DC 1 – Geometry of extensor varieties

Project Title: Geometry of extensor varieties
Advisors: B. Mourrain (Inria), A. Bernardi (UniTN) Mentor: F. Lucca (BlueTensor)
Objectives: Decomposition of tensors is an important challenge in scientific computing, with applications in many domains including data analysis, statistics, engineering, quantum information, physics, chemistry. The goal of the project is to investigate and develop new approaches for tensor decomposition, based on flat extension properties. Such an approach appears to be particularly powerful to address the challenging problem of decomposition of tensors of high rank. Geometric properties of extensor varieties and of the corresponding determinantal varieties will be analyzed. We will consider both degree and dimension extensions (Inria). This analysis will be exploited to devise new efficient algorithms for tensor rank decomposition (Inria, UniTN). These decomposition algorithms will be tested to recover hidden information in data, in Machine Learning problems for data analysis, such as topic model reconstruction (BlueTensor) and in entanglement problems for Quantum Information Analysis (UniTN).
Expected Results: New theoretical and methodological progresses for the decomposition of tensors of high rank. Proof-of-concept implementation showing the performance and impact of the approach and validation on examples. Tensor-based solutions for problems in data analysis and entanglement in quantum information.
Planned secondment(s): with A. Bernardi, UniTN (M22-31) to work on the geometry of flat extensions of tensors and algorithms developments; with F. Lucca (BlueTensor) (M32-33) to work on applications of tensor decomposition methods via flat extensions to data analysis on handwritten texts and topic models for Recommendation Engines.
Joint degree: Université Côte d’Azur, University of Trento

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