  In this paper we propose a framework to leverage Lie group symmetries on
arbitrary spaces exploiting \textit{algebraic signal processing} (ASP). We show
that traditional group convolutions are one particular instantiation of a more
general Lie group algebra homomorphism associated to an algebraic signal model
rooted in the Lie group algebra $L^{1}(G)$ for given Lie group $G$. Exploiting
this fact, we decouple the discretization of the Lie group convolution
elucidating two separate sampling instances: the filter and the signal. To
discretize the filters, we exploit the exponential map that links a Lie group
with its associated Lie algebra. We show that the discrete Lie group filter
learned from the data determines a unique filter in $L^{1}(G)$, and we show how
this uniqueness of representation is defined by the bandwidth of the filter
given a spectral representation. We also derive error bounds for the
approximations of the filters in $L^{1}(G)$ with respect to its learned
discrete representations. The proposed framework allows the processing of
signals on spaces of arbitrary dimension and where the actions of some elements
of the group are not necessarily well defined. Finally, we show that multigraph
convolutional signal models come as the natural discrete realization of Lie
group signal processing models, and we use this connection to establish
stability results for Lie group algebra filters. To evaluate numerically our
results, we build neural networks with these filters and we apply them in
multiple datasets, including a knot classification problem.
