Third-Order Advantages: a Step Forward

Due to the sophisticated analytical instrumentation higher-order data acquisition\resolution appeared at the horizon [1]. In a sequel, several multi-linear algorithms have been proposed to decompose these data tensors [2]. As data acquisitions provide more information for a sample, the required algorithm will be more complex to truly unravel data hidden information [2]. Although, Booksh and Kowalski in a corner-stone paper highlighted the advantages of higher order data sets, but third order advantages left out the famous table in that paper [3]. Oliveri et al. documented that “No additional analytical advantages appear to be known for third-order data processing” [1]. Hence, the main question is that doe it really worth to gather higher order data sets.It is highlighted that third-order decomposition bears all advantages of second order decompositions together with higher sensitivity, selectivity and improved algorithmic resolution of highly collinear third-order data [1]. In this contribution, using direct visualization of the possible solution for a quadrilinear model like four-way PARAFAC, it is emphasized that unique resolution of four-way data arrays in the presence of rank-overlap is possible. To do this, possible solutions of simulated four-way data sets were calculated for the first time in the literature. This can be considered as additional benefit of higher order data resolution, third order advantages. Finally, this finding can be further confirmed using the generalization of the Kruskal’s in-equality to N-way data arrays relying on theory and simulation are interwoven and mutually supportive [4].