A Novel k-Means Framework via Constrained Relaxation and Spectral Rotation

Owing to its simplicity, the traditional $k$ - means (Lloyd heuristic) clustering method plays a vital role in a variety of machine-learning applications. Disappointingly, the Lloyd heuristic is prone to local minima. In this article, we propose $k$ - mRSR, which converts the sum-of-squared error (SSE) (Lloyd) into a combinatorial optimization problem and incorporates a relaxed trace maximization term and an improved spectral rotation term. The main advantage of $k$ - mRSR is that it only needs to solve the membership matrix instead of computing the cluster centers in each iteration. Furthermore, we present a nonredundant coordinate descent method that brings the discrete solution infinitely close to the scaled partition matrix. Two novel findings from the experiments are that $k$ - mRSR can further decrease (increase) the objective function values of the $k$ - means obtained by Lloyd (CD), while Lloyd (CD) cannot decrease (increase) the objective function obtained by $k$ - mRSR. In addition, the results of extensive experiments on 15 datasets indicate that $k$ - mRSR outperforms both Lloyd and CD in terms of the objective function value and outperforms other state-of-the-art methods in terms of clustering performance.