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Condition Number and Geometry of Tensor decompositions

Speaker: 
Paul Breiding
Institution: 
MPI Leipzig
Schedule: 
Thursday, October 26, 2017 - 16:30 to 18:00
Location: 
A-134
Abstract: 

The canonical polyadic decomposition (CPD) of a tensor is a direct generalization of the singular value decomposition from matrices to tensors (i.e., higher dimensional arrays of numbers). Computing the CPD of a tensor is a hard problem. State-of-the-art algorithms solve the CPD by numerically solving an optimization problem (tensorlab.net). The state-of-the-art algorithms use a variant of Newton's method for this problem. The performance of the latter depends on the condition number of the tensor rank decomposition. This condition number has some interesting geometric interpretations, which I aim at explaining. One interesting aspect is that the condition number connects the numerics of the CPD with the Abo-Ottaviani-Peterson conjecture on defective secant varieties. Moreover, the geometric point of view led us to define another optimization problem for the CPD, which we solve by using Riemannian optimization methods. Experiments show that our algorithm outperforms state-of-the-art methods on the smale scale. This is joint work with Nick Vannieuwenhoven (KU Leuven).

If time permits, I want to quickly show another project: "JuliaHomotopyContinuation", a numerical solver for (square) systems of polynomial equations in Julia (github.com/JuliaHomotopyContinuation).

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