relativ wenigen, genetischen oder molekularen Markern und können daher nur eine grobe Unterteilung der Tumoren liefern. Die zunehmende Verfügbarkeit von Multi-Omics-Daten für Krebspatienten ermöglicht die Neudefinition von fundierteren Krebssubtypen, die wiederum zu spezifischeren Behandlungen für Krebspatienten führen könnten. In dieser Dissertation identifizieren wir neue, potentielle Krebssubtypen basierend auf Multi-Omics-Daten. Hierfür verwenden wir unüberwachtes Multiple Kernel Learning, welches in der Lage ist mehrere Datentypen miteinander zu kombinieren. Drei Herausforderungen des unüberwachten Multiple Kernel Learnings werden adressiert: Robustheit, Anwendbarkeit und Interpretierbarkeit. Zunächst zeigen wir, dass die zusätzliche Regularisierung des Multiple Kernel Learning Frameworks zur Implementierung verschiedener Dimensionsreduktionstechniken die Stabilität der identifizierten Patientengruppen erhöht. Diese Robustheit ist besonders vorteilhaft für Datensätze mit einer geringen Anzahl von Proben. Zweitens passen wir die Zielfunktion der kernbasierten Hauptkomponentenanalyse an, um eine integrative Version dieser weit verbreiteten Dimensionsreduktionstechnik zu ermöglichen. Drittens verbessern wir die Interpretierbarkeit von kernbasierten Lernprozeduren, indem wir verwendete Merkmale in homogene Gruppen unterteilen bevor wir die Daten integrieren. Mit Hilfe dieser Gruppen definieren wir eine Bewertungsfunktion, die die weitere Auswertung der biologischen Eigenschaften von Patientengruppen erleichtert. Alle drei Verfahren werden an realen Krebsdaten getestet. Den Vergleich unserer Methodik mit etablierten Methoden weist nach, dass unsere Arbeit neue und nützliche Möglichkeiten bietet, um integrative Patientengruppen zu identifizieren und Einblicke in medizinisch relevante Eigenschaften von Krebssubtypen zu erhalten

From Statistical Relational to Neurosymbolic Artificial Intelligence: a Survey

This survey explores the integration of learning and reasoning in two different fields of artificial intelligence: neurosymbolic and statistical relational artificial intelligence. Neurosymbolic artificial intelligence (NeSy) studies the integration of symbolic reasoning and neural networks, while statistical relational artificial intelligence (StarAI) focuses on integrating logic with probabilistic graphical models. This survey identifies seven shared dimensions between these two subfields of AI. These dimensions can be used to characterize different NeSy and StarAI systems. They are concerned with (1) the approach to logical inference, whether model or proof-based; (2) the syntax of the used logical theories; (3) the logical semantics of the systems and their extensions to facilitate learning; (4) the scope of learning, encompassing either parameter or structure learning; (5) the presence of symbolic and subsymbolic representations; (6) the degree to which systems capture the original logic, probabilistic, and neural paradigms; and (7) the classes of learning tasks the systems are applied to. By positioning various NeSy and StarAI systems along these dimensions and pointing out similarities and differences between them, this survey contributes fundamental concepts for understanding the integration of learning and reasoning.Comment: To appear in Artificial Intelligence. Shorter version at IJCAI 2020 survey track, https://www.ijcai.org/proceedings/2020/0688.pd