Bicompletions of distance matrices

In the practice of information extraction, the input data are usually arranged into pattern matrices, and analyzed by the methods of linear algebra and statistics, such as principal component analysis. In some applications, the tacit assumptions of these methods lead to wrong results. The usual reason is that the matrix composition of linear algebra presents information as flowing in waves, whereas it sometimes flows in particles, which seek the shortest paths. This wave-particle duality in computation and information processing has been originally observed by Abramsky. In this paper we pursue a particle view of information, formalized in *distance spaces*, which generalize metric spaces, but are slightly less general than Lawvere's *generalized metric spaces*. In this framework, the task of extracting the 'principal components' from a given matrix of data boils down to a bicompletio}, in the sense of enriched category theory. We describe the bicompletion construction for distance matrices. The practical goal that motivates this research is to develop a method to estimate the hardness of attack constructions in security.Comment: 20 pages, 5 figures; appeared in Springer LNCS vol 7860 in 2013; v2 fixes an error in Sec. 2.3, noticed by Toshiki Kataok

Towards concept analysis in categories: limit inferior as algebra, limit superior as coalgebra

While computer programs and logical theories begin by declaring the concepts of interest, be it as data types or as predicates, network computation does not allow such global declarations, and requires *concept mining* and *concept analysis* to extract shared semantics for different network nodes. Powerful semantic analysis systems have been the drivers of nearly all paradigm shifts on the web. In categorical terms, most of them can be described as bicompletions of enriched matrices, generalizing the Dedekind-MacNeille-style completions from posets to suitably enriched categories. Yet it has been well known for more than 40 years that ordinary categories themselves in general do not permit such completions. Armed with this new semantical view of Dedekind-MacNeille completions, and of matrix bicompletions, we take another look at this ancient mystery. It turns out that simple categorical versions of the *limit superior* and *limit inferior* operations characterize a general notion of Dedekind-MacNeille completion, that seems to be appropriate for ordinary categories, and boils down to the more familiar enriched versions when the limits inferior and superior coincide. This explains away the apparent gap among the completions of ordinary categories, and broadens the path towards categorical concept mining and analysis, opened in previous work.Comment: 22 pages, 5 figures and 9 diagram

On the fuzzy concept complex

Every relation between posets gives rise to an adjunction, known as a Galois connection, between the corresponding power sets. Formal concept analysis (FCA) studies the fixed points of these adjunctions, which can be interpreted as latent “concepts” [20], [19]. In [47] Pavlovic defines a generalisation of posets he calls proximity sets (or proxets), which are equivalent to the generalised metric spaces of Lawvere [37], and introduces a form of quantitative concept analysis (QCA) which provides a different viewpoint from other approaches to fuzzy concept analysis (for a survey see [4]). The nucleus of a fuzzy relation between proxets is defined in terms of the fixed points of a naturally arising adjunction based on the given relation, generalising the Galois connections of formal concept analysis. By giving the unit interval [0, 1] an appropriate category structure it can be shown that proxets are simply [0, 1]-enriched categories and the nuclues of a proximity relation between proxets is a generalisation of the notion of the Isbell completion of an enriched category. We prove that the sets of fixed points of an adjunction arising from a fuzzy relation can be given the structure of complete idempotent semimodules and show that they are isomorphic to tropical convex hulls of point configurations in tropical projective space, in which addition and scalar multiplication are replaced with pointwise minima and addition, respectively. We show that some the results of Develin and Sturmfels on tropical convex sets [13] can be applied to give the nucleus of a proximity relation the structure of a cell complex, which we term the fuzzy concept complex. We provide a formula for counting cells of a given dimension in generic situations. We conclude with some thoughts on computing the fuzzy concept complex using ideas from Ardila and Develin’s work on tropical oriented matroids [1]

Bicompletions of Distance Matrices

In the practice of information extraction, the input data are usually arranged into pattern matrices, and analyzed by the methods of linear algebra and statistics, such as principal component analysis. In some applications, the tacit assumptions of these methods lead to wrong results. The usual reason is that the matrix composition of linear algebra presents information as flowing in waves, whereas it sometimes flows in particles, which seek the shortest paths. This wave-particle duality in computation and information processing has been originally observed by Abramsky. In this paper we pursue a particle view of information, formalized in distance spaces, which generalize metric spaces, but are slightly less general than Lawvere’s generalized metric spaces. In this framework, the task of extracting the ‘principal components’ from a given matrix of data boils down to a bicompletion, in the sense of enriched category theory. We describe the bicompletion construction for distance matrices. The practical goal that motivates this research is to develop a method to estimate the hardness of attack constructions in security