Minimum Sobolev Norm schemes and applications in image processing

Abstract

This paper describes an extension of the Minimum Sobolev Norm interpolation scheme to an approximation scheme. A fast implementation of the MSN interpolation method using the methods for Hierarchical Semiseparable (HSS) matrices is described and experimental results are provided. The approximation scheme is introduced along with a numerically stable solver. Several numerical results are provided comparing the interpolation scheme, the approximation scheme and Thin Plate Splines. A method to decompose images into smooth and rough components is presented. A metric that could be used to distinguish edges and textures in the rough component is also introduced. Suitable examples are provided for both the above.

Impossibility of Fast Stable Approximation of Analytic Functions from Equispaced Samples

“…One variant of this idea explored by the first author (unpublished) is to fit data on the equispaced n-grid by a polynomial of degree d n − 1, using the extra d + 1 − n coefficients to minimize a norm that measures the degree of oscillations. A similar method has been developed by Chandrasekaran et al under the name of the Minimal Sobolev Norm scheme [10]. Perhaps the first contribution of this kind was by Boyd in 1992 [4].…”

An analysis of training and generalization errors in shallow and deep networks

“…Theorems 3.3 and 4.2 suggest that the approximation error given by the trigonometric polynomials constructed there gives an estimate on the support of the marginal distribution of the training and test data. In [4], the approximation errors of the convergent bounded interpolatory polynomials constructed in [3] were used for texture detection and segmentation of images. What would be the analogues for understanding the nature of the data using the approximation errors obtained here?…”

“…The emphasis here is on an estimation of how much over-parametrization is necessary to get theoretical guarantees. 4. What bounds on the generalization error can be guaranteed by the solution of the (global) regularization scheme at each point in the test data (rather than a global error estimate), compared to the nearest neighbor in the training data?…”

Higher Order Numerical Discretization Methods with Sobolev Norm Minimization

“…The MSN interpolation scheme computes an interpolating polynomial of order higher than the number of interpolating nodes, whose sobolev norm is minimum [1] [2]. Traditional lagrange interpolation uses a polynomial of order equal to the number of interpolating nodes.…”

“…This paper introduces a higher order finite difference scheme for solving two dimensional elliptic PDEs based on Minimum Sobolev Norm (MSN) interpolation [1] [2]. The MSN scheme is a higher order interpolation idea, that suppresses Runge Oscillations by minimizing an appropriately set up Sobolev Norm of the interpolant.…”