Use cases of Holder continuity in Machine Learning domain part7
- L1-optimal linear programming estimatorfor periodic frontier functions with Holder continuous derivative(arXiv)
Author : Alexander Nazin, Stephane Girard
Abstract : We propose a new estimator based on a linear programming method for smooth frontiers of sample points. The derivative of the frontier function is supposed to be Holder continuous.The estimator is defined as a linear combination of kernel functions being sufficiently regular, covering all the points and whose associated support is of smallest surface. The coefficients of the linear combination are computed by solving a linear programming problem. The L1- error between the estimated and the true frontier functionsis shown to be almost surely converging to zero, and the rate of convergence is proved to be optimal
2.Holder Continuous Solutions of Active Scalar Equations (arXiv)
Author : Philip Isett, Vlad Vicol
Abstract : We consider active scalar equations ∂tθ+∇⋅(uθ)=0, where u=T[θ] is a divergence-free velocity field, and T is a Fourier multiplier operator with symbol m. We prove that when m is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with Hölder regularity C1/9−t,x. In fact, every integral conserving scalar field can be approximated in D′ by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when the multiplier m is odd, weak limits of solutions are solutions, so that the h-principle for odd active scalars may not be expected
