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Giacomo Kahn

Publications and source records attributed to Giacomo Kahn.

8 recordsLinked to original sources
arXiv2022

Some Notes on Polyadic Concept Analysis

Despite the popularity of Formal Concept Analysis (FCA) as a mathematical framework for data analysis, some of its extensions are still considered arcane. Polyadic Concept Analysis (PCA) is one of the most promising yet understudied of these extensions. This formalism offers many interesting open questions but is hindered in its dissemination by complex notations and a lack of agreed-upon basic definitions. In this paper, we discuss in a mostly informal way the fundamental differences between FCA and PCA in the relation between contexts, conceptual structures, and rules. We identify open questions, present partial results on the maximal size of concept n-lattices and suggest new research directions.

cs.DM↗
arXiv2021

Subgraph Signal Processing

Graph signal processing, like the graph Fourier transform, requires the full graph signal at every vertex of the graph. However, in practice, only signals at a subset of vertices may be available. We propose a subgraph signal processing framework that relates a graph shift operator or filter on a subgraph with a filter on the ambient graph through an operator loss. It allows us to define shift operators for the subgraph signal, which has a meaningful interpretation and relation to mixtures of shift invariant filters restricted to different subgraphs of the ambient graph (which we call semi shift invariant). This leads to a notion of frequency domain for the subgraph signal consistent in some sense with that of the full graph signal. We apply the subgraph signal processing machinery to several applications and demonstrate the utility of this framework in cases where only partial graph signals are observed.

eess.SP↗
arXiv2020

Signal processing on simplicial complexes

Theoretical development and applications of graph signal processing (GSP) have attracted much attention. In classical GSP, the underlying structures are restricted in terms of dimensionality. A graph is a combinatorial object that models binary relations, and it does not directly model complex n-ary relations. One possible high dimensional generalization of graphs are simplicial complexes. They are a step between the constrained case of graphs and the general case of hypergraphs. In this paper, we develop a signal processing framework on simplicial complexes, such that we recover the traditional GSP theory when restricted to signals on graphs. It is worth mentioning that the framework works much more generally, though the focus of the paper is on simplicial complexes. We demonstrate how to perform signal processing with the framework using numerical examples.

eess.SP↗
arXiv2019

Bisplit graphs satisfy the Chen-Chvátal conjecture

In this paper, we give a lengthy proof of a small result! A graph is bisplit if its vertex set can be partitioned into three stable sets with two of them inducing a complete bipartite graph. We prove that these graphs satisfy the Chen-Chvátal conjecture: their metric space (in the usual sense) has a universal line (in an unusual sense) or at least as many lines as the number of vertices.

cs.DM↗
arXiv2018

On-demand Relational Concept Analysis

Formal Concept Analysis and its associated conceptual structures have been used to support exploratory search through conceptual navigation. Relational Concept Analysis (RCA) is an extension of Formal Concept Analysis to process relational datasets. RCA and its multiple interconnected structures represent good candidates to support exploratory search in relational datasets, as they are enabling navigation within a structure as well as between the connected structures. However, building the entire structures does not present an efficient solution to explore a small localised area of the dataset, for instance to retrieve the closest alternatives to a given query. In these cases, generating only a concept and its neighbour concepts at each navigation step appears as a less costly alternative. In this paper, we propose an algorithm to compute a concept and its neighbourhood in extended concept lattices. The concepts are generated directly from the relational context family, and possess both formal and relational attributes. The algorithm takes into account two RCA scaling operators. We illustrate it on an example.

cs.DB↗
arXiv2018

Introducer Concepts in n-Dimensional Contexts

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 often too large to be efficiently computed, and too complex to be browsed. For this reason, the Galois Sub-Hierarchy, a restriction of the concept lattice to introducer concepts, has been introduced as a smaller alternative. In this paper, we generalise the Galois Sub-Hierarchy to n-lattices, conceptual structures obtained from multidimensional data in the same way that concept lattices are obtained from binary relations.

cs.AI↗
arXiv2018

Average Size of Implicational Bases

Implicational bases are objects of interest in formal concept analysis and its applications. Unfortunately, even the smallest base, the Duquenne-Guigues base, has an exponential size in the worst case. In this paper, we use results on the average number of minimal transversals in random hypergraphs to show that the base of proper premises is, on average, of quasi-polynomial size.

cs.AI↗