Tuesday December 6th - 1:30pm
Bernard Mourrain
INRIA - GALAAD
An algebraic-geometric view on tensor decomposition problems
Tensors are often used to collect data according to different "modes" or dimensions. Decomposing them is often used to extract intrinsic information hidden in this data. This problem appears in many domains such as signal processing, data analysis, complexity analysis, phylogenetic, ...
The presentation will start with few examples of such practical problems where tensor decomposition is an important ingredient. Next, we will describe different notions of ranks associated to a tensor and some of their surprising properties.
Revisiting the approach of J.J.Sylvester for the decomposition of binary forms, from a dual point of view, we will describe how it extends to general forms and related it to some recent developments of the decomposition problem for symmetric and multi-homogeneous tensors.
From a geometric point of view, we will see how algebraic varieties such as secant varieties, cactus varieties and the Hilbert scheme of points appear naturally in this framework. From an algebraic point of view, we will see how solving truncated moment problems related to Hankel matrices help computing such a decomposition. An algorithm based on an extension of Sylvester approach will be described and illustrated.
