On the best rank-1 approximation of higher-order supersymmetric tensors

Recently the problem of determining the best, in the least-squares sense, rank-1 approximation to a higher-order tensor was studied and an iterative method that extends the wellknown power method for matrices was proposed for its solution. This higher-order power method is also proposed for the special but important class of supersymmetric tensors, with no change. A simplified version, adapted to the special structure of the supersymmetric problem, is deemed unreliable, as its convergence is not guaranteed. The aim of this paper is to show that a symmetric version of the above method converges under assumptions of convexity (or concavity) for the functional induced by the tensor in question, assumptions that are very often satisfied in practical applications. The use of this version entails significant savings in computational complexity as compared to the unconstrained higher-order power method. Furthermore, a novel method for initializing the iterative process is developed which has been observed to yield an estimate that lies closer to the global optimum than the initialization suggested before. Moreover, its proximity to the global optimum is a priori quantifiable. In the course of the analysis, some important properties that the supersymmetry of a tensor implies for its square matrix unfolding are also studied

On the robustness of the linear prediction method for blind channel identification with respect to effective channel undermodeling/overmodeling

Summarization: We study the performance of the linear prediction (LP) method for blind channel identification when the true channel is of order M, whereas the channel model is of order m, with m<M. By partitioning the true channel into the mth-order significant part and the unmodeled tails, we show that the LP method furnishes an approximation to the mth-order significant part. The closeness depends on the diversity of the mth-order significant part and the size of the unmodeled tails. Furthermore, we show that two frequently encountered claims concerning the LP method, namely, that (a) the method is robust with respect to channel overmodeling and (b) the performance of the method depends critically on the size of the first impulse response term, are not correct in realistic scenariosPresented on: IEEE Transactions on Signal Processin

Blind channel approximation: effective channel order determination

Summarization: A common assumption of blind channel identification methods is that the order of the true channel is known. This information is not available in practice, and we are obliged to estimate the channel order by applying a rank detection procedure to an “overmodeled ” data covariance matrix. Information theoretic criteria have been widely suggested approaches for this task. We check the quality of their estimates in the context of order estimation of measured microwave radio channels and confirm that they are very sensitive to variations in the SNR and the number of data samples. This fact has prohibited their successful application for channel order estimation and has created some confusion concerning the classification into underand over-modeled cases. Recently, it has been shown that blind channel approximation methods should attempt to model only the significant part of the channel composed of the “large ” impulse response terms because efforts toward modeling “small ” leading and/or trailing terms lead to effective overmodeling, which is generically ill-conditioned and, thus, should be avoided. This can be achieved by applying blind identification methods with model order equal to the order of the significant part of the true channel called the effective channel order. Toward developing an efficient approach for the detection of the effective channel order, we use numerical analysis arguments. The derived criterion provides a “maximally stable ” decomposition of the range space of an “overmodeled ” data covariance matrix into signal and noise subspaces. It is shown to be robust to variations in the SNR and the number of data samples. Furthermore, it provides useful effective channel order estimates, leading to sufficiently good blind approximation/equalization of measured real-world microwave radio channelsΠαρουσιάστηκε στο: IEEE Transactions Signal Processin