  Transform-domain least mean squares (LMS) adaptive filters encompass the
class of algorithms where the input data are subjected to a data-independent
unitary transform followed by a power normalization stage as preprocessing
steps. Because conventional transformations are not data-dependent, this
preconditioning procedure was shown theoretically to improve the convergence of
the LMS filter only for certain classes of input data. However, in reality if
the class of input data is not known beforehand, it is difficult to decide
which transformation to use. Thus, there is a need to devise a learning
framework to obtain such a preconditioning transformation using input data
prior to applying on the input data. It is hypothesized that the underlying
topology of the data affects the selection of the transformation. With the
input modeled as a weighted graph that mimics neuronal interactions, PrecoG
obtains the desired transform by recursive estimation of the graph Laplacian
matrix. Additionally, we show the efficacy of the transform as a generalized
split preconditioner on a linear system of equations and in Hebb-LMS settings.
In terms of the improvement of the condition number after applying the
transformation, PrecoG performs significantly better than the existing
state-of-the-art techniques that involve unitary and non-unitary transforms.
