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.
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.
Quantitative Concept Analysis
Formal Concept Analysis (FCA) begins from a context, given as a binary relation between some objects and some attributes, and derives a lattice of concepts, where each concept is given as a set of objects and a set of attributes, such that the first set consists of all objects that satisfy all attributes in the second, and vice versa. Many applications, though, provide contexts with quantitative information, telling not just whether an object satisfies an attribute, but also quantifying this satisfaction. Contexts in this form arise as rating matrices in recommender systems, as occurrence matrices in text analysis, as pixel intensity matrices in digital image processing, etc. Such applications have attracted a lot of attention, and several numeric extensions of FCA have been proposed. We propose the framework of proximity sets (proxets), which subsume partially ordered sets (posets) as well as metric spaces. One feature of this approach is that it extracts from quantified contexts quantified concepts, and thus allows full use of the available information. Another feature is that the categorical approach allows analyzing any universal properties that the classical FCA and the new versions may have, and thus provides structural guidance for aligning and combining the approaches.
Quantifying and qualifying trust: Spectral decomposition of trust networks
In a previous FAST paper, I presented a quantitative model of the process of trust building, and showed that trust is accumulated like wealth: the rich get richer. This explained the pervasive phenomenon of adverse selection of trust certificates, as well as the fragility of trust networks in general. But a simple explanation does not always suggest a simple solution. It turns out that it is impossible to alter the fragile distribution of trust without sacrificing some of its fundamental functions. A solution for the vulnerability of trust must thus be sought elsewhere, without tampering with its distribution. This observation was the starting point of the present paper. It explores a different method for securing trust: not by redistributing it, but by mining for its sources. The method used to break privacy is thus also used to secure trust. A high level view of the mining methods that connect the two is provided in terms of similarity networks, and spectral decomposition of similarity preserving maps. This view may be of independent interest, as it uncovers a common conceptual and structural foundation of mathematical classification theory on one hand, and of the spectral methods of graph clustering and data mining on the other hand.
A semantical approach to equilibria and rationality
Game theoretic equilibria are mathematical expressions of rationality. Rational agents are used to model not only humans and their software representatives, but also organisms, populations, species and genes, interacting with each other and with the environment. Rational behaviors are achieved not only through conscious reasoning, but also through spontaneous stabilization at equilibrium points.
Formal theories of rationality are usually guided by informal intuitions, which are acquired by observing some concrete economic, biological, or network processes. Treating such processes as instances of computation, we reconstruct and refine some basic notions of equilibrium and rationality from the some basic structures of computation.
It is, of course, well known that equilibria arise as fixed points; the point is that semantics of computation of fixed points seems to be providing novel methods, algebraic and coalgebraic, for reasoning about them.
Dynamics, robustness and fragility of trust
Trust is often conveyed through delegation, or through recommendation. This makes the trust authorities, who process and publish trust recommendations, into an attractive target for attacks and spoofing. In some recent empiric studies, this was shown to lead to a remarkable phenomenon of adverse selection: a greater percentage of unreliable or malicious web merchants were found among those with certain types of trust certificates, then among those without. While such findings can be attributed to a lack of diligence in trust authorities, or even to conflicts of interest, our analysis of trust dynamics suggests that the public trust networks would probably remain vulnerable even if the trust authorities were perfectly diligent. The reason is that the main processes of trust building, under reasonable modeling assumptions, naturally lead to power-law distributions of trust: the rich get richer, the trusted attract more trust. The evolutionary processes governed by such distributions, ubiquitous in nature, are known to be robust with respect to random failures, but vulnerable to adaptive attacks. We recommend some ways to decrease this vulnerability, and suggest some ideas for exploration.
Network as a computer: ranking paths to find flows
We explore a simple mathematical model of network computation, based on Markov chains. Similar models apply to a broad range of computational phenomena, arising in networks of computers, as well as in genetic, and neural nets, in social networks, and so on. The main problem of interaction with such spontaneously evolving computational systems is that the data are not uniformly structured. An interesting approach is to try to extract the semantical content of the data from their distribution among the nodes. A concept could then be identified by finding the community of nodes that share it. The task of data structuring is thus reduced to the task of finding the network communities, as groups of nodes that together perform some non-local data processing. Towards this goal, we extend the ranking methods from nodes to paths. This allows us to extract some information about the likely flow biases from the available static information about the network.
On quantum statistics in data analysis
Originally, quantum probability theory was developed to analyze statistical phenomena in quantum systems, where classical probability theory does not apply, because the lattice of measurable sets is not necessarily distributive. On the other hand, it is well known that the lattices of concepts, that arise in data analysis, are in general also non-distributive, albeit for completely different reasons. In his recent book, van Rijsbergen (2004) argues that many of the logical tools developed for quantum systems are also suitable for applications in information retrieval. I explore the mathematical support for this idea on an abstract vector space model, covering several forms of data analysis (information retrieval, data mining, collaborative filtering, formal concept analysis…), and roughly based on an idea from categorical quantum mechanics (Abramsky & Coecke 2004; Coecke & Pavlovic 2007). It turns out that quantum (i.e., noncommutative) probability distributions arise already in this rudimentary mathematical framework. We show that a Bell-type inequality (Bell 1964) must be satisfied by the standard similarity measures, if they are used for preference predictions. The fact that already a very general, abstract version of the vector space model yields simple counterexamples for such inequalities seems to be an indicator of a genuine need for quantum statistics in data analysis.