A note on hypercircle inequality for data error with l 1 $l^{1}$ norm

Abstract

Abstract In our previous work, we have extended the hypercircle inequality (HI) to situations where the data error is known. Furthermore, the most recent result is applied to the problem of learning a function value in the reproducing kernel Hilbert space. Specifically, a computational experiment of the method of hypercircle, where the data error is measured with the l p $l^{p}$ norm ( 1 < p ≤ ∞ ) $(1< p \leq \infty )$ , is compared to the regularization method, which is a standard method of the learning problem. Despite this breakthrough, there is still a significant aspect of data error measure with the l 1 $l^{1}$ norm to consider in this issue. In this paper, we do not only explore the hypercircle inequality for the data error measured with the l 1 $l^{1}$ norm, but also provide an unexpected application of hypercircle inequality for only one data error to the l ∞ $l^{\infty}$ minimization problem, which is a dual problem in this case.

Keywords

• Hypercircle inequality
• Convex optimization
• Noise data