Abstract.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.Key words. supersymmetric tensors, rank-1 approximation, higher-order power method, higherorder singular value decomposition AMS subject classifications. 15A18, 15A57, 15A69PII. S0895479801387413 Introduction.A tensor of order N is an N -way array, i.e., its entries are accessed via N indices.1 For example, a scalar is a tensor of order 0, a vector is a tensor of order 1, and a matrix is a second-order tensor. Tensors find applications in such diverse fields as physics, signal processing, data analysis, chemometrics, and psychology [4].The notion of rank can also be defined for tensors of order higher than 2. The way this is done is via an extension of the well-known expansion of a matrix in a sum of rank-1 terms. Thus, the rank, R, of an N th-order tensor T is the minimum number of rank-1 tensors that sum up to T . A rank-1 tensor of order N is given by the generalized outer product of N vectors, u (i) , i = 1, 2, . . . , N, i.e., its (i 1 , i 2 , . . . , i N ) entry

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Abstract-The Akaike information criterion (AIC) and the minimum description length (MDL) are two well-known criteria for model order selection in the additive white noise case. Our aim is to study the influence on their behavior of a large gap between the signal and the noise eigenvalues and of the noise eigenvalue dispersion. Our results are mostly qualitative and serve to explain the behavior of the AIC and the MDL in some cases of great practical importance. We show that when the noise eigenvalues are not clustered sufficiently closely, then the AIC and the MDL may lead to overmodeling by ignoring an arbitrarily large gap between the signal and the noise eigenvalues. For fixed number of data samples, overmodeling becomes more likely for increasing the dispersion of the noise eigenvalues. For fixed dispersion, overmodeling becomes more likely for increasing the number of data samples. Undermodeling may happen in the cases where the signal and the noise eigenvalues are not well separated and the noise eigenvalues are clustered sufficiently closely. We illustrate our results by using simulations from the effective channel order determination area.Index Terms-Akaike information criterion, minimum description length criterion.

Abslraci --It is shown that a large class of real-coefficient doubly-complementary IIR transfer function pairs can be implemented by means of a single complex allpass filter. For a real input sequence, the real part of the output sequence corresponds to the output of one of the transfer functions G(z) (foi example, lowpass), whereas the imaginary part of the output sequence corresponds to its "complementary" filter H('z) (for example, highpass). The resulting implementation is structurally lossless, and hence the implementations of G(z) and H(z) have very low passband sensitivity. Numerical design examples are included, and a typical numerical example shows that the new implementation with 4 bits per multiplier is considerably better than a direct form implementation with 9 bits per multiplier. Multirate filter bank applications (quadrature mirror filtering) are outlined.

Discrete unitary transforms are extensively used in many signal processing applications, and in the development of fast algorithms Kronecker products have proved quite useful.In this semitutorial paper, we briefly review properties of Kronecker products and direct sums of matrices, which provide a compact notation in treating patterned matrices. A generalized matrix product, which inherits some useful algebraic properties from the standard Kronecker product and allows a large class of discrete unitary transforms to be generated from a single recursion formula, is then introduced. The notation is intimately related to sparse matrix factorizations, and examples are included illustrating the utility of the new notation in signal processing applications. Finally, some novel characteristics of Hadamard transforms and polyadic permutations are derived in the framework of Kronecker products.

We re-examine a fixed-point algorithm proposed by Hyvarinen for independent component analysis, wherein local convergence is proved subject to an ideal signal model using a square invertible mixing matrix. Here, we derive step-size bounds which ensure monotonic convergence to a local extremum for any initial condition. Our analysis does not assume an ideal signal model but appeals rather to properties of the contrast function itself, and so applies even with noisy data and/or more sources than sensors. The results help alleviate the guesswork that often surrounds step-size selection when the observed signal does not fit an idealized model.

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