DC 9-Algorithms for Tensor Decomposition

Project Title: Algorithms for Tensor Decomposition
Advisors: A. Bernardi (UniTN), A. Oneto (UniTN), B. Mourrain (Inria) Mentor: Federico Lucca (BlueTensor)
Objectives: The objective of the project is from the academic side it intends to investigate and develop new methods and algorithms for tensor decomposition (TD), and from the industrial side the implementation of these techniques requires an intermediary figure with the scientific reality to bring these techniques to the forefront of Machine Learning sectors. We aim at producing new algorithms for the ”rank” type decomposition which is strongly connected with secant varieties of Segre and Veronese varieties (in the symmetrical case). The study of other types of tensor decomposition will also be addressed, in particular that of “cactus” type is an extremely deep geometric algebraic problem linked to minimal apolar zero-dimensional scheme and which finds applications in complexity theory and big data analysis. The main goal is to implement both symbolic and numerical stable algorithms for the computation of the rank of a given tensor with particular attention to large tensors encoding data clouds or noisy data from which to extract significant information (crucial for Optical Character Recognition (OCR) and Recommendation Engine (RE) problems posed by the partner company). The mid term objective is to work on symbolic and numerical algorithms for the so-called ”cactus rank”. This will be the first needed step to realize the main objective. The importance for a student of writing algorithms for the computation of the cactus rank and the rank of a tensor is double: from the purely scientific side, these are profound problems of commutative algebra that will allow the doctoral student to be immediately internationally competitive, from the application they are refined and useful tools both in the study of big data and in quantum physics.
Expected Results: : Implementation of algorithms for the computation of the cactus rank and of the rank of a tensor in their applications of Artificial Intelligence. To evaluate the Tensor Decomposition applications on actual customers data collected from the field as a possible option to exploit new areas where to boost new disruptive technologies.
Planned secondment(s): with B. Mourrain at Inria (M23-33) for the implementation and test of a new robust and efficient tensor decomposition algorithm; with F. Lucca at BlueTensor (M21-22) to work on improving the state-of-the-art technology for OCR with better data extraction from handwritten texts (for instance on transportation documents) and improving the profiling systems (needed for Recommendation Engines) with more accurate and faster algorithms.
Joint degree: University of Trento, Université Côte d’Azur

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