A Deep Monotone Approximation Operator Based on the Best Quadratic Lower Bound of Convex Functions


IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E91-A   No.8   pp.1858-1866
Publication Date: 2008/08/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e91-a.8.1858
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Signal Processing)
monotone approximation,  attracting mapping,  subgradient projection,  convex feasibility problem,  adaptive filtering,  

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This paper presents a closed form solution to a problem of constructing the best lower bound of a convex function under certain conditions. The function is assumed (I) bounded below by -ρ, and (II) differentiable and its derivative is Lipschitz continuous with Lipschitz constant L. To construct the lower bound, it is also assumed that we can use the values ρ and L together with the values of the function and its derivative at one specified point. By using the proposed lower bound, we derive a computationally efficient deep monotone approximation operator to the level set of the function. This operator realizes better approximation than subgradient projection which has been utilized, as a monotone approximation operator to level sets of differentiable convex functions as well as nonsmooth convex functions. Therefore, by using the proposed operator, we can improve many signal processing algorithms essentially based on the subgradient projection.