Section: New Results

Collaborative and distributed algorithms

From Internet to large research infrastructures, the volume of data generated by our societies is continuously increasing. A deluge faced by the producers of these data as well as their users. The big data issue is a significant scientific challenge that requires deep investigations in both engineering and fundamental science. Low-rank matrix factorization has received a particular attention in recent years, since it is fundamental to a variety of mining tasks that are increasingly being applied to massive datasets. In large applications, matrix factorizations can involve matrices with billions of entries. At this massive scale, distributed algorithms for matrix factorization are essential to achieve reasonable performance [43] . However, in many disciplines, data inherently has more than two axes of variation and can be arranged as tensors (i.e. multi-way arrays). Computing tensor decompositions of multi-way datasets is particularly useful to extract hidden patterns and structure in data analytics problems. Specifically, CPD (Canonical Polyadic Decomposition) also known as PARAFAC (Parallel factor analysis) is an extension of a low rank matrix decomposition to tensors. In [26] , we have introduced a fully distributed method to compute the CPD of a large-scale data tensor across a network of machines with limited computation resources. The proposed approach is based on collaboration between the machines in the network across the three modes of the data tensor. Such a multi-modal collaboration allows an essentially unique reconstruction of the factor matrices in an efficient way. We provide an analysis of the computation and communication cost of the proposed scheme and address the problem of minimizing communication costs while maximizing the use of available computation resources.

The problem of source localization consists in finding, with one or several agents possibly cooperating with each other, the point or the spatial region from which a quantity of interest is being emitted. Source-seeking agents can be fixed sensors, that collect and exchange some information about the signal field and try to identify the position of the source (or the smallest region in which it is included), or moving devices equipped with one or more sensors, that physically reach the source in an individual or cooperative way. This research area is attracting a rapidly increasing interest, in particular in applications where the agents have limited or no position information and GPS navigation is not available, as in underwater navigation or in cave exploration: for instance, source localization is relevant to many applications of vapor emitting sources such as explosive detection, drug detection, sensing leakage or hazardous chemicals, pollution sensing and environmental studies. Other fields of interest are sound source localization, heat source localization and vent sources in underwater field. Techniques present in literature either are based on a specific knowledge of the solution of the diffusion process, or make use of an extremum-seeking approach, exciting the system with a periodic signal so as to explore the field and collect enough information to reconstruct the gradient of the quantity of interest. Our approach [13] lies in the computation of derivatives (potentially of any order) from Poisson integrals that, for isotropic diffusive source in steady-state, whose solution satisfies the Laplace equation, allows for a gradient search with a small computation load (derivatives are computed by integrals) and without requiring any knowledge of the closed-form solution, avoiding in the same time extremum-seeking oscillations; this has the additional advantage of an intrinsic high-frequency filtering, that makes the method robust to measurement noise. We also propose a distributed version of this algorithm [28] , where agents communicate in order to reconstruct gradient information from local pointwise measurements, and a control law combines the two objectives of formation control (to have a circular formation, so that measurements are taken around circle) and gradient ascent (so as to move towards the source); differently from previous literature [41] , the moving agents do not need to know their absolute position, but only relative bearing angle of their neighbours.