Blind Multilinear Identification

Abstract

International audienceWe discuss a technique that allows blind recovery of signals or blind identification of mixtures in instances where such recovery or identification were previously thought to be impossible: (i) closely located or highly correlated sources in antenna array processing, (ii) highly correlated spreading codes in CDMA radio communication, (iii) nearly dependent spectra in fluorescent spectroscopy. This has important implications --- in the case of antenna array processing, it allows for joint localization and extraction of multiple sources from the measurement of a noisy mixture recorded on multiple sensors in an entirely deterministic manner. In the case of CDMA, it allows the possibility of having a number of users larger than the spreading gain. In the case of fluorescent spectroscopy, it allows for detection of nearly identical chemical constituents. The proposed technique involves the solution of a bounded coherence low-rank multilinear approximation problem. We show that bounded coherence allows us to establish existence and uniqueness of the recovered solution. We will provide some statistical motivation for the approximation problem and discuss greedy approximation bounds. To provide the theoretical underpinnings for this technique, we develop a corresponding theory of sparse separable decompositions of functions, including notions of rank and nuclear norm that specialize to the usual ones for matrices and operators but applies to also hypermatrices and tensors

CP decomposition approach to blind separation for DS-CDMA system using a new performance index

“…In fact, the set of tensors of rank at most ξ is not closed if ξ > 1. Examples of the lack of closeness have been provided in the literature [15,16], which suffice to prove it. In other words, it may happen that for a given tensor, and for any rank-r approximation of it, there always exists another better rank-r approximation.…”

“…Define the three coherences μ A , μ B , and μ C associated with the matrices A, B, and C, respectively. It has been indeed shown in [15,17] that under the constraint:…”

“…According to the results in [15,16], the infimum of (10) may not be reachable. In fact, the set of tensors of rank at most ξ is not closed if ξ > 1.…”

Tensors : A brief introduction

“…These include: (i) impose orthogonality between columns of factor matrices [20] -in Blind Source Separation, this takes the form of a spatial prewhitening; (ii) impose orthogonality between decomposable tensors [45]; (iii) prevent divergence by bounding coefficients λ r [61], [54]; (iv) if the tensor is nonnegative, use a nonnegative CP [54]; (v) impose a minimal angle between columns of factor matrices [55]; (vi) compute an exact CP of another tensor 8 , which has undergone a multilinear compression via truncated HOSVD [21], [11]; (vii) compute another decomposition where the core tensor is block diagonal instead of diagonal [26] [79]; (viii) compute a Joint Approximate Diagonalization (JAD) of matrix slices, which may be viewed as another decomposition where the core tensor is not diagonal [62], [87], [89], [2], [86], [51], [20], [30], [56], [69], [14], as depicted in Figure 1. The drawbacks of this family of approaches, which become more and more popular, are three-fold.…”

Statistical efficiency of structured CPD estimation applied to Wiener-Hammerstein modeling

“…As it requires only mild assumptions for being essentially unique, the CPD provides means for blindly and jointly identifying the components of multilinear models, which arise in many real-world applications; see [1][2][3] for some examples.…”