A Generalized Constraint for Achieving the Unique Solution in Self Modeling Curve Resolution Methods based on Duality Principle

There is a natural duality between the row and column vector spaces of the data matrix using minimal constraints. First time Henry [١٥] has introduced the duality principle in chemometrics and later Rajkó [١٦] showed that there is a natural duality between the row and column vector spaces of a bilinear data matrix using only the non-negativity property of data set. It is remarkable that this mathematical relation between row and column space provides an efficient tool to transfer the information of the considered space to the dual space. The duality concept is a general principle that can be formulated in an easy way. Simply the duality principle states that each data point corresponds to one special directed hyper-plane in the dual space. This relation can be used for any point in the considered space. Thus, a unique solution as a point fixes a unique directed hyper-plane as its dual subspace. Hence the conditions for achieving the unique solution can be studied simply based on this general principle. Undoubtedly, the necessary condition to get a unique solution is the definition of the directed hyper-plane in the dual space. So according to this concept, there is a generalized constraint for achieving the unique solution. We have shown that implying constraints like trilinearity, known values and local rank constraint can be interpreted based on the duality principle. Similarly, we have explored that other conditions which result in the unique solution such as extracting the net analyte signal and resolution theorem conditions just fulfill the duality principle.