16 research outputs found
FCA2VEC: Embedding Techniques for Formal Concept Analysis
Embedding large and high dimensional data into low dimensional vector spaces
is a necessary task to computationally cope with contemporary data sets.
Superseding latent semantic analysis recent approaches like word2vec or
node2vec are well established tools in this realm. In the present paper we add
to this line of research by introducing fca2vec, a family of embedding
techniques for formal concept analysis (FCA). Our investigation contributes to
two distinct lines of research. First, we enable the application of FCA notions
to large data sets. In particular, we demonstrate how the cover relation of a
concept lattice can be retrieved from a computational feasible embedding.
Secondly, we show an enhancement for the classical node2vec approach in low
dimension. For both directions the overall constraint of FCA of explainable
results is preserved. We evaluate our novel procedures by computing fca2vec on
different data sets like, wiki44 (a dense part of the Wikidata knowledge
graph), the Mushroom data set and a publication network derived from the FCA
community.Comment: 25 page
Discovering Implicational Knowledge in Wikidata
Knowledge graphs have recently become the state-of-the-art tool for
representing the diverse and complex knowledge of the world. Examples include
the proprietary knowledge graphs of companies such as Google, Facebook, IBM, or
Microsoft, but also freely available ones such as YAGO, DBpedia, and Wikidata.
A distinguishing feature of Wikidata is that the knowledge is collaboratively
edited and curated. While this greatly enhances the scope of Wikidata, it also
makes it impossible for a single individual to grasp complex connections
between properties or understand the global impact of edits in the graph. We
apply Formal Concept Analysis to efficiently identify comprehensible
implications that are implicitly present in the data. Although the complex
structure of data modelling in Wikidata is not amenable to a direct approach,
we overcome this limitation by extracting contextual representations of parts
of Wikidata in a systematic fashion. We demonstrate the practical feasibility
of our approach through several experiments and show that the results may lead
to the discovery of interesting implicational knowledge. Besides providing a
method for obtaining large real-world data sets for FCA, we sketch potential
applications in offering semantic assistance for editing and curating Wikidata
Counting Proper Mergings of Chains and Antichains
A proper merging of two disjoint quasi-ordered sets and is a
quasi-order on the union of and such that the restriction to and
yields the original quasi-order again and such that no elements of and
are identified. In this article, we consider the cases where and
are chains, where and are antichains, and where is an antichain and
is a chain. We give formulas that determine the number of proper mergings
in all three cases, and introduce two new bijections from proper mergings of
two chains to plane partitions and from proper mergings of an antichain and a
chain to monotone colorings of complete bipartite digraphs. Additionally, we
use these bijections to count the Galois connections between two chains, and
between a chain and a Boolean lattice respectively.Comment: 36 pages, 15 figures, 5 table
Scaling Dimension
Conceptual Scaling is a useful standard tool in Formal Concept Analysis and
beyond. Its mathematical theory, as elaborated in the last chapter of the FCA
monograph, still has room for improvement. As it stands, even some of the basic
definitions are in flux. Our contribution was triggered by the study of concept
lattices for tree classifiers and the scaling methods used there. We extend
some basic notions, give precise mathematical definitions for them and
introduce the concept of scaling dimension. In addition to a detailed
discussion of its properties, including an example, we show theoretical bounds
related to the order dimension of concept lattices. We also study special
subclasses, such as the ordinal and the interordinal scaling dimensions, and
show for them first results and examples
Formal Context Generation using Dirichlet Distributions
We suggest an improved way to randomly generate formal contexts based on
Dirichlet distributions. For this purpose we investigate the predominant way to
generate formal contexts, a coin-tossing model, recapitulate some of its
shortcomings and examine its stochastic model. Building up on this we propose
our Dirichlet model and develop an algorithm employing this idea. By comparing
our generation model to a coin-tossing model we show that our approach is a
significant improvement with respect to the variety of contexts generated.
Finally, we outline a possible application in null model generation for formal
contexts.Comment: 16 pages, 7 figure
Drawing Order Diagrams Through Two-Dimension Extension
Order diagrams are an important tool to visualize the complex structure of
ordered sets. Favorable drawings of order diagrams, i.e., easily readable for
humans, are hard to come by, even for small ordered sets. Many attempts were
made to transfer classical graph drawing approaches to order diagrams. Although
these methods produce satisfying results for some ordered sets, they
unfortunately perform poorly in general. In this work we present the novel
algorithm DimDraw to draw order diagrams. This algorithm is based on a relation
between the dimension of an ordered set and the bipartiteness of a
corresponding graph.Comment: 16 pages, 12 Figure
Proceedings of the International Workshop "What can FCA do for Artificial Intelligence?" (FCA4AI 2014)
International audienceThis is the third edition of the FCA4AI workshop, whose first edition was organized at ECAI 2012 Conference (Montpellier, August 2012) and second edition was organized at IJCAI 2013 Conference (Beijing, August 2013, see http://www.fca4ai.hse.ru/). Formal Concept Analysis (FCA) is a mathematically well-founded theory aimed at data analysis and classification that can be used for many purposes, especially for Artificial Intelligence (AI) needs. The objective of the workshop is to investigate two main main issues: how can FCA support various AI activities (knowledge discovery, knowledge representation and reasoning, learning, data mining, NLP, information retrieval), and how can FCA be extended in order to help AI researchers to solve new and complex problems in their domain
Towards Ordinal Data Science
Order is one of the main instruments to measure the relationship between objects in (empirical) data. However, compared to methods that use numerical properties of objects, the amount of ordinal methods developed is rather small. One reason for this is the limited availability of computational resources in the last century that would have been required for ordinal computations. Another reason - particularly important for this line of research - is that order-based methods are often seen as too mathematically rigorous for applying them to real-world data. In this paper, we will therefore discuss different means for measuring and ‘calculating’ with ordinal structures - a specific class of directed graphs - and show how to infer knowledge from them. Our aim is to establish Ordinal Data Science as a fundamentally new research agenda. Besides cross-fertilization with other cornerstone machine learning and knowledge representation methods, a broad range of disciplines will benefit from this endeavor, including, psychology, sociology, economics, web science, knowledge engineering, scientometrics