Consistent Discretization and Minimization of the L1 Norm on Manifolds

TL;DR

This paper addresses the inconsistency of common L1 norm discretizations on manifolds, proposing two alternative methods that improve stability and accuracy in shape analysis applications.

Contribution

It introduces two new discretization strategies for the L1 norm on manifolds, replacing the naive sampled vector approach with an iteratively-reweighted l2 norm for better consistency.

Findings

Proposed discretizations are more stable and accurate.

The method simplifies to eigendecomposition problems.

Improves shape analysis tasks like compressed manifold modes.

Abstract

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.