Local high-order regularization and applications to hp-methods

No . 38 / 2014 Local high - order regularization and applications to hp - methods ( extended version )

We develop a regularization operator based on smoothing on a locally defined length scale. This operator is defined on L and has approximation properties that are given by the local regularity of the function it is applied to and the local length scale. Additionally, the regularized function satisfies inverse estimates commensurate with the approximation orders. By combining this operator with ...

for “ Local High - order Regularization on Data Manifolds ”

We introduce the basics of Riemannian geometry, and the corresponding notation and symbol conventions used in the main paper. For a comprehensive introduction, we refer to [Lee, 1997, Jost, 2011, Hein et al., 2007] from which our exposition has been developed. Then, we show the proof of proposition 1 in the main paper and analyze the speed of the convergence of our RKHS norm-based energy estima...

High-Order Regularization on Graphs

The Laplace-Beltrami operator for graphs has been been widely used in many machine learning issues, such as spectral clustering and transductive inference. Functions on the nodes of a graph with vanishing Laplacian are called harmonic functions. In differential geometry, the Laplace-de Rham operator generalizes the Laplace-Beltrami operator. It is a differential operator on the exterior algebra...

High Order Methods for High-Frequency Scattering Applications

Local Behavior of Sparse Analysis Regularization: Applications to Risk Estimation

In this paper, we aim at recovering an unknown signal x0 from noisy measurements y = Φx0 +w, where Φ is an ill-conditioned or singular linear operator and w accounts for some noise. To regularize such an ill-posed inverse problem, we impose an analysis sparsity prior. More precisely, the recovery is cast as a convex optimization program where the objective is the sum of a quadratic data fidelit...