# Towards Continuous Consistency Axiom

arXiv: https://arxiv.org/abs/2202.06015 · HTML: https://mlsanity.com/paper/2202.06015/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 53 references that are in the map.
- First posted on arXiv 12 Feb 2022.

## Builds on (most cited first, 10 of 10)

- [Dual Student: Breaking the Limits of the Teacher in Semi-supervised Learning](https://mlsanity.com/paper/1909.01804/) (2019)
- [Heavy hitters via cluster-preserving clustering](https://mlsanity.com/paper/1604.01357/) (2016)
- [Axioms for graph clustering quality functions](https://mlsanity.com/paper/1308.3383/) (2013)
- [Possibility results for graph clustering: A novel consistency axiom](https://mlsanity.com/paper/1806.06142/) (2018)
- [Towards an Axiomatic Approach to Hierarchical Clustering of Measures](https://mlsanity.com/paper/1508.03712/) (2015)
- [On the Consistency of k-means++ algorithm](https://mlsanity.com/paper/1702.06120/) (2017)
- [Subspace Clustering through Sub-Clusters](https://mlsanity.com/paper/1811.06580/) (2018)
- [Communities in Preference Networks: Refined Axioms and Beyond](https://mlsanity.com/paper/1603.03543/) (2016)
- [Validity of Clusters Produced By kernel-k-means With Kernel-Trick](https://mlsanity.com/paper/1701.05335/) (2017)
- [Axiomatizing logics of fuzzy preferences using graded modalities](https://mlsanity.com/paper/1909.07674/) (2019)

## Built on by (most cited first, 4 of 4)

- [A Clustering Preserving Transformation for k-Means Algorithm Output](https://mlsanity.com/paper/2202.10455/) (2022)
- [How To Overcome Richness Axiom Fallacy](https://mlsanity.com/paper/2210.15507/) (2022)
- [High-Dimensional Wide Gap k-Means Versus Clustering Axioms](https://mlsanity.com/paper/2211.17036/) (2022)
- [Wide Gaps and Clustering Axioms](https://mlsanity.com/paper/2308.03464/) (2023)

## Abstract (arXiv)

Development of new algorithms in the area of machine learning, especially clustering, comparative studies of such algorithms as well as testing according to software engineering principles requires availability of labeled data sets. While standard benchmarks are made available, a broader range of such data sets is necessary in order to avoid the problem of overfitting. In this context, theoretical works on axiomatization of clustering algorithms, especially axioms on clustering preserving transformations are quite a cheap way to produce labeled data sets from existing ones. However, the frequently cited axiomatic system of Kleinberg:2002, as we show in this paper, is not applicable for finite dimensional Euclidean spaces, in which many algorithms like k-means, operate. In particular, the so-called outer-consistency axiom fails upon making small changes in datapoint positions and inner-consistency axiom is valid only for identity transformation in general settings. Hence we propose an alternative axiomatic system, in which Kleinberg's inner consistency axiom is replaced by a centric consistency axiom and outer consistency axiom is replaced by motion consistency axiom. We demonstrate that the new system is satisfiable for a hierarchical version of k-means with auto-adjusted k, hence it is not contradictory. Additionally, as k-means creates convex clusters only, we demonstrate that it is possible to create a version detecting concave clusters and still the axiomatic system can be satisfied. The practical application area of such an axiomatic system may be the generation of new labeled test data from existent ones for clustering algorithm testing. %We propose the gravitational consistency as a replacement which does not have this deficiency.