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The L1 norm has been tremendously popular in signal and image processing inthe past two decades due to its sparsity-promoting properties. More recently,its generalization to non-Euclidean domains has been found useful in shapeanalysis applications. For example, in conjunction with the minimization of theDirichlet energy, it was shown to produce a compactly supported quasi-harmonicorthonormal basis, dubbed as compressed manifold modes. The continuous L1 normon the manifold is often replaced by the vector l1 norm applied to sampledfunctions. We show that such an approach is incorrect in the sense that it doesnot consistently discretize the continuous norm and warn against itssensitivity to the specific sampling. We propose two alternativediscretizations resulting in an iteratively-reweighed l2 norm. We demonstratethe proposed strategy on the compressed modes problem, which reduces to asequence of simple eigendecomposition problems not requiring non-convexoptimization on Stiefel manifolds and producing more stable and accurateresults.

Consistent Discretization and Minimization of the L1 Norm on Manifolds