International Workshop Logic Matters (LM-2021)

Formal Concept Analysis (FCA) mines concepts from relational and posetal contexts, capturing the correlations between some given sets of objects and attributes. Latent Semantic Analysis (LSA) mines concepts from numeric contexts, capturing cumulative correlations as data matrices, and processing them by linear operations. The family of algorithms used in recommender systems behind social networks, news aggregators, and web-commerce marketplaces are largely based on linear algebra. The linearity tacitly imposes the assumption that the dominant concepts are independent. The existing correlations thus lead to amplifying correlated concepts. The feedback caused by the amplified recommendations accounts for at least some instances of the pernicious phenomenon of "echo chambers", which has been deluding network communities and hampering network societies. Tracking the concept dependencies through data amounts to refining the linear operations on matrices of numbers to functional operations on matrices of sets. The matrix completions leading to the concept lattices of FCA and the concept spaces of LSA should then lift to distributor completions in a putative Categorical Concept Analysis. This turns out to be an interesting technical task. I will provide an account of the tight bicompletion process extracting from arbitrary distributors ("matrices of sets") the concept categories, as common generalizations of concept lattices and concept spaces.