Vanguard: Postdoc in Network Tensor Completion

About Mohammed VI Polytechnic University (UM6P):

Located at the heart of the Green City of Benguerir, Mohammed VI Polytechnic University (UM6P), a higher education institution with an international standard, was established to serve Morocco and the African continent and to advance applied research and innovation. This unique university, with state-of-the-art infrastructure, has woven an extensive academic and research network, and its recruitment process is seeking outstanding academics and professionals to promote Morocco and Africa’s innovation ecosystem. 

About the department

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. For more information about our Center, please visit our webpage: https://vanguard.um6p.ma/

Offer description:

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.

Job responsibilities

•             Research and Development: Conduct research to develop novel algorithms and methodologies for tensor completion in multidimensional networks. This includes exploring optimization techniques, tensor decompositions, and incorporating auxiliary information for more accurate completion.

•             Algorithm Design: Design and implement algorithms for tensor completion, considering the unique challenges posed by sparse and multidimensional network data. This involves developing efficient and scalable algorithms that can handle large-scale datasets.

•             Tensor Analysis: Analyze the structure and properties of multidimensional networks represented as tensors. Investigate different measures and metrics for characterizing network connectivity and relationships.

•             Sparse Optimization: Address the challenge of sparse optimization for tensors by integrating randomized algorithms into optimization frameworks. Study the theoretical foundations of randomized algorithms in the context of sparse tensor operations and apply them to real-world datasets.

•             Parallel Computing: Explore opportunities for parallelism in the tensor completion process to enhance computational efficiency. Investigate parallel algorithms and architectures that can exploit the inherent parallelism in tensor operations.

•             Collaboration: Collaborate with interdisciplinary teams including computer scientists, statisticians, and domain experts to apply tensor completion techniques to real-world applications, especially in the case of social sciences. This involves effective communication and coordination to ensure the successful integration of mathematical methods into practical systems.

•             Publication and Dissemination: Publish research findings in top-tier journals and present results at conferences and workshops. Disseminate knowledge and contribute to the academic community by sharing insights and methodologies developed during the course of the project.

•             Mentorship and Training: Provide mentorship and guidance to graduate students and junior researchers involved in related projects. Share expertise and knowledge in applied mathematics, tensor analysis, and network science to foster the professional development of team members.

Qualifications and experience essential

PhD in Applied Mathematics in the fields of Numerical Linear Algebra, or equivalent. Prior experience on the subject is highly desired.