Relations Between Higher Order TV Regularization and Support Vector Regression

A Note on the Dual Treatment of Higher-Order Regularization Functionals

“…This problem which is just the Fenchel dual of (2) In [25] we have examined higher order TV regularization in one dimension from a different point of view, namely with respect to its relation to spline interpolation with variable knots and to SVR with discrete spline kernels. Finishing [25], we became aware of its close relation to Legendre-Fenchel dualization techniques.…”

“…Finishing [25], we became aware of its close relation to Legendre-Fenchel dualization techniques. Since this was indeed the motivation to write this paper, we briefly want to explain the relation to [25]. By adding an appropriate last row toD N,1 ∈ R N −1,N , we introduce the Toeplitz matrix …”

“…For tube methods in more than one dimension, we refer to [11]. In [25], we have shown that for given F ∈ R N , the solution U of (6) is a discrete spline of degree 2m − 1 which interpolates F ± λ at its spline knots. For discrete splines, we refer to [16].…”

“…For the onedimensional setting, numerical experiments are already contained in [25]. Therefore, we restrict our attention to two dimensions.…”

A Guide to the TV Zoo

“…The second order total variation has been investigated and applied in [68,105,171,176], duality properties have been investigated in [175]. The extension to higher orders is obvious, but so far hardly investigated, in particular in applications, probably also due to the difficulty of treating higher-order functionals.…”