  The L1 norm has been tremendously popular in signal and image processing in
the past two decades due to its sparsity-promoting properties. More recently,
its generalization to non-Euclidean domains has been found useful in shape
analysis applications. For example, in conjunction with the minimization of the
Dirichlet energy, it was shown to produce a compactly supported quasi-harmonic
orthonormal basis, dubbed as compressed manifold modes. The continuous L1 norm
on the manifold is often replaced by the vector l1 norm applied to sampled
functions. We show that such an approach is incorrect in the sense that it does
not consistently discretize the continuous norm and warn against its
sensitivity to the specific sampling. We propose two alternative
discretizations resulting in an iteratively-reweighed l2 norm. We demonstrate
the proposed strategy on the compressed modes problem, which reduces to a
sequence of simple eigendecomposition problems not requiring non-convex
optimization on Stiefel manifolds and producing more stable and accurate
results.
