Portail HAL de l'Institut Polytechnique de Paris
Not a member yet
28935 research outputs found
Sort by
The facial weak order in finite Coxeter groups
No full textInternational audienceWe investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W. We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bjo ̈rner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of its classes
Symmetric Chain Decompositions and the Strong Sperner Property for Noncrossing Partition Lattices
No full textInternational audienceWe prove that the noncrossing partition lattices associated with the complex reflection groups G(d, d, n) for d, n ≥ 2 admit a decomposition into saturated chains that are symmetric about the middle ranks. A consequence of this result is that these lattices have the strong Sperner property, which asserts that the cardinality of the union of the k largest antichains does not exceed the sum of the k largest ranks for all k ≤ n. Subsequently, we use a computer to complete the proof that any noncrossing partition lattice associated with a well-generated complex reflection group is strongly Sperner, thus affirmatively answering a special case of a question of D. Armstrong. This was previously established only for the Coxeter groups of type A and B
The number of corner polyhedra graphs
No full textInternational audienceCorner polyhedra were introduced by Eppstein and Mumford (2014) as the set of simply connected 3D polyhedra such that all vertices have non negative integer coordinates, edges are parallel to the coordinate axes and all vertices but one can be seen from infinity in the direction (1, 1, 1). These authors gave a remarkable characterization of the set of corner polyhedra graphs, that is graphs that can be skeleton of a corner polyhedron: as planar maps, they are the duals of some particular bipartite triangulations, which we call hereafter corner triangulations.In this paper we count corner polyhedral graphs by determining the generating function of the corner triangulations with respect to the number of vertices: we obtain an explicit rational expression for it in terms of the Catalan gen- erating function. We first show that this result can be derived using Tutte's classical compositional approach. Then, in order to explain the occurrence of the Catalan series we give a direct algebraic decomposition of corner triangu- lations: in particular we exhibit a family of almond triangulations that admit a recursive decomposition structurally equivalent to the decomposition of binary trees. Finally we sketch a direct bijection between binary trees and almond triangulations. Our combinatorial analysis yields a simpler alternative to the algorithm of Eppstein and Mumford for endowing a corner polyhedral graph with the cycle cover structure needed to realize it as a polyhedral graph
Interpreting deep glucose predictive models for diabetic people using RETAIN
No full textInternational audienceProgress in the biomedical field through the use of deep learning is hindered by the lack of interpretability of the models. In this paper, we study the RETAIN architecture for the forecasting of future glucose values for diabetic people. Thanks to its two-level attention mechanism, the RETAIN model is interpretable while remaining as efficient as standard neural networks.We evaluate the model on a real-world type-2 diabetic population and we compare it to a random forest model and a LSTM-based recurrent neural network. Our results show that the RETAIN model outperforms the former and equals the latter on common accuracy metrics and clinical acceptability metrics, thereby proving its legitimacy in the context of glucose level forecasting. Furthermore, we propose tools to take advantage of the RETAIN interpretable nature. As informative for the patients as for the practitioners, it can enhance the understanding of the predictions made by the model and improve the design of future glucose predictive models
An automated quantification of the transmural myocardial infarct extent using cardiac DE-MR images
No full textInternational audienceEvaluating myocardial viability is an important prognostic factor in the follow-up of infarctions. Delayed Enhancement magnetic resonance (DE-MR) imaging allows precise delineation of the infarct transmural extent. Visual interpretation is the most commonly used method to assess the myocardial infarction (MI) transmural extent. This study proposes to automate the segmentation of the (DE) images prior to the estimation of the extent of infarcted tissue. Indeed the segmentation of the myocardium was performed using cine contraction images which present a high contrast between cavity and myocardium. After the segmentation, the segmental transmurality is estimated on a conventional five point scale. A head to head comparison was performed between visual and quantitative analysis of infarct transmurality on DE-MR imaging. Results on 921 sub-segments (9 patients) showed an absolute agreement of 80% and a relative agreement (with one point difference) of 97%