# Areas of interest

### Inpainting of missing data in audio spectrograms

A variant of the Nonnegative Matrix Factorization (NMF) called Convex NMF
(CNMF) has been extended to estimate missing coefficients in STFT
representation of audio signals. The dictionary matrix is constrained to be a
convex linear combination of a known matrix of examples, i.e, the combination
coefficients forming each dictionary element are nonnegative and sum to 1.
Compared to the fully unsupervised NMF setting, it is a source of supervision
that may guide learning based on this additional data: in particular, an
interesting case of CNMF consists in auto-encoding the data themselves, by
defining the dictionary as the data matrix. This problem has been show of
interest to application such as transcription, and might be useful to improve
reconstruction in source separation techniques.

### Duality between graphs and time series

Many complex systems, whether physical, biological or social, can be naturally
represented as networks, i.e., a set of relationships between entities. Network
science has been widely developed to study such objects, generally supported by
a graph structure. Based on prior work to transform graph into time series
using multidimensional scaling, a technique used to reduce the dimensionality
of data, a framework to transform graph into a collection of signals has been
developed in order to study networks using a signal processing
approach. Spectral analysis of this collection of signals associates a
frequency pattern, obtained from signals, with a specific graph structure of
the underlying graph.

### Analysis of temporal networks

Temporal networks are common object to describe networks with a time
evolution. The extension of the duality between graph and signals to the
temporal case is easily achieved by considering temporal networks as
discrete-time sequences of graph snapshots. At each time step, the
transformation from graph to signals is performed: a temporal collection of
signals is then obtained, and thus temporal frequency patterns. Tracking these
patterns along time allows for the tracking of the graph structure. This is
achieved by using NMF, which extracts automatically the relevant frequency
patterns, as well as the associated activation coefficients over time. It is
then possible to retrieve from each frequency pattern the underlying structure
in the graph domain, and then by using the associated activation coefficients,
to track the topology of the temporal network over time. Results showed that
the method is able to successfully extract different types structures, such as
for instance organization in communities or regular structures, as well as
mixture of these structures.

### Study of the Lyon's Bike Sharing System

Many big cities in the world propose a bike sharing system in which bikes are
made available at any time for short trips. In Lyon,
the

Vélo’v program has been
deployed since May 2005 and consists of 350 stations spread over all the
agglomeration in which bikes can be hired or returned back. Thanks to a
partnership with the operator JCDecaux and the “Grand Lyon" City Hall,
anonymized data for the year 2011 were made available to us. The study of such
system has numerous goals as for instance viability of the system, its
integration in the transportation scheme of the city or the social behaviors
linked to the bike use. Study of Vélo'v system as a complex network is part of
a broader project called "Vél'innov" and funded by the ANR.

### Relabeling the vertices following the structure

A new heuristic has been developed to find a labeling which follows the
structure of the graph. The heuristic is a two-step algorithm: the first step
consists of browsing the graph to find a set of paths which follow the
structure of the graph, using a Jaccard-based index to jump from one vertex to
the next one. The second step consists of merging all the previously obtained
paths, based on a greedy approach that extends a partial solution by inserting
a new path at the position that minimizes the cyclic bandwidth sum, a classical
graph labeling problem.