Weighted Minkowski Distance

• Weighted Minkowski Distance is a generalization of standard Minkowski norms that incorporates nonnegative, normalized weights to adjust the contribution of each feature.
• It enables robust clustering through adaptive weight and exponent mechanisms, mitigating noisy dimensions and alleviating the curse of dimensionality.
• Scalable implementations via explicit kernel feature mappings allow efficient computation and practical integration into advanced k-means extensions.

The weighted Minkowski distance is a generalization of the standard Minkowski (or $L_p$) norms that introduces nonnegative, typically normalized, weight parameters—one per feature dimension or coordinate—to modulate the contribution of each feature when computing inter-point distances. It is central to modern clustering algorithms where features vary significantly in relevance or noise characteristics and is especially prominent in adaptive extensions of $k$-means, both in input and (explicit) kernel-induced feature spaces (Aradnia et al., 2020, Amorim et al., 26 Mar 2026). This distance, parametrized by weights $w = (w_1, \ldots, w_d)$ and exponent $p>0$, interpolates between selective and uniform feature usage, enables robust clustering, and can alleviate the “curse of dimensionality” via fractional norms and power-mean aggregation mechanisms.

1. Mathematical Definition and Norm Properties

Let $x, y \in \mathbb{R}^d$ and $w=(w_1,\dots,w_d)$ be a vector of nonnegative feature weights summing to one. The weighted Minkowski distance of order $p>0$ is

$$d_{p,w}(x,y) = \left( \sum_{j=1}^d w_j\,|x_j-y_j|^p \right)^{1/p}.$$

In feature spaces induced by explicit (approximate) kernel mappings $\phi: \mathbb{R}^D \to \mathbb{R}^{D'}$, the distance is

$$D_{w,p}\bigl(\phi(x),\phi(y)\bigr) = \biggl(\sum_{j=1}^{D'}(w_j)^{p} |\phi_j(x) - \phi_j(y)|^p\biggr)^{1/p}.$$

This formulation covers the Euclidean ($k$0, $k$1), Manhattan ($k$2), and Chebyshev ($k$3) distances as special cases, but allows arbitrary exponents—including fractional $k$4—and arbitrary nonnegative weights.

Weighted Minkowski distances are not metrics for all $k$5; for $k$6 they are quasinorms (sometimes termed “fractional norms”), lacking convexity but providing useful discriminatory power in high dimension (Aradnia et al., 2020).

2. Weight and Exponent Learning Mechanisms

Feature relevance is encoded through the $k$7 weights. In clustering formulations such as Minkowski-weighted $k$8-means (mwk-means) and its kernel variants, each cluster $k$9 may be endowed with its own weight vector $w = (w_1, \ldots, w_d)$0. These weights are adapted in closed form in each iteration: $w = (w_1, \ldots, w_d)$1 where $w = (w_1, \ldots, w_d)$2 is within-cluster dispersion of feature $w = (w_1, \ldots, w_d)$3 (Aradnia et al., 2020). Thus, features with lower dispersion receive more weight, effectively downweighting noisy or uninformative dimensions.

The Minkowski exponent $w = (w_1, \ldots, w_d)$4 governs the selectivity versus uniformity of feature usage. Empirical or data-driven selection is achieved by maximizing the contrast $w = (w_1, \ldots, w_d)$5 or maximizing relative variance $w = (w_1, \ldots, w_d)$6, both of which often peak for $w = (w_1, \ldots, w_d)$7—sometimes even $w = (w_1, \ldots, w_d)$8 (Aradnia et al., 2020).

3. The Power-Mean Aggregation and Feature Selectivity

The weighted Minkowski clustering objective can be reformulated as a sum of power means of within-cluster dispersions: $w = (w_1, \ldots, w_d)$9 where $p>0$0 is the $p>0$1-dispersion of feature $p>0$2 in cluster $p>0$3, $p>0$4 is the power mean of order $p>0$5, and $p>0$6 (Amorim et al., 26 Mar 2026).

This structure leads to:

• Low $p>0$7 ($p>0$8): $p>0$9 approaches $x, y \in \mathbb{R}^d$0, making the objective focus on compact (informative) features, favoring sparse, selective usage.
• Large $x, y \in \mathbb{R}^d$1 ($x, y \in \mathbb{R}^d$2): $x, y \in \mathbb{R}^d$3 approaches the geometric or uniform mean, yielding nearly equal weighting of all features.

Suppression of noisy or high-dispersion features is explicit: for large $x, y \in \mathbb{R}^d$4 relative to other $x, y \in \mathbb{R}^d$5, $x, y \in \mathbb{R}^d$6 tends to zero as a power law $x, y \in \mathbb{R}^d$7 (Amorim et al., 26 Mar 2026).

