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Transform-domain least mean squares (LMS) adaptive filters encompass theclass of algorithms where the input data are subjected to a data-independentunitary transform followed by a power normalization stage as preprocessingsteps. Because conventional transformations are not data-dependent, thispreconditioning procedure was shown theoretically to improve the convergence ofthe LMS filter only for certain classes of input data. However, in reality ifthe class of input data is not known beforehand, it is difficult to decidewhich transformation to use. Thus, there is a need to devise a learningframework to obtain such a preconditioning transformation using input dataprior to applying on the input data. It is hypothesized that the underlyingtopology of the data affects the selection of the transformation. With theinput modeled as a weighted graph that mimics neuronal interactions, PrecoGobtains the desired transform by recursive estimation of the graph Laplacianmatrix. Additionally, we show the efficacy of the transform as a generalizedsplit preconditioner on a linear system of equations and in Hebb-LMS settings.In terms of the improvement of the condition number after applying thetransformation, PrecoG performs significantly better than the existingstate-of-the-art techniques that involve unitary and non-unitary transforms.

PrecoG: an efficient unitary split preconditioner for the transform-domain LMS filter via graph Laplacian regularization