31 research outputs found

    Blind Multilinear Identification

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    We 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 apply to also hypermatrices and tensors.Comment: 20 pages, to appear in IEEE Transactions on Information Theor

    Tensor polyadic decomposition for antenna array processing

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    International audienceIn the present framework, a tensor is understood as a multi-way array of complex numbers indexed by three (or more) indices. The decomposition of such tensors into a sum of decomposable (i.e. rank-1) terms is called ''Polyadic Decomposition'' (PD), and qualified as ''canonical'' (CPD) if it is unique up to trivial indeterminacies. The idea is to use the CPD to identify the location of radiating sources in the far-field from several sensor subarrays, deduced from each other by a translation in space. The main difficulty of this problem is that noise is present, so that the measurement tensor must be fitted by a low-rank approximate, and that the infimum of the distance between the two is not always reached. Our contribution is three-fold. We first propose to minimize the latter distance under a constraint ensuring the existence of the minimum. Next, we compute the Cram{é}r-Rao bounds related to the localization problem, in which nuisance parameters are involved (namely the translations between subarrays). Then we demonstrate that the CPD-based localization algorithm performs better than ESPRIT when more than 2 subarrays are used, performances being the same for 2 subarrays. Some inaccuracies found in the literature are also pointed out

    Combinatorial methods for the spectral p-norm of hypermatrices

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    The spectral pp-norm of rr-matrices generalizes the spectral 22-norm of 22-matrices. In 1911 Schur gave an upper bound on the spectral 22-norm of 22-matrices, which was extended in 1934 by Hardy, Littlewood, and Polya to rr-matrices. Recently, Kolotilina, and independently the author, strengthened Schur's bound for 22-matrices. The main result of this paper extends the latter result to rr-matrices, thereby improving the result of Hardy, Littlewood, and Polya. The proof is based on combinatorial concepts like rr-partite rr-matrix and symmetrant of a matrix, which appear to be instrumental in the study of the spectral pp-norm in general. Thus, another application shows that the spectral pp-norm and the pp-spectral radius of a symmetric nonnegative rr-matrix are equal whenever prp\geq r. This result contributes to a classical area of analysis, initiated by Mazur and Orlicz around 1930. Additionally, a number of bounds are given on the pp-spectral radius and the spectral pp-norm of rr-matrices and rr-graphs.Comment: 29 pages. Credit has been given to Ragnarsson and Van Loan for the symmetrant of a matri

    Performance Index for Tensor Polyadic Decompositions

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    International audienceIt is proposed to isolate the computation of the scaling matrix in CP tensor decompositions. This has two implications. First, the conditioning of the problem shows up explicitly, and could be controlled via a constraint on the so-called coherences. Second, a performance measure concerning only the factor matrices can be exactly calculated, and does not present the optimistic bias of the minimal error generally utilized in the literature. In fact, for tensors of order dd, it suffices to solve a degree-2 polynomial system in dd variables. We subsequently give an explicit solution when d=3

    Joint Tensor Compression for Coupled Canonical Polyadic Decompositions

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    International audienceTo deal with large multimodal datasets, coupled canonical polyadic decompositions are used as an approximation model. In this paper, a joint compression scheme is introduced to reduce the dimensions of the dataset. Joint compression allows to solve the approximation problem in a compressed domain using standard coupled decomposition algorithms. Computational complexity required to obtain the coupled decomposition is therefore reduced. Also, we propose to approximate the update of the coupled factor by a simple weighted average of the independent updates of the coupled factors. The proposed approach and its simplified version are tested with synthetic data and we show that both do not incur substantial loss in approximation performance

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

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    Accepted for publication in the Proceedings of the European Signal Processing Conference (EUSIPCO) 2015.International audienceThe computation of a structured canonical polyadic decomposition (CPD) is useful to address several important modeling problems in real-world applications. In this paper, we consider the identification of a nonlinear system by means of a Wiener-Hammerstein model, assuming a high-order Volterra kernel of that system has been previously estimated. Such a kernel, viewed as a tensor, admits a CPD with banded circulant factors which comprise the model parameters. To estimate them, we formulate specialized estimators based on recently proposed algorithms for the computation of structured CPDs. Then, considering the presence of additive white Gaussian noise, we derive a closed-form expression for the Cramer-Rao bound (CRB) associated with this estimation problem. Finally, we assess the statistical performance of the proposed estimators via Monte Carlo simulations, by comparing their mean-square error with the CRB
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