The investigation of efficient algorithms to solve the PARAFAC problem is a current and active topic of research [15][16]. For the sake of simplicity, the simplest algorithm is described here, based on alternative least squares (ALS) [13]. ALS consists in iteratively estimating one of the three matrices entering in the factorization assuming that the other two are fixed. This way, the minimization problem turns out linear in the parameters and is easily solved by least-squares. The algorithm is described in a step-by-step manner as follows: 1) Step I = 0: initialize a i( ) L , ai( ) R and ( ) [ ] i τ D 2) Step i+1 : update a i( ) L , ai( ) R and ( ) [ ] i τ D as a) 1 ( ) ( ) ( ) 2 ( ) ( ) ( 1) [ ] [ ] [ ] ˆ [ ] [ ] ˆ [ ] argmin − ∈ ∈ ∈ + = − = ∑ ∑ ∑ H aHi T a i H aHi T a yy T a i i a a yy a i a τ τ τ τ τ τ τ τ τ D R R D R R D R L D R L L (9) b) ˆ [ ] [ ] [ ] [ ] [ ] ˆ [ ] argmin ( ) 1 ( ) ( ) 2 ( ) ( ) ( 1) τ τ τ τ τ τ τ τ τ a yy aH i T H a i T aHi H T a i a i a yy a i a D L R D L L D R L D R R R ∑ ∑ ∑ ∈ − ∈ ∈ + = − = (10) c) 1 1 * ( ) ( ) 1 * ( ) 2 1 ( ) ( 1) 2 2 2 [ ] ~ [ ] ~ [ ] ~ [ ] ~ [ ] ~ [ ] ~ argmin − = = = + = − = ∑ ∑ ∑ mK j H a i a t i mK j H a i a yy mK j a t i a yy a i j j j j j j a R L L R R L R R DR L D D (11) where, in the last line, 1 1 1 , 1, , 0 0 1, [ ] [ ] [ ] [ ] [ ] ~ K mK j K mK a yy j K a yy j mK a yy j a yy a yy IR j × ∈ = τ τ τ τ R R R R R L M M L , 1 0 0 2 2 1 2 1 2 2 2 2 , 1, , and , 1,..., [ ] ~ K n n n a n n n j j C j mK C j K K × × ∈ Σ Σ Σ Σ = = ∈ = τ τ τ τ L M M L O D 0 R 0 R R 3) Stop iterating if the relative errors a a i i ( ) ( 1) L L− = , a a i i ( ) ( 1) R R− + and a a i i ( ) ( 1) D D− + are all smaller than a predefined value. At this point, it is important to mention a few remarks about the convergence of the ALS algorithm, which also happens to be one the limitations of the suggested approach. • First it is clear that there is no guarantee of convergence to a global minimum. Hence, the final estimates will strongly depend on the quality of the starting values. It has been observed by the authors that reasonable estimates are returned after initializing matrix D with the resonance frequencies obtained from a simple peak-picking method, with assumed zero damping ratios. Matrices aL and aR have been initialized randomly, although better strategies could perhaps be investigated such as using results from SO-BSS. 182
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