DC 3 – Tropicalization of moment relaxations for tensor decompositions

Project Title: Tropicalization of moment relaxations for tensor decompositions
Advisors: M. Skomra (CNRS), S. Kuhlmann (UKON), Mentor: M. Gabay (Artelys)
Objectives: The problem of computing tensor decompositions by moment relaxations and weighted sums of squares (SOS) has been extensively studied in the last decade. At the same time, another certificate of positivity, relying on representation of positive polynomials as sums of nonnegative circuit polynomials (SONC) proved to be very interesting. Both SOS and SONC representations were successfully applied in polynomial optimisation. The idea for SONC certificates emerged from the amoeba theory, which provides a bridge between the algebraic geometry and the tropical geometry. The aim of this project is to improve the methods used for solving moment relaxations using various techniques coming from tropical geometry. A first research direction is to develop an analog of the bounded sums of squares (BSOS) hierarchy obtained by replacing the SOS condition by the SONC condition and apply it to tensor decomposition problems. One important question is to investigate the practical convergence rate of this hierarchy. A second research direction is to investigate the tropical properties of the Hankel matrices that arise in the moment relaxations. This could lead to the development of new tropical scaling techniques that improve the performance of numerical algorithms used for finding tensor decompositions. A third research direction is to deepen our understanding of moment relaxations for infinite-dimensional problems by studying the truncated moment problem in symmetric tensor algebras.
Expected Results: New hierarchies of relaxations that compute tensor decompositions. Developing tropical Positivstellensatze and tropical scaling techniques. Practical implementations of the obtained hierarchies and their validation on examples.
Planned secondment(s): A 10 month secondment (M23-32) at UKON to collaborate with S. Kuhlmann and a 2 month secondment (M33-34) at Artelys to work on implementations and applications of the new hierarchies.
Joint degree: Université Toulouse III-Paul Sabatier, University of Konstanz

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