Related papers: Scaling Dimension
We present a novel approach for data set scaling based on scale-measures from formal concept analysis, i.e., continuous maps between closure systems, and derive a canonical representation. Moreover, we prove said scale-measures are lattice…
The notion of concept has been studied for centuries, by philosophers, linguists, cognitive scientists, and researchers in artificial intelligence (Margolis & Laurence, 1999). There is a large literature on formal, mathematical models of…
Dimension reduction of data sets is a standard problem in the realm of machine learning and knowledge reasoning. They affect patterns in and dependencies on data dimensions and ultimately influence any decision-making processes. Therefore,…
Lattices and their order diagrams are an essential tool for communicating knowledge and insights about data. This is in particular true when applying Formal Concept Analysis. Such representations, however, are difficult to comprehend by…
Formal concept analysis has grown from a new branch of the mathematical field of lattice theory to a widely recognized tool in Computer Science and elsewhere. In order to fully benefit from this theory, we believe that it can be enriched…
Lattices are a commonly used structure for the representation and analysis of relational and ontological knowledge. In particular, the analysis of these requires a decomposition of a large and high-dimensional lattice into a set of…
Measurement is a fundamental building block of numerous scientific models and their creation. This is in particular true for data driven science. Due to the high complexity and size of modern data sets, the necessity for the development of…
In the 1960's, four famous scaling relations were developed which relate the six standard critical exponents describing continuous phase transitions in the thermodynamic limit of statistical physics models. They are well understood at a…
Formal definitions of quantities, quantity spaces, dimensions and dimension groups are introduced. Based on these concepts, a theoretical framework and a practical algorithm for dimensional analysis are developed, and examples of…
Formal Concept Analysis (FCA) allows to analyze binary data by deriving concepts and ordering them in lattices. One of the main goals of FCA is to enable humans to comprehend the information that is encapsulated in the data; however, the…
Formal Concept Analysis has proven to be an effective method of restructuring complete lattices and various algebraic domains. In this paper, the notions of attribute continuous formal context and continuous formal concept are introduced by…
Concept lattices are well-known conceptual structures that organise interesting patterns-the concepts-extracted from data. In some applications, such as software engineering or data mining, the size of the lattice can be a problem, as it is…
Embedding spaces contain interpretable dimensions indicating gender, formality in style, or even object properties. This has been observed multiple times. Such interpretable dimensions are becoming valuable tools in different areas of…
Formal Concept Analysis (FCA) offers several tools for qualitative data analysis. One possibility is to group objects that share common attributes together and get a concept lattice that describes the data. Quite often the size of this…
Formal Concept Analysis (FCA) is a mathematical theory based on the formalization of the notions of concept and concept hierarchies. It has been successfully applied to several Computer Science fields such as data mining,software…
Model complexity is an important factor to consider when selecting among graphical models. When all variables are observed, the complexity of a model can be measured by its standard dimension, i.e. the number of independent parameters. When…
The Ordered Set Theory is a branch of Mathematics that studies partially ordered sets (usually posets) and lattices. The meaning of dimension is one of the main parts of this eld. Dimensions of partially ordered sets and lattices have been…
With the growing popularity of general-purpose Large Language Models (LLMs), comes a need for more global explanations of model behaviors. Concept-based explanations arise as a promising avenue for explaining high-level patterns learned by…
Large Language Models (LLMs) have benefited enormously from scaling, yet these gains are bounded by five fundamental limitations: (1) hallucination, (2) context compression, (3) reasoning degradation, (4) retrieval fragility, and (5)…
When considering perceptions, the observation scale and resolution are closely related properties. There is consensus in considering resolution as the density of elementary pieces of information in a specified information space.…