Projection Onto A Simplex

TL;DR

This paper introduces a fast, simple algorithm for projecting vectors onto the canonical simplex, leveraging Moreau's identity to reduce the problem to a univariate convex minimization with explicit candidate solutions.

Contribution

It provides a novel, efficient method for simplex projection by explicitly identifying candidate solutions and ensuring the unique minimizer can be computed directly.

Findings

The algorithm is fast and simple to implement.

It reduces the problem to a univariate convex minimization.

Explicitly computes at most n candidate solutions.

Abstract

This mini-paper presents a fast and simple algorithm to compute the projection onto the canonical simplex $\triangle^n$. Utilizing the Moreau's identity, we show that the problem is essentially a univariate minimization and the objective function is strictly convex and continuously differentiable. Moreover, it is shown that there are at most n candidates which can be computed explicitly, and the minimizer is the only one that falls into the correct interval.