Career Opportunity: Postdoc in Tensor Network Completion

Vanguard works on the development of innovative and interdisciplinary applied research projects. From technological innovation to the transfer of research to industry, Vanguard has also the mission of developing an ecosystem of related start-ups.

There are many systems of interest to scientists that are composed of individual parts or components linked together in some way. Examples include the Internet, a collection of computers linked by data connections, human societies, which are collections of people linked by acquaintance or social interaction, transportation systems and biological interactions. These systems are represented as networks.

A network is a set of objects that are connected to each other in some fashion. Mathematically, a network is represented by a graph, which is a collection of nodes that are connected to each other by edges. The nodes represent the objects of the network and the edges represent relationships between objects. A common way to represent a graph is to use the adjacency matrix associated with the graph.

However, adjacency matrices only model networks with one kind of objects or relations between the objects. Many real world networks have a multidimensional nature such as networks that contain multiple connections. For instance, transport networks in a country when considering different means of transportation. The train and bus routes are different types of connections and should in some models be represented by different kinds of edges. These kind of situations can be modeled using multilayer networks which emphasize the different kind or levels, known as layers, of connections between the elements of the network and the interactions between these levels as well.

In order to capture the structure and complexity of relationships between the nodes of networks with a mul-tidimensional nature, tensors are used to represent these kind of networks. For example, the transport network mentioned earlier would be represented by a 4th order tensor A 2RN_L_N_L  where L is the number of the layers (transportation means) and N is the number of nodes (stations or stops). Using convenient tensor products, the goal is to define measures to analyze different multidimensional networks based on their adjacency tensors.

However, collecting all the interactions in the systems and sometimes even observing all the components is a challenging task. In most cases, only a sample of a network is observed. Therefore, network completion needs to be addressed. Matrix completion methods have proved to be efficient when reconstructing a non fully observed data. These methods can be applied to complete or predict links in a network. However, missing information in a network can include both missing edges and nodes which makes classical matrix completion method insufficient. However,we may collect other information and features about the elements of the network. Therefore, side information about the nodes along with the observed edges need to be exploited.

The problem of network completion arrises also for applications where the network has a multidimensional representation such as multiplexes and multilayer networks. Since multidimensional networks can be represented by tensors, one can think of applying tensor completion methods which have proved to be efficient in many applications such as image and video reconstruction. However, the same issue arises, tensor completion methods can not be directly applied to recover the links of the network giving the fact that the data is sparse most of the time. We aim to use auxiliary information about the multiplex and multilayer networks alongside with the observed links in order to predict or reconstruct the missing links. The first step is to explore different optimization methods using low rank tensor minimization and tensor decompositions paired with auxiliary information in order to recover missing links in a multilayer network with connected components.

An important constraint in network completion is that the factorization must only capture the non zero entries of the tensor. The remaining entries are treated as missing values, not actual zeros as is often the case in sparse tensor and matrix operations. Therefore, the next step in this project is to address sparse optimization for tensors. We propose the integration of randomized algorithms into sparse optimization frameworks for the purpose of completing multidimensional networks by studying the theoretical foundations behind randomized algorithms in the context of sparse optimization and applications in real world data sets. We are also interested in exploring opportunities for parallelism of the completion process, highlighting the potential for significant speedup in computations.

Applications must contain: 

1. A cover letter indicating the position applied for and the main research interests. 
2. A detailed CV. 
3. Research and Teaching statements. 
4. Contact information of 3 references (applicants are assumed to have obtained their references consent to be contacted for this matter). 

The shortlisted candidates will be invited to meet the university selection committee. 

For those interested in applying, please visit the following link to submit your application: [Postdoc in Tensor Network Completion]