Source-linked AI summary
Learning Order Forest for Qualitative-Attribute Data Clustering
Mingjie Zhao, Sen Feng, Yiqun Zhang, Mengke Li, Yang Lu, Yiu-ming Cheung
TL;DR
Qualitative attributes lack explicit distance structures, especially when nominal values have no known semantic order. COForest jointly learns tree-like distance structures and clusters, and experiments report superior clustering performance across benchmark comparisons. The method is evaluated on static qualitative data with the true number of clusters given.
Problem
Qualitative values do not inherently provide a distance structure, and existing methods may rely on explicit semantic ordering unavailable for nominal attributes.
Method
COForest alternates between inferring graph structures from the current partition and clustering with the resulting graph distances, using minimal spanning trees to form an order forest.
Results
Extensive experiments report superior clustering performance, with significance tests, ablation studies, and visual comparisons supporting the method's effectiveness.
Takeaways & Limitations
The learned tree-like structures are concise and interpretable representations of implicit qualitative-data distributions, while the parameter-free procedure can be applied to various datasets.
Takeaways & Limitations
The method is demonstrated for static qualitative data when the true number of clusters is given; future work targets streaming mixed-type data with unknown imbalanced clusters.
Abstract
from arXiv · showhide
Clustering is a fundamental approach to understanