A study on the differential strategy of some iterative trilinear decomposition algorithms: PARAFAC‐ALS, ATLD, SWATLD, and APTLD

Abstract

This study presents an in‐depth discussion of the differential properties of various iterative trilinear decomposition algorithms, including Parallel Factor Analysis‐Alternating Least Squares (PARAFAC‐ALS), Alternating Trilinear Decomposition (ATLD), Self‐Weighted Alternating Trilinear Decomposition (SWATLD), and Alternating Penalty Trilinear Decomposition (APTLD). The shape of each algorithm's objective function (“convex” or “strictly convex”) is related to the algorithm's sensitivity to the estimated component number of the trilinear system. Different situations near the objective solution are analyzed both theoretically and numerically. The wall of perturbation generated by deviations in the iterative steps prevents the PARAFAC algorithm from achieving the objective solution when the component number is overestimated. This may explain, from a calculational perspective, why the PARAFAC algorithm could not obtain the objective solution or any equivalent thereto (although equivalents might still be chemically meaningful optimal solutions). The different effects of deviation and residual on the algorithms are demonstrated by numerical analysis in this paper. The convergence rate can be improved by the use of high‐performance computing strategy of the specific algorithm. The concept of solution set discussed in this paper complements the theory of the uniqueness of trilinear decomposition. Copyright © 2014 John Wiley & Sons, Ltd.

Solving signal instability to maintain the second-order advantage in the resolution and determination of multi-analytes in complex systems by modeling liquid chromatography–mass spectrometry data using alternating trilinear decomposition method assisted with piecewise direct standardization

“…ATLD holds the fastest convergence, which is attributed to the operation based on sliced matrices with less size and two other major strategies. One is the truncated least-squares method, which uses the tolerance to truncate the small singular values in the singular value decomposition; the other is the operation of selecting diagonal elements, which also makes ATLD retaining trilinearity property indeed and be insensitive to the overestimated number of components [54,55]. As long as the selected number of components is equal to or greater than the underlying real number of components, the ATLD method can provide correct decomposed profiles, and the excess components always appear in the shapes of noises which are easily recognized.…”

“…Therefore, ATLD method became the best candidate to analyze the three-way large-scale LC-MS experimental data. More detailed information for ATLD method can be found in the previously published references [44,[54][55][56][57].…”

Second-order calibration method applied to process three-way excitation–emission-kinetic fluorescence data: A novel tool for real-time quantitative analysis of the lactone hydrolysis of irinotecan in human plasma

“…However, it is worth noting that there is not a single versatile method that is appropriate for all situations because every method has its own advantages and disadvantages [32], and the fields of application of these methods are not the same. Some reports have compared the properties of these methods from the perspectives of theory and application [33][34][35][36][37][38].…”

Identification and Determination of Oil Pollutants Based on 3-D Fluorescence Spectrum Combined with Self-weighted Alternating Trilinear Decomposition Algorithm

“…It can quantitatively measure the specific components in the existence of interferents [16][17][18]. But only estimating the component number correctly can we get the accurate result [19,20]. The self-weighted alternating trilinear decomposition (SWATLD) algorithm, which has been developed in recent years, is insensitive to component numbers, requires fewer iterations and has high stability [20][21][22].…”

“…But only estimating the component number correctly can we get the accurate result [19,20]. The self-weighted alternating trilinear decomposition (SWATLD) algorithm, which has been developed in recent years, is insensitive to component numbers, requires fewer iterations and has high stability [20][21][22]. Hu and Yin applied PARAFAC-alternative least squares (PARAFAC-ALSs) and SWATLD to measure Vancomycin and Cephalexin in human plasma, the experimental result showed that SWATLD algorithm had good effect on the determination of complex analysis of drugs in plasma [23].…”