Linear Multifractional Stable Motion: fine path properties

Linear Multifractional Stable Motion (LMSM), denoted by $\{Y(t):t\in\R\}$, has been introduced by Stoev and Taqqu in 2004-2005, by substituting to the constant Hurst parameter of a classical Linear Fractional Stable Motion (LFSM), a deterministic function $H(\cdot)$ depending on the time variable $t$; we always suppose $H(\cdot)$ to be continuous and with values in (1/\al,1), also, in general we restrict its range to a compact interval. The main goal of our article is to make a comprehensive study of the local and asymptotic behavior of $\{Y(t):t\in\R\}$; to this end, one needs to derive fine path properties of $\{X(u,v) : (u,v)\in\R \times (1/\alpha,1)\}$, the field generating the latter process (i.e. one has $Y(t)=X(t,H(t))$ for all $t\in\R$). This leads us to introduce random wavelet series representations of $\{X(u,v) : (u,v)\in\R \times (1/\alpha,1)\}$ as well as of all its pathwise partial derivatives of any order with respect to $v$. Then our strategy consists in using wavelet methods. Among other things, we solve a conjecture of Stoev and Taqqu, concerning the existence for LMSM of a modification with almost surely continuous paths; moreover we provides some bounds for the local H\"older exponent of LMSM: namely, we obtain a quasi-optimal global modulus of continuity for it, and also an optimal local one. It is worth noticing that, even in the quite classical case of LFSM, the latter optimal local modulus of continuity provides a new result which was unknown so far

PAC-Bayesian Majority Vote for Late Classifier Fusion

A lot of attention has been devoted to multimedia indexing over the past few years. In the literature, we often consider two kinds of fusion schemes: The early fusion and the late fusion. In this paper we focus on late classifier fusion, where one combines the scores of each modality at the decision level. To tackle this problem, we investigate a recent and elegant well-founded quadratic program named MinCq coming from the Machine Learning PAC-Bayes theory. MinCq looks for the weighted combination, over a set of real-valued functions seen as voters, leading to the lowest misclassification rate, while making use of the voters' diversity. We provide evidence that this method is naturally adapted to late fusion procedure. We propose an extension of MinCq by adding an order- preserving pairwise loss for ranking, helping to improve Mean Averaged Precision measure. We confirm the good behavior of the MinCq-based fusion approaches with experiments on a real image benchmark.Comment: 7 pages, Research repor

Explainable cardiac pathology classification on cine MRI with motion characterization by semi-supervised learning of apparent flow

We propose a method to classify cardiac pathology based on a novel approach to extract image derived features to characterize the shape and motion of the heart. An original semi-supervised learning procedure, which makes efficient use of a large amount of non-segmented images and a small amount of images segmented manually by experts, is developed to generate pixel-wise apparent flow between two time points of a 2D+t cine MRI image sequence. Combining the apparent flow maps and cardiac segmentation masks, we obtain a local apparent flow corresponding to the 2D motion of myocardium and ventricular cavities. This leads to the generation of time series of the radius and thickness of myocardial segments to represent cardiac motion. These time series of motion features are reliable and explainable characteristics of pathological cardiac motion. Furthermore, they are combined with shape-related features to classify cardiac pathologies. Using only nine feature values as input, we propose an explainable, simple and flexible model for pathology classification. On ACDC training set and testing set, the model achieves 95% and 94% respectively as classification accuracy. Its performance is hence comparable to that of the state-of-the-art. Comparison with various other models is performed to outline some advantages of our model

Joint continuity of the local times of fractional Brownian sheets

Let $B^H=\{B^H(t),t\in{{\mathbb{R}}_+^N}\}$ be an $(N,d)$-fractional Brownian sheet with index $H=(H_1,...,H_N)\in(0,1)^N$ defined by $B^H(t)=(B^H_1(t),...,B^H_d(t)) (t\in {\mathbb{R}}_+^N),$ where $B^H_1,...,B^H_d$ are independent copies of a real-valued fractional Brownian sheet $B_0^H$. We prove that if $d<\sum_{\ell=1}^NH_{\ell}^{-1}$, then the local times of $B^H$ are jointly continuous. This verifies a conjecture of Xiao and Zhang (Probab. Theory Related Fields 124 (2002)). We also establish sharp local and global H\"{o}lder conditions for the local times of $B^H$. These results are applied to study analytic and geometric properties of the sample paths of $B^H$.Comment: Published in at http://dx.doi.org/10.1214/07-AIHP131 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org