Characterization of image sets: the Galois Lattice approach

This paper presents a new method for supervised image classification. One or several landmarks are attached to each class, with the intention of characterizing it and discriminating it from the other classes. The different features, deduced from image primitives, and their relationships with the sets of images are structured and organized into a hierarchy thanks to an original method relying on a mathematical formalism called Galois (or Concept) Lattices. Such lattices allow us to select features as landmarks of specific classes. This paper details the feature selection process and illustrates this through a robotic example in a structured environment. The class of any image is the room from which the image is shot by the robot camera. In the discussion, we compare this approach with decision trees and we give some issues for future research

Modelling lexical databases with formal concept analysis.

This paper provides guidelines and examples for visualising lexical relations using Formal Concept Analysis. Relations in lexical databases often form trees, imperfect trees or poly-hierarchies which can be embedded into concept lattices. Many-to-many relations can be represented as concept lattices where the values from one domain are used as the formal objects and the values of the other domain as formal attributes. This paper further discusses algorithms for selecting meaningful subsets of lexical databases, the representation of complex relational structures in lexical databases and the use of lattices as basemaps for other lexical relations

An extension of Tamari lattices

For any finite path $v$ on the square grid consisting of north and east unit steps, starting at (0,0), we construct a poset Tam$(v)$ that consists of all the paths weakly above $v$ with the same number of north and east steps as $v$. For particular choices of $v$, we recover the traditional Tamari lattice and the $m$-Tamari lattice. Let $\overleftarrow{v}$ be the path obtained from $v$ by reading the unit steps of $v$ in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam$(v)$ is isomorphic to the dual of the poset Tam$(\overleftarrow{v})$. We do so by showing bijectively that the poset Tam$(v)$ is isomorphic to the poset based on rotation of full binary trees with the fixed canopy $v$, from which the duality follows easily. This also shows that Tam$(v)$ is a lattice for any path $v$. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height $n$, is a partition of the (smaller) lattices Tam$(v)$, where the $v$ are all the paths on the square grid that consist of $n-1$ unit steps. We explain possible connections between the poset Tam$(v)$ and (the combinatorics of) the generalized diagonal coinvariant spaces of the symmetric group.Comment: 18 page

Steps Towards Achieving Distributivity in Formal Concept Analysis

International audienceIn this paper we study distributive lattices in the framework of Formal Concept Analysis (FCA). The main motivation comes from phylogeny where biological derivations and parsimonious trees can be represented as median graphs. There exists a close connection between distributive lattices and median graphs. Moreover, FCA provides efficient algorithms to build concept lattices. However, a concept lattice is not necessarily distributive and thus it is not necessarily a median graph.In this paper we investigate possible ways of transforming a concept lattice into a distributive one, by making use Birkhoff’s representation of distributive lattices. We detail the operation that transforms a reduced context into a context of a distributive lattice. This allows us to reuse the FCA algorithmic machinery to build and to visualize distributive concept lattices, and then to study the associated median graphs

Continuous-time quantum walk on integer lattices and homogeneous trees

This paper is concerned with the continuous-time quantum walk on Z, Z^d, and infinite homogeneous trees. By using the generating function method, we compute the limit of the average probability distribution for the general isotropic walk on Z, and for nearest-neighbor walks on Z^d and infinite homogeneous trees. In addition, we compute the asymptotic approximation for the probability of the return to zero at time t in all these cases.Comment: The journal version (save for formatting); 19 page

Percolation on hyperbolic lattices

The percolation transitions on hyperbolic lattices are investigated numerically using finite-size scaling methods. The existence of two distinct percolation thresholds is verified. At the lower threshold, an unbounded cluster appears and reaches from the middle to the boundary. This transition is of the same type and has the same finite-size scaling properties as the corresponding transition for the Cayley tree. At the upper threshold, on the other hand, a single unbounded cluster forms which overwhelms all the others and occupies a finite fraction of the volume as well as of the boundary connections. The finite-size scaling properties for this upper threshold are different from those of the Cayley tree and two of the critical exponents are obtained. The results suggest that the percolation transition for the hyperbolic lattices forms a universality class of its own.Comment: 17 pages, 18 figures, to appear in Phys. Rev.