A Non-Parametric Constrained Regression Algorithm for Dual Norm Computation and its Applications

Abstract

Dual norms are useful to modelize smooth and oscillating patterns for decomposingan image into a geometrical component and a textured component.As the dual space of Banach space W1,p0 (Ω) defining (1, p) norm, W−1,q(Ω)corresponds to its dual norm which is an effective description for oscillatingpatterns(smaller for oscillating patterns than smooth patterns) with noassumption of noise process and model(i.e. various types of uncorrelatedand correlated noise). Scholars define the dualnorm of total variation(TV ) as G norm which is the limit of (−1, q) normwith q = ∞. Combined with priors for non-oscillating components, G norm and (−1, q) norm canplay an important role in various image decomposition tasks. However traditionalcomputation of G norm uses projection algorithm and dichotomywhile traditional computation for (−1, q) norm uses steepest descent withouttheoretically guaranteed convergence rate. These methods are inefficientand inaccurate. In this paper, we propose a non-parametricconstrained regression algorithm for G norm and (−1, q) norm computationvia adding lq norm linear regression module(q ∈ [2,∞]) and non-parametricregularization module. Compared with the traditional algorithm, the proposedalgorithm has a theoretically guaranteed bound for faster convergenceto a (1 + ϵ)-approximate solution. With the proposed algorithm, we discussits potential application to calibrate the parameter of traditional dual normbased image decomposition. Since TV can be replaced with other traditionaland deep image priors, the proposed algorithm is general and usefulwith potential applications in many areas.

Keywords: dual norms, G norm, (−1, q) norm, lq norm linear regression, iterativereweighted least squares, non-parametric regularization, geometricconvergence