## Section: New Results

### Collaborative and distributed algorithms

#### Finite-time average consensus

Participants : A. Kibangou [Contact person] , T.-M. D. Tran.

Nowadays, several distributed estimation algorithms are based on the average consensus concept. Average consensus can be reached using a linear iterations scheme where each node repeatedly updates its value as a weighted linear combination of its own value and those of its neighbors; the main benefit of using a linear iterations scheme is that, at each time-step, each node only has to transmit a single value to each of its neighbors. Based on such a scheme, several algorithms have been proposed in the literature; however, in the most proposed algorithms the weights are chosen so that all the nodes asymptotically converge to the same value. Sometimes, consensus can be embedded as a step of more sophisticated distributed; obviously, asymptotic convergence is not suitable for these kinds of distributed methods, and therefore it is interesting to address the question of exact consensus in finite-time. For time-invariant network topologies and in the perfect information exchange case, i.e., without channel noise nor quantization, we have shown that the finite-time average consensus problem can be solved as a matrix factorization problem with joint diagonalizable matrices depending on the graph Laplacian eigenvalues; moreover, the number of iterations is equal to the number of distinct nonzero eigenvalues of the graph Laplacian matrix. The design of such a protocol requires the knowledge of the Laplacian spectrum, which can be carried out in a distributed way (see Section 6.1.1 . The matrix factorization problem is solved in a distributed way, in particular a learning method was proposed and the optimization problem was solved by means of distributed gradient backpropagation algorithms. The factor matrices are not necessarily symmetric and the number of these factor matrices is exactly equal to the diameter of the graph [30] .

#### Linear consensus in large-scale geometric graphs

Participants : F. Garin [Contact person] , E. Lovisari [Lund] , S. Zampieri [Padova] .

Traditional analysis of linear average-consensus algorithms studies, for a given communication graph, the convergence rate, given by the essential spectral radius of the transition matrix (i.e., the second largest eigenvalues' modulus). For many graph families, such analysis predicts a performance which degrades when the number of agents grows, basically because spreading information across a larger graph requires a longer time; however, when considering other well-known quadratic performance indices (involving all the eigenvalues of the transition matrix), the scaling law with respect to the number of agents can be different. This is consistent with the fact that, in many applications, for example in estimation problems, it is natural to expect that a larger number of cooperating agents has a positive, not a negative effect on performance. It is natural to use a different performance measure when the algorithm is used for different purposes, e.g., within a distributed estimation or control algorithm. We are interested in evaluating the effect of the topology of the communication graph on performance, in particular for large-scale graphs. We have focused on graph families which can describe sensor networks, and hence have geometric constraints, namely nodes can be connected only with nearby nodes in the sense of Euclidean distance [16] .

#### Distributed computation methods for multidimensional data

Participants : A. Kibangou [Contact person] , A. de Almeida [UFC Brazil] .

In [19] , we consider the issue of distributed computation of tensor decompositions. A central unit observing a global data tensor assigns different data sub-tensors to several computing nodes grouped into clusters. The goal is to distribute the computation of a tensor decomposition across the different computing nodes of the network, which is particularly useful when dealing with large-scale data tensors. However, this is only possible when the data sub-tensors assigned to each computing node in a cluster satisfies minimum conditions for uniqueness. By allowing collaboration between computing nodes in a cluster, we show that average consensus based estimation is useful to yield unique estimates of the factor matrices of each data sub-tensor. Moreover, an essentially unique reconstruction of the global factor matrices at the central unit is possible by allowing the subtensors assigned to different clusters to overlap in one or several modes. The proposed approach is useful to a number of distributed tensor-based estimation problems in signal and data processing.

#### Collaborative source seeking

Participants : C. Canudas de Wit [Contact person] , R. Fabbiano, F. Garin, Y. Gaudfrin, J. Dumon.

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 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 low sensitive to measurement noise. This work is the topic of the Ph.D. of Ruggero Fabbiano, and is described in papers under review.

Moreover, a hardware implementation of the source-seeking algorithm has been done during the internship of Yvan Gaudfrin, at Gipsa-Lab with the support of Jonathan Dumon. A description of the setup and videos of the source-seeking robot are available online: http://necs.inrialpes.fr/pages/platforms.php