Section: New Results

Graphs & Signal Processing

Participants : Paulo Gonçalves [correspondant] , Éric Fleury, Christophe Crespelle.

Signal Processing on Graphs

Semi-Supervised Learning for Graph to Signal Mapping: a Graph Signal Wiener Filter Interpretation [14] . We investigate a graph to signal mapping with the objective of analyzing intricate structural properties of graphs with tools borrowed from signal processing. We successfully use a graph-based semi-supervised learning approach to map nodes of a graph to signal amplitudes such that the resulting time series is smooth and the procedure efficient and scalable. Theoretical analysis of this method reveals that it essentially amounts to a linear graph-shift-invariant filter with the a priori knowledge put into the training set as input. Further analysis shows that we can interpret this filter as a Wiener filter on graphs. We finally build upon this interpretation to improve our results.

Graphs

(Nearly-)tight bounds on the contiguity and linearity of cographs [6] . In this paper we show that the contiguity and linearity of cographs on $n$ vertices are both $O(logn)$. Moreover, we show that this bound is tight for contiguity as there exists a family of cographs on $n$ vertices whose contiguity is $\Omega (logn)$. We also provide an $\Omega (logn/loglogn)$ lower bound on the maximum linearity of cographs on $n$ vertices. As a by-product of our proofs, we obtain a min-max theorem, which is worth of interest in itself, stating equality between the rank of a tree and the minimum height of one of its path partitions.

Signal processing

Analysis of intrapartum foetal heart rate (FHR), enabling early detection of foetal acidosis to prevent asphyxia and labour adverse outcomes, remains a challenging signal processing task. In this direction, we carried out a series of works to characterize the fetal heart rate variability with specific attributes able to discriminate between healthy fetuses and fetuses presenting a risk of brain injury. Last year, we investigated two different approaches:

Nearest-Neighbor based Wavelet Entropy Rate Measures for Intrapartum Fetal Heart Rate Variability [23] .

Firstly, we showed that a k-nearest neighbor procedure yields estimates for entropy rates that are robust and well-suited to FHR variability. Secondly, we experimentally proved that entropy rates measured on multiresolution wavelet coefficients permit to improve classification performance.

Impacts of labour first and second stages on Hurst parameter based intrapartum FHR analysis [22] .

In this study, we proposed to quantify the FHR temporal dynamics with a Hurst exponent estimated within a wavelet framework. Analyses performed over a large (3049 records) and well documented database revealed that the evolution of the Hurst exponent during delivery, is significantly different for healthy fetuses and for acidotic fetuses.