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Symmetric eigenvalue decompositions for symmetric tensors

Algebra & Discrete Mathematics

Speaker: Lek-Heng Lim, UC Berkeley
Location: 2112 MSB
Start time: Thu, Jan 29 2009, 2:10PM

The spectral theorem for symmetric matrices that one learns in college has two natural generalizations to symmetric tensors. Viewed as a homogeneous form of degree k, a symmetric tensor of order k may be decomposed into a linear combination of k-powers of linear forms, ie. order-k rank-1 symmetric tensors. Alternatively one may also decompose a symmetric tensor into a multilinear combination of points on a Stiefel manifold. If A = USU' is the eigenvalue decomposition of a symmetric matrix A, then the two generalizations preserve the "diagonality" of S and the orthogonality of U respectively. The geometry that underlies the first decomposition is well-known, namely, that of secant varieties to the Veronese variety; the geometry that underlies the second decomposition is that of symmetric subspace varieties. We will discuss properties of these decompositions and applications to signal processing and machine learning. We will also examine a recent algorithm of Comon-Mourrain-Tsigaridas that provides a method to compute the first decomposition, in essence "effective Alexander-Hirschowitz". This talk will feature joint work with P. Comon, J. Morton, B. Mourrain, B. Savas.

Note that this seminar is on a Thursday, not the regular Friday of the Algebra & Discrete Math seminar.