On the Objective and Feature Weights of Minkowski Weighted k-Means

Abstract

The Minkowski weighted k-means (mwk-means) algorithm extends classical k-means by incorporating feature weights and a Minkowski distance. Despite its empirical success, its theoretical properties remain insufficiently understood. We show that the mwk-means objective can be expressed as a power-mean aggregation of within-cluster dispersions, with the order determined by the Minkowski exponent p. This formulation reveals how p controls the transition between selective and uniform use of features. Using this representation, we derive bounds for the objective function and characterise the structure of the feature weights, showing that they depend only on relative dispersion and follow a power-law relationship with dispersion ratios. This leads to explicit guarantees on the suppression of high-dispersion features. Finally, we establish convergence of the algorithm and provide a unified theoretical interpretation of its behaviour.

Figures (2)

• Figure 1: Sorted feature weights for different values of $p$. Smaller $p$ yields sparse weight distributions, while larger $p$ produces near-uniform weights, consistent with Proposition \ref{['prop:weight_structure']}.
• Figure 2: Values of the normalised objective across datasets and runs for different values of $p$. The shaded region denotes the theoretical bounds $[0,1]$ established in Theorem \ref{['thm:mwk_bounds']}. All observed values lie within this interval, confirming that the objective remains within the derived bounds as $p$ varies.

Theorems & Definitions (21)

• Proposition 1
• proof
• Theorem 1
• proof
• Lemma 1
• proof
• Definition 1
• Lemma 2
• proof
• Corollary 1