A Tetradic Decomposition of 4 Th - Order Tensors . Application to the Source Separation Problem

Two results are presented on a SVD-like decomposition of 4th-order tensors. This is motivated by an array processing problem: consider an array of m sensors listening at n independent narrow band sources; the 4th-order cumulants of the array output form a 4th-order rank-deecient symmetric tensor which has a tetradic structure. Finding a tetradic decomposition of this tensor is equivalent to identify the spatial transfert function of the system which is a matrix whose knowledge allows to recover the source signals. We rst show that when a 4th-order tensor is a sum of independant tetrads, this tetradic structure is essentially unique. This is to be contrasted with the second order case, where it is weel known that dyadic decompositions are not unique unless some constraints are put on the dyads (like orthogonality, for instance). Hence this rst result is equivalent to an identiiability property. Our second result is that (under a `locality' condition), symmetric and rank-n 4th-order tensors necessarily are a sum of n tetrads. This result is needed because the sample cumu-lant tensor being only an approximation to the true cumulant tensor, is not exactly a sum of tetrads. Our result implies that the sample cumulants can bèenhanced' to the closest tetradic cumulants by alternatively forcing their rank-deeciency and symmetry. A simple algorithm is described to this eeect. Its output is an enhanced statistic, from which blind identiication is obtained deterministically. This leads to a source separation algorithm based only on the 4th-order cumulants, which is equivalent to robust statistic matching without the need for an explicit optimization procedure.