Hyperspectral images segmentation: a proposal

Hyper-Spectral Imaging (HIS) also known as chemical or spectroscopic imaging is an emerging technique that combines imaging and spectroscopy to capture both spectral and spatial information from an object. Hyperspectral images are made up of contiguous wavebands in a given spectral band. These images provide information on the chemical make-up profile of objects, thus allowing the differentiation of objects of the same colour but which possess make-up profile. Yet, whatever the application field, most of the methods devoted to HIS processing conduct data analysis without taking into account spatial information.Pixels are processed individually, as an array of spectral data without any spatial structure. Standard classification approaches are thus widely used (k-means, fuzzy-c-means hierarchical classification...). Linear modelling methods such as Partial Least Square analysis (PLS) or non linear approaches like support vector machine (SVM) are also used at different scales (remote sensing or laboratory applications). However, with the development of high resolution sensors, coupled exploitation of spectral and spatial information to process complex images, would appear to be a very relevant approach. However, few methods are proposed in the litterature. The most recent approaches can be broadly classified in two main categories. The first ones are related to a direct extension of individual pixel classification methods using just the spectral dimension (k-means, fuzzy-c-means or FCM, Support Vector Machine or SVM). Spatial dimension is integrated as an additionnal classification parameter (Markov fields with local homogeneity constrainst [5], Support Vector Machine or SVM with spectral and spatial kernels combination [2], geometrically guided fuzzy C-means [3]...). The second ones combine the two fields related to each dimension (spectral and spatial), namely chemometric and image analysis. Various strategies have been attempted. The first one is to rely on chemometrics methods (Principal Component Analysis or PCA, Independant Component Analysis or ICA, Curvilinear Component Analysis...) to reduce the spectral dimension and then to apply standard images processing technics on the resulting score images i.e. data projection on a subspace. Another approach is to extend the definition of basic image processing operators to this new dimensionality (morphological operators for example [1, 4]). However, the approaches mentioned above tend to favour only one description either directly or indirectly (spectral or spatial). The purpose of this paper is to propose a hyperspectral processing approach that strikes a better balance in the treatment of both kinds of information....Cet article présente une stratégie de segmentation d’images hyperspectrales liant de façon symétrique et conjointe les aspects spectraux et spatiaux. Pour cela, nous proposons de construire des variables latentes permettant de définir un sous-espace représentant au mieux la topologie de l’image. Dans cet article, nous limiterons cette notion de topologie à la seule appartenance aux régions. Pour ce faire, nous utilisons d’une part les notions de l’analyse discriminante (variance intra, inter) et les propriétés des algorithmes de segmentation en région liées à celles-ci. Le principe générique théorique est exposé puis décliné sous la forme d’un exemple d’implémentation optimisé utilisant un algorithme de segmentation en région type split and merge. Les résultats obtenus sur une image de synthèse puis réelle sont exposés et commentés

Bounds for self-stabilization in unidirectional networks

A distributed algorithm is self-stabilizing if after faults and attacks hit the system and place it in some arbitrary global state, the systems recovers from this catastrophic situation without external intervention in finite time. Unidirectional networks preclude many common techniques in self-stabilization from being used, such as preserving local predicates. In this paper, we investigate the intrinsic complexity of achieving self-stabilization in unidirectional networks, and focus on the classical vertex coloring problem. When deterministic solutions are considered, we prove a lower bound of $n$ states per process (where $n$ is the network size) and a recovery time of at least $n(n-1)/2$ actions in total. We present a deterministic algorithm with matching upper bounds that performs in arbitrary graphs. When probabilistic solutions are considered, we observe that at least $\Delta + 1$ states per process and a recovery time of $\Omega(n)$ actions in total are required (where $\Delta$ denotes the maximal degree of the underlying simple undirected graph). We present a probabilistically self-stabilizing algorithm that uses $\mathtt{k}$ states per process, where $\mathtt{k}$ is a parameter of the algorithm. When $\mathtt{k}=\Delta+1$, the algorithm recovers in expected $O(\Delta n)$ actions. When $\mathtt{k}$ may grow arbitrarily, the algorithm recovers in expected O(n) actions in total. Thus, our algorithm can be made optimal with respect to space or time complexity

