Support vector regression for the solution of linear integral equations

Abstract

The aim of this paper is to establish a support vector regression method for semi-discrete ill-posed problems. We consider the equation Af = g, where a linear integral operator A is known; discrete measurements of the right-hand side are observed and a solution f * is sought. For the reconstruction, instead of a standard square loss function, Vapnik's -intensive function is used to measure the distance between Af and g. This avoids an overfitting to disturbed data and guarantees additional stability given that the cut-off parameter is chosen appropriately. The resulting solution procedure can be formulated as a quadratic program. Besides the method, a Sobolev error analysis and a parameter strategy for the regularization parameters are provided. The results are substantiated with a numerical example.

Multiscale Support Vector Approach for Solving Ill-Posed Problems

“…We remark here that the minimizer of (9) falls into the space H Φ,X automatically and hence avoid the artificial assumption given in [23] that the exact solution f * should belong to H Y .…”

“…3. At the same time, we present the single-level SVA approximate solution, following the same approach in [23]. One can observe the improvement by the l 2 and l ∞ errors for multiscale method.…”

“…We refer [23] that the coefficients are determined by a quadratic minimization problem, where the symmetric collocation matrix with entries…”

“…to obtain an error estimate between the exact solution and approximate one (see Corollary 4.9 in [23]). Nevertheless such a choice is not reasonable in our multiscale SVA, in the sense that each norm Es δ, * k 2 Φ k shall be known a priori.…”

“…Recently, an extension of the TP scheme to support vector approach (SVA) has been investigated by Krebs [23] in which the Vapnik -intensive distance function was used to replace the l 2 loss function in the data-fitting term of (2) to greatly reduce the influence of small data errors. It is noted that the Vapnik function plays an important role in the framework of statistical learning theory.…”