CANDECOMP/PARAFAC model order selection based on Reconstruction Error in the presence of Kronecker structured colored noise

Canonical Decomposition (CANDECOMP) also known as Parallel Factor Analysis (PARAFAC) is a well-known multiway model in high-dimensional data modeling. Approaches that use CANDECOMP/PARAFAC for parametric modeling of a noisy observation require an estimate of the number of signal components (rank) of the data as well. In real applications, the true model of data is unknown and model order selection is a challenging step of these algorithms. In addition, considering noise samples with correlation in different dimensions makes the model order selection even more challenging. Model order selection methods generally minimize a criterion to find the optimum model order. In this paper, we propose using the Reconstruction error, which is the error between the reconstructed data and the unavailable noiseless data, for a range of possible ranks, and use an estimate of this error as the desired criterion for order selection. Furthermore, we propose using the CORCONDIA measure for determining the range of possible model orders. In the presence of the colored noise with Kronecker structure, our proposed algorithm performs the multidimensional prewhitening prior to the model order selection. In addition, our method is able to estimate the noise covariance through an iterative algorithm when no prior information about the noise covariance is available. Simulation results show that the proposed method can be effectively exploited for robustly detecting the true rank of the observed tensor even in mid and low SNRs (i.e. 0-10 dB). It also has an advantage over the state-of-the-art methods, such as different variants of CORCONDIA, by having a better Probability of Detection (PoD) with almost no extra computational overhead after the CANDECOMP/PARAFAC decomposition.