Comments on "Iteratively Re-weighted Algorithm for Fuzzy c-Means"

TL;DR

This paper offers a simpler derivation of the IRW-FCM algorithm for Fuzzy c-Means, revealing it as a Majorization Minimization method, which allows for a more efficient single-loop implementation.

Contribution

It presents a new, straightforward MM-based derivation of IRW-FCM, eliminating the need for auxiliary variables and enabling faster algorithm execution.

Findings

The IRW-FCM steps are equivalent to a Majorization Minimization algorithm.

A single inner loop suffices to decrease the Fuzzy c-means objective.

The proposed derivation simplifies and speeds up the IRW-FCM algorithm.

Abstract

In this comment, we present a simple alternate derivation to the IRW-FCM algorithm presented in "Iteratively Re-weighted Algorithm for Fuzzy c-Means" for Fuzzy c-Means problem. We show that the iterative steps derived for IRW-FCM algorithm are nothing but steps of the popular Majorization Minimization (MM) algorithm. The derivation presented in this note is much simpler and straightforward and, unlike the derivation of IRW-FCM, the derivation here does not involve introduction of any auxiliary variable. Moreover, by showing the steps of IRW-FCM as the MM algorithm, the inner loop of the IRW-FCM algorithm can be eliminated and the algorithm can be effectively run as a "single loop" algorithm. More precisely, the new MM-based derivation deduces that a single inner loop of IRW-FCM is sufficient to decrease the Fuzzy c-means objective function, which speeds up the IRW-FCM algorithm.