21 feb 2017 -- 13:30
Aula Tricerri, DiMaI, Firenze
Abstract.
I start this talk with the definition of best rank k approximation problem for the case of matrices and symmetric matrices. With this purpose I define the distance function and its critical points. Moreover, I give a geometric interpretation for each of these cases in terms of the Segre and Vernoese variety and the eigenvectors. In the case of symmetric matrices I give explicitly Eckart- Young theorem. For an arbitrary real algebraic variety can be defined the EDdegree, which is the number of critical points of the euclidean distance function. (More details in 1 and 2). I give also the definition of symmetric tensors in terms of homogeneous polynomial. I explain the best rank 1 approximation problem by defining the distance function, its critical points and the relation with the eigenvectors. Moreover I point out the differences between matrices and tensors. Then I focus on the special case of binary forms (homogeneous polynomials in two variables f(x,y)). In this case the eigenvectors of f can be defined as the roots of the differential operator D(f) = yfx −xfy. Geometrically the binary forms of rank 1 can be seen as points on the rational normal curve (Cd). Finally, concerning binary forms, I give some new results collected in 3. We give an interpreta- tion in terms of tangent and normal space of the critical points of the distance function restricted to the k−secant variety of the rational normal curve (σkCd). Moreover, we define the singular space Hf as the hyperplane killed by the differential operator D(f). We also prove that the critical points of the form ki=1 lid, where li are linear forms, of the distance function restricted to σkCd belong to the singular space Hf . Furthermore, for the case k = 1 we prove that Hf coincide with the span of the eigenvectors of f. References 1 J. Draisma. E. Horobeţ, G. Ottaviani, B. Sturmfels, R. Thomas, The Euclidean Distance Degree of an Algebraic Variety, Found. Comput. Math. 16 (2016), no. 1, 99-149 2 G. Ottaviani, R. Paoletti, A Geometric Perspective on the Singular Value Decomposition, Rend. Istit. Mat. Univ. Trieste, Volume 47, 107- 125, 2015. 3 G. Ottaviani, A. Tocino, Best rank k approximation for binary forms, on preparation.