# Efficient Kirszbraun Extension with Applications to Regression

## TL;DR

This paper presents a novel regression framework between Hilbert spaces using Kirszbraun's extension theorem, offering improved computational efficiency and empirical performance in supervised learning tasks.

## Contribution

It introduces the first application of Kirszbraun's extension to supervised learning, with a new MWU algorithm that improves runtime and performance.

## Key findings

- Quadratic runtime improvement over existing methods
- Significant empirical performance gains
- Effective decomposition into training and prediction stages

## Abstract

We introduce a framework for performing regression between two Hilbert spaces. This is done based on Kirszbraun's extension theorem, to the best of our knowledge, the first application of this technique to supervised learning. We analyze the statistical and computational aspects of this method. We decompose this task into two stages: training (which corresponds operationally to smoothing/regularization) and prediction (which is achieved via Kirszbraun extension). Both are solved algorithmically via a novel multiplicative weight updates (MWU) scheme, which, for our problem formulation, achieves a quadratic runtime improvement over the state of the art. Our empirical results indicate a dramatic improvement over standard off-the-shelf solvers in our setting.