4. Alternating Optimization in $x, y \in \mathbb{R}^d$8-Means Frameworks

Clustering algorithms based on weighted Minkowski distances typically alternate three steps:

1. Assignment: Points assigned to nearest cluster using $x, y \in \mathbb{R}^d$9.
2. Center Update: Cluster centers recomputed as Minkowski centers, minimizing $w=(w_1,\dots,w_d)$0 for each $w=(w_1,\dots,w_d)$1.
3. Weight Update: Weights updated via $w=(w_1,\dots,w_d)$2, normalized to sum to one.

Each step strictly decreases the objective, guaranteeing convergence in a finite number of steps due to the finiteness of possible partitions (Aradnia et al., 2020, Amorim et al., 26 Mar 2026).

When implemented in explicit kernel feature spaces, the algorithm bypasses $w=(w_1,\dots,w_d)$3 kernel matrix computation by employing finite-dimensional mappings, thus scaling linearly in both $w=(w_1,\dots,w_d)$4 and $w=(w_1,\dots,w_d)$5 for distance and center computations, and enabling easy feature-weight learning (Aradnia et al., 2020).

5. Fractional Norms and Distance Concentration Effects

Distance concentration—the phenomenon wherein distances between points in high dimensions become nearly identical—can severely degrade clustering reliability. For $w=(w_1,\dots,w_d)$6, the expected contrast $w=(w_1,\dots,w_d)$7 decays as dimension increases, leading to clustering ambiguity.

By contrast, setting $w=(w_1,\dots,w_d)$8 (especially $w=(w_1,\dots,w_d)$9) in the weighted Minkowski norm reverses this trend: the contrast grows as $p>0$0, enhancing discrimination between clusters in high-dimensional regimes (Aradnia et al., 2020). This adjustable discrimination is a principal reason for the revival of non-Euclidean, and fractional, norms in modern unsupervised learning.

6. Theoretical Guarantees and Suppression of Noisy Features

The power-mean structure allows precise theoretical bounds: $p>0$1 and for the weighted clustering objective,

$p>0$2

(Amorim et al., 26 Mar 2026). As $p>0$3, noisy features are assigned vanishingly small weights; as $p>0$4, weights approach equality, sacrificing selective suppression. Therefore, weighted Minkowski clustering provides explicit, parameter-controlled guarantees on noise and outlier feature suppression.

7. Computational Benefits via Explicit Feature Mapping

In kernelized $p>0$5-means, direct computation of the Gram matrix for $p>0$6 samples incurs $p>0$7 memory and time per iteration. Replacing implicit kernels with explicit, approximate finite-dimensional feature maps (e.g., via Fourier series sampling for homogeneous additive kernels) yields embeddings $p>0$8 that satisfy $p>0$9, reducing memory to $$d_{p,w}(x,y) = \left( \sum_{j=1}^d w_j\,|x_j-y_j|^p \right)^{1/p}.$$0 and per-distance computation to $$d_{p,w}(x,y) = \left( \sum_{j=1}^d w_j\,|x_j-y_j|^p \right)^{1/p}.$$1 (Aradnia et al., 2020). This enables scalability to large datasets while allowing for explicit feature weighting and integration of advanced $$d_{p,w}(x,y) = \left( \sum_{j=1}^d w_j\,|x_j-y_j|^p \right)^{1/p}.$$2-means extensions in the transformed space.

Algorithm	Memory Complexity	Distance Computation	Weight Learning
Kernel $$d_{p,w}(x,y) = \left( \sum_{j=1}^d w_j\,|x_j-y_j|^p \right)^{1/p}.$$3-means	$$d_{p,w}(x,y) = \left( \sum_{j=1}^d w_j\,|x_j-y_j|^p \right)^{1/p}.$$4	$$d_{p,w}(x,y) = \left( \sum_{j=1}^d w_j\,|x_j-y_j|^p \right)^{1/p}.$$5	Indirect
Explicit KMWK-means	$$d_{p,w}(x,y) = \left( \sum_{j=1}^d w_j\,|x_j-y_j|^p \right)^{1/p}.$$6	$$d_{p,w}(x,y) = \left( \sum_{j=1}^d w_j\,|x_j-y_j|^p \right)^{1/p}.$$7	Direct (vector)

This choice yields efficient clustering, robust selectivity through the weight/exponent mechanism, and practical applicability to nonlinearly separable or high-dimensional data (Aradnia et al., 2020).

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Weighted Minkowski distances, through adaptable exponents and learned weights, offer a unified and theoretically grounded extension to classical distance measures—supporting robust, interpretable, and scalable unsupervised learning both in the original and kernel-embedded feature domains.