Projections onto hyperbolas or bilinear constraint sets in Hilbert spaces

TL;DR

This paper derives a comprehensive formula for projecting onto hyperbolic and bilinear constraint sets in Hilbert spaces, which are crucial in many machine learning models involving bilinear constraints.

Contribution

It provides the first complete formula for projections onto bilinear constraint sets in general Hilbert spaces, advancing mathematical tools for machine learning applications.

Findings

Explicit projection formulas for hyperbolas in Hilbert spaces

Enhanced understanding of bilinear constraint sets in high-dimensional spaces

Potential applications in optimization algorithms for machine learning

Abstract

Sets of bilinear constraints are important in various machine learning models. Mathematically, they are hyperbolas in a product space. In this paper, we give a complete formula for projections onto sets of bilinear constraints or hyperbolas in a general Hilbert space.