Myriam Rajih

Enhanced Line Search: A Novel Method to Accelerate PARAFAC

The ALS algorithm, used to fit the Parafac model, sometimes needs a large number of iterations before converging. The slowness in convergence can be due to the large size of the data, or to the presence of degeneracies, etc. Several methods have been proposed to speed up the algorithm, some of which are compression [3], and Line Search [2]. In this paper, after a description of Parafac, we will present a novel method for speeding up the algorithm that shows better results in simulations compared to the existing methods, especially in the case of degeneracy. The paper gives an application of the method to blindly identify the mixing matrix of an Under-Determined Mixture (UDM), but it can be applied to any N-way decomposition problem.

Blind identification of under-determined mixtures based on the characteristic function

Linear mixtures of independent random variables (the so-called sources) are sometimes referred to as under-determined mixtures (UDM) when the number of sources exceeds the dimension of the observation space. The algorithms proposed are able to identify algebraically a UDM using the second characteristic function (c.f.) of the observations, without any need of sparsity assumption on sources. In fact, by taking higher order derivatives of the multivariate c.f. core equation, the blind identification problem is shown to reduce to a tensor decomposition. With only two sensors, the first algorithm only needs a SVD. With a larger number of sensors, the second algorithm executes an alternating least squares (ALS) algorithm. The joint use of statistics of different orders is possible, and a LS solution can be computed. Identifiability conditions are stated in each of the two cases. Computer simulations eventually demonstrate performances in the absence of sparsity, and emphasize the interest in using jointly derivatives of different orders.

Blind Identification of Complex Under-Determined Mixtures

Linear Mixtures of independent random variables (the so-called sources) are sometimes referred to as Under-Determined Mixtures (UDM) when the number of sources exceeds the dimension of the observation space. The algorithm proposed is able to identify algebraically a complex mixture of complex sources. It improves an algorithm proposed by the authors for mixtures received on a single sensor, also based on characteristic functions. Computer simulations demonstrate the ability of the algorithm to identify mixtures with typically 3 complex sources received on 2 sensors.

A Deterministic Blind Receiver for MIMO OFDM Systems

In this paper a deterministic parallel factor (PARAFAC) receiver is proposed, for multiple input multiple output (MIMO) orthogonal frequency division multiplexing (OFDM) systems. We show that the received signal forms a 4-way tensor whose dimensions are space, time, and frequency, and can be written as a sum of tensor products (over paths and users) of four tensors representing the channel (for two of them), the symbols, and the modulation. Parameters of this model are identified via an alternating least squares (ALS) algorithm, called DEBRE, whose identifiability conditions are pointed out

Alternating Least Squares Identification of Under-Determined Mixtures Based on the Characteristic Function

Algorithm ALESCAF (alternating least squares identification based on the characteristic function) uses the derivatives of the second characteristic function (c.f.) of observations, without any need of sparsity assumption on sources, but assuming their statistical independence. ALESCAF was already proposed by the authors in P. Comon and M. Rajih (2005), where only one derivative order was considered. In this paper, new versions of ALESCAF are proposed, that jointly use derivatives of different orders. We also propose ALESCAS, a new algorithm that uses the knowledge of source c.f.s. Computer simulations demonstrate that both algorithms accelerate the convergence

Blind identification of under-determined complex mixtures of independent sources

This paper is devoted to under-determined linear mixtures of independent random variables (i.e. with more inputs than outputs). Blind identifiability of general under-determined mixtures is first discussed, and the maximum number of sources is given, depending on the hypotheses assumed. Then an algorithm proposed by Taleb, essentially usable for 2-dimensional mixtures, is extended to the complex field. A procedure is proposed in order to avoid the enormous increase in complexity. Computer simulations demonstrate the ability of the algorithm to identify mixtures of N QPSK sources received on 1 or 2 sensors.

Blind identification of under-determined mixtures based on the characteristic function: influence of the knowledge of source PDF's

When the number of inputs (sources) is larger than the number of outputs (observations), linear mixtures are referred to as Under-Determined (UDM). The algorithms proposed here aim at identifying UDM using the second characteristic function (c.f.) of observations, without any need of sparsity assumption on sources, but assuming their statistical independence. The first algorithm, already proposed by the authors in P. Comon and M. Rajih (2005), assumes that the source c.f.s are unknown. In this paper, a variant of the algorithm is described, which allows to take into account the knowledge of source c.f.s. Performances of both algorithms are compared based on computer simulations

Enhanced line search applied to blind channel identification identifiability conditions

The ALS algorithm, used to fit the PARAFAC model, sometimes needs a large number of iterations before converging. The slowness in convergence can be due to the large size of the data, or to the presence of degeneracies, etc. Several methods have been proposed to speed up the algorithm, some of which are compression (R. Bro and C.A. Andersson, 1998), and line search (R. Bro, 1998). In M. Rajih and P. Comon (2005) presents a novel method for speeding up the algorithm, enhanced line search (ELS), that shows better results in simulations compared to the existing methods, especially in the case of degeneracy. This paper gives an application of ELS to blindly identify the mixing matrix of an under-determined mixture (UDM): algorithm ALESCAF, and states the identifiability conditions based on ALESCAF