Waring, tangential and cactus decompositions

Abstract

We revise the famous algorithm for symmetric tensor decomposition due to Brachat, Comon, Mourrain and Tsidgaridas. Afterwards, we generalize it in order to detect possibly different decompositions involving points on the tangential variety of a Veronese variety. Finally, we produce an algorithm for cactus rank and decomposition, which also detects the support of the minimal apolar scheme and its length at each component. Contents 1. Introduction 1.1. Novel contribution 1.2. Structure of the paper 2. Preliminaries 2.1. Notation 2.2. Algebraic tools 2.3. Apolarity 3. Waring decomposition algorithm 3.1. Problem reformulation 3.2. Choice of the basis 3.3. Minimal Waring rank to test 3.4. Waring decomposition algorithm 4. Algorithm advantages 4.1. Essential variables 4.2. The starting r 4.3. The requirements on B 4.4. Looking at eigenvectors 5. The tangential case 5.1. Generalizing previous results 5.2. Tangential decomposition algorithm 5.3. Some examples 6. The cactus rank 6.1. Evaluation of the cactus rank 6.2. Some examples 7. Conclusions and further work 7.1. How to choose the h's 7.2. Choice of the basis 7.3. Cactus decomposition

Sparse Interpolation in Terms of Multivariate Chebyshev Polynomials

“…Identifying the support of a linear form on a polynomial ring K[x] already has applications in optimization, tensor decomposition and cubature [1,2,9,13,15,36,37]. How to take advantage of symmetry in some of these applications appears in [14,21,52].…”

“…The usual Hankel or mixed Hankel-Toepliz matrices that appeared in the literature on sparse interpolation [8,35] are actually the matrices of the Hankel operator mentioned above in the different univariate polynomial bases considered. The recovery of the support of a linear form with this type of technique also appears in optimization, tensor decomposition and cubature [2,9,13,15,36,37]. We present new developments to take advantage of the invariance or semi-invariance of the linear form.…”

The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition

“…We will show a more modern reformulation of the same algorithm presented by G. Comas and M. Seiguer in [27] and a more efficient one presented in [28]; see Section 2.3.1. In Section 2.3.2, we will tackle the more general case of the computation of the symmetric-rank of any homogeneous polynomial, and we will show the only theoretical algorithm (to our knowledge) that is able to do so, which was developed by J. Brachat, P. Comon, B. Mourrain and E. Tsigaridas in [30] with its reformulation [32,31].…”

“…The original proof of this proposition can be found in [30]; for a more detailed and expanded presentation, see [32,31]. Theorem 2.86 (Brachat, Comon, Mourrain, Tsigaridas [30]).…”

“…Algorithm 2.92 (Brachat, Comon, Mourrain, Tsigaridas, [30,32,31].). Here is the idea of algorithm presented in [30].…”