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Messungen an Quantenobjekten haben eine logische Struktur. Diese schließt die klassische Logik ein, geht jedoch weit darüber hinaus. Das wesentliche Ziel dieses Lehrbuchs ist es, die mathematischen Werkzeuge der Quantenlogik zu vermitteln und technische Anwendungsmöglichkeiten aufzuzeigen. Die dafür erforderlichen mathematischen Sachverhalte werden anhand von Beispielen so erläutert und motiviert, dass sie für angehende Ingenieure und Informatiker verständlich sind. Die vorliegende Auflage wurde korrigiert und um zahlreiche Anwendungen, zusätzliche Beispiele und Beweise erweitert sowie um ein neues Beispiel zur Mustererkennung ergänzt.
This article provides a detailed explanation of the BBQ-Tree, a unified logic-based model that integrates both classical Decision Trees and Quantum-Logic Decision Trees into a generalized framework for classification and regression. As it combines these paradigms, the BBQ-Tree effectively addresses problems with both linear and curved decision boundaries while prioritizing interpretability. We provide a detailed description of the underlying concepts, a possible training algorithm, experimental evaluations and the incorporation of regression functionality, broadening its applicability beyond classification tasks. Strategies for efficient training and model optimization are also presented. Experimental results demonstrate that the BBQ-Tree produces compact, interpretable models capable of revealing data trends, while achieving accuracy comparable to Decision Trees. Furthermore, its new regression capabilities highlight its versatility and performance across a wider range of tasks.
Reducing feature dimensionality in spectroscopic data is crucial for efficient analysis and classification. Using all available features for classification typically results in an unacceptably high runtime and poor accuracy. Popular feature extraction methods, such as principal component analysis (PCA), linear discriminant analysis (LDA), and autoencoders, reduce feature dimensionality by extracting latent features that can be challenging to interpret. To enable better human interpretation of the classification model, we avoid extraction methods and instead propose applying feature selection methods. In this work, we develop an innovative PCA-based feature selection method for spectroscopic data, providing an essential subset of the original features. As an important advantage, no prior knowledge about the characteristic signals of the respective target substance is required. In this proof-of-concept study, the proposed method is initially characterized using simulated Raman and infrared absorption datasets. From the top five PCA eigenvectors of spectroscopic data, we identify a set of three top peaks each at specific wavenumbers (features). The compact set of selected features is then used for classification tasks applying a decision tree. Based on two well-defined spectroscopic datasets, our study demonstrates that our new method of PCA-based peak finding outperforms selected other approaches with regard to interpretability and accuracy. For both investigated datasets, accuracies greater than 97% are achieved. Our approach shows large potential for accurate classification combined with interpretability in further scenarios involving spectroscopic datasets.
Consider a binary classification problem solved using a feed-forward artificial neural network (ANN). Let the ANN be composed of a ReLU layer and several linear layers (convolution, sum-pooling, or fully connected). We assume the network was trained with high accuracy. Despite numerous suggested approaches, interpreting an artificial neural network remains challenging for humans. For a new method of interpretation, we construct a bridge between a simple ANN and logic. As a result, we can analyze and manipulate the semantics of an ANN using the powerful tool set of logic. To achieve this, we decompose the input space of the ANN into several network partition cells. Each network partition cell represents a linear combination that maps input values to a classifying output value. For interpreting the linear map of a partition cell using logic expressions, we suggest minterm values as the input of a simple ANN. We derive logic expressions representing interaction patterns for separating objects classified as 1 from those classified as 0. To facilitate an interpretation of logic expressions, we present them as binary logic trees.
This study proposes the BBQ-Tree, a new logic-based classifier that combines the two concepts of classical Decision Trees and Quantum-Logic Decision Trees into a generalized model. It thus creates a method that has the power to solve classification problems that incorporate both curved and linear decision boundaries, with a particular focus on interpretability. In addition to the model itself, ways for its efficient training are discussed. Our experimental evaluation demonstrates that our approach is able to produce models that remain compact and provide good insights over trends in data while maintaining an accuracy not worse than Decision Trees alone.
Das wesentliche Ziel dieses Lehrbuchs ist es, die mathematischen Werkzeuge zur Modellierung kognitiver Strukturen und Prozesse auf der Grundlage der klassischen Logik und der Quantenlogik zu entwickeln. Die dafür erforderlichen mathematischen Sachverhalte werden so dargestellt und anhand von Beispielen motiviert, dass sie für angehende Ingenieure und Informatiker verständlich sind.
- Kompakte logische Darstellung der mathematischen Werkzeuge zur Modellierung kognitiver Strukturen und Prozesse
- Beispiele erläutern die Anwendung im Engineering
Zielgruppen sind insbesondere Studierende der Ingenieurwissenschaften und der Informatik, aber auch Studierende der Mathematik oder der Physik können durch den anwendungsbezogenen Blick ihren Horizont erweitern.
The quantum-logic inspired decision tree (QLDT) is based on quantum logic concepts and input values from the unit interval whereas the traditional decision tree is based on Boolean values. The logic behind the QLDT obeys the rules of a Boolean algebra. The QLDT is appropriate for classification problems where for a class decision several input values interact gradually with each other. The QLDT construction for a classification problem with n input attributes requires the computation of 2n minterms. The QLDT+ method, however, uses a heuristic for obtaining a QLDT with much smaller computational complexity. As result, the QLDT+ method can be applied to classification problems with a higher number of input attributes.
Homogeneity of persons in a social network is based on the
similarity of their attributes. Traditional clustering algorithms like
hierarchical (agglomerative) clustering or DBSCAN take distances
between objects as input and find clusters of objects. Distance
functions need to satisfy the triangle inequality (TI) property, but
sometimes TI is violated and, in addition, not all attributes do have
the same influence on the network and thus may affect the network and
compromise the quality of resulting clusters. We present an adaptive
clustering-based quantitative weighting approach that is completely
embedded in logic. To facilitate the user interaction with the system,
we exploit the concept of relevance feedback. The approach takes user
feedback as input to improve the quality of clusters and finds
meaningful clusters where TI does not hold. In addition, it has the
capability of providing the user alternative possible feedbacks that
can be fulfilled. To test the approach, we evaluate a clustering
distance regarding an ideal solution. Experiments demonstrate the
benefits of our approach.
Besides a good prediction a classifier is to give an explanation how the input data is related to the classification result. There is a general agreement that logic expressions provide a better explanation than other methods like SVM, logistic regression, and neural networks. However, a classifier based on Boolean logic needs to map continuous data to Boolean values which can cause a loss of information. In contrast, we design a quantum-logic-inspired classifier where continuous data are directly processed and the laws of the Boolean algebra are maintained. As a result from our approach we obtain a CQQL condition which provides good insights into the relation of input features to the class decision. Furthermore, our experiment shows a good prediction accuracy.
Besides a good prediction a classifier is to give an explanation how input data is related to the classification result. Decision trees are very popular classifiers and provide a good trade-off between accuracy and explainability for many scenarios. Its split decisions correspond to Boolean conditions on single attributes. In cases when for a class decision several attribute values interact gradually with each other, Boolean-logic-based decision trees are not appropriate. For such cases we propose a quantum-logic inspired decision tree (QLDT) which is based on sums and products on normalized attribute values. In contrast to decision trees based on fuzzy logic a QLDT obeys the rules of the Boolean algebra.