Symmetric indefinite triangular factorization revealing the rank profile matrix

We present a novel recursive algorithm for reducing a symmetric matrix to a triangular factorization which reveals the rank profile matrix. That is, the algorithm computes a factorization $\mathbf{P}^T\mathbf{A}\mathbf{P} = \mathbf{L}\mathbf{D}\mathbf{L}^T$ where $\mathbf{P}$ is a permutation matrix, $\mathbf{L}$ is lower triangular with a unit diagonal and $\mathbf{D}$ is symmetric block diagonal with $1{\times}1$ and $2{\times}2$ antidiagonal blocks. The novel algorithm requires $O(n^2r^{\omega-2})$ arithmetic operations. Furthermore, experimental results demonstrate that our algorithm can even be slightly more than twice as fast as the state of the art unsymmetric Gaussian elimination in most cases, that is it achieves approximately the same computational speed. By adapting the pivoting strategy developed in the unsymmetric case, we show how to recover the rank profile matrix from the permutation matrix and the support of the block-diagonal matrix. There is an obstruction in characteristic $2$ for revealing the rank profile matrix which requires to relax the shape of the block diagonal by allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient. This relaxed decomposition can then be transformed into a standard $\mathbf{P}\mathbf{L}\mathbf{D}\mathbf{L}^T\mathbf{P}^T$ decomposition at a negligible cost

Bandits Warm-up Cold Recommender Systems

We address the cold start problem in recommendation systems assuming no contextual information is available neither about users, nor items. We consider the case in which we only have access to a set of ratings of items by users. Most of the existing works consider a batch setting, and use cross-validation to tune parameters. The classical method consists in minimizing the root mean square error over a training subset of the ratings which provides a factorization of the matrix of ratings, interpreted as a latent representation of items and users. Our contribution in this paper is 5-fold. First, we explicit the issues raised by this kind of batch setting for users or items with very few ratings. Then, we propose an online setting closer to the actual use of recommender systems; this setting is inspired by the bandit framework. The proposed methodology can be used to turn any recommender system dataset (such as Netflix, MovieLens,...) into a sequential dataset. Then, we explicit a strong and insightful link between contextual bandit algorithms and matrix factorization; this leads us to a new algorithm that tackles the exploration/exploitation dilemma associated to the cold start problem in a strikingly new perspective. Finally, experimental evidence confirm that our algorithm is effective in dealing with the cold start problem on publicly available datasets. Overall, the goal of this paper is to bridge the gap between recommender systems based on matrix factorizations and those based on contextual bandits

Resolution of the finite Markov moment problem

We expose in full detail a constructive procedure to invert the so--called "finite Markov moment problem". The proofs rely on the general theory of Toeplitz matrices together with the classical Newton's relations

The 0-1 inverse maximum stable set problem

Given an instance of a weighted combinatorial optimization problem and its feasible solution, the usual inverse problem is to modify as little as possible (with respect to a fixed norm) the given weight system to make the giiven feasible solution optimal. We focus on its 0-1 version, which is to modify as little as possible the structure of the given instance so that the fixed solution becomes optimal in the new instance. In this paper, we consider the 0-1 inverse maximum stable set problem against a specific (optimal or not) algorithm, which is to delete as few vertices as possible so that the fixed stable set S* can be returned as a solution by the given algorithm in the new instance. Firstly, we study the hardness and approximation results of the 0-1 inverse maximum stable set problem against the algorithms. Greedy and 2-opt. Secondly, we identify classes of graphs for which the 0-1 inverse maximum stable set problem can be polynomially solvable. We prove the tractability of the problem for several classes of perfect graphs such as comparability graphs and chordal graphs.Combinatorial inverse optimization, maximum stable set problem, NP-hardness, performance ratio, perfect graphs.