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Decomposition of quantics in sums of powers of linear forms
 Signal Processing
, 1996
"... Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent complementary use of HighOrder Statistics (HOS). However, very few special purpose tools are at disposal for manipulating such objects in engineering prob ..."
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Cited by 80 (23 self)
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Symmetric tensors of order larger than two arise more and more often in signal and image processing and automatic control, because of the recent complementary use of HighOrder Statistics (HOS). However, very few special purpose tools are at disposal for manipulating such objects in engineering problems. In this paper, the decomposition of a symmetric tensor into a sum of simpler ones is focused on, and links with the theory of homogeneous polynomials in several variables (i.e. quantics) are pointed out. This decomposition may be seen as a formal extension of the Eigen Value Decomposition (EVD), known for symmetric matrices. By reviewing the state of the art, quite surprising statements are emphasized, that explain why the problem is much more complicated in the tensor case than in the matrix case. Very few theoretical results can be applied in practice, even for cubics or quartics, because proofs are not constructive. Nevertheless in the binary case, we have more freedom to devise numerical algorithms. Keywords. Tensors, Polynomials, Diagonalization, EVD, HighOrder Statistics, Cumulants. 1
Blind channel identification and extraction of more sources than sensors
, 1998
"... It is often admitted that a static system with more inputs (sources) than outputs (sensors, or channels) cannot be blindly identified, that is, identified only from the observation of its outputs, and without any a priori knowledge on the source statistics but their independence. By resorting to Hig ..."
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Cited by 23 (6 self)
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It is often admitted that a static system with more inputs (sources) than outputs (sensors, or channels) cannot be blindly identified, that is, identified only from the observation of its outputs, and without any a priori knowledge on the source statistics but their independence. By resorting to HighOrder Statistics, it turns out that static MIMO systems with fewer outputs than inputs can be identified, as demonstrated in the present paper. The principle, already described in a recent rather theoretical paper, had not yet been applied to a concrete blind identification problem. Here, in order to demonstrate its feasibility, the procedure is detailed in the case of a 2sensor 3source mixture; a numerical algorithm is devised, that blindly identifies a 3input 2output mixture. Computer results show its behavior as a function of the data length when sources are QPSKmodulated signals, widely used in digital communications. Then another algorithm is proposed to extract the 3 sources from the 2 observations, once the mixture has been identified. Contrary to the first algorithm, this one assumes that the sources have a known discrete distribution. Computer experiments are run in the case of three BPSK sources in presence of Gaussian noise.
Blind Identification and Source Separation In 2 x 3 Underdetermined Mixtures
, 2004
"... Underdetermined mixtures are characterized by the fact that they have more inputs than outputs, or, with the antenna array processing terminology, more sources than sensors. The problem addressed is that of identifying and inverting the mixture, which obviously does not admit a linear inverse. Id ..."
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Cited by 15 (4 self)
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Underdetermined mixtures are characterized by the fact that they have more inputs than outputs, or, with the antenna array processing terminology, more sources than sensors. The problem addressed is that of identifying and inverting the mixture, which obviously does not admit a linear inverse. Identification is carried out with the help of tensor canonical decompositions. On the other hand, the discrete distribution of the sources is utilized for performing the source extraction, the underdetermined mixture being either known or unknown. The results presented in this paper are limited to 2dimensional mixtures of 3 sources.
Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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Cited by 11 (4 self)
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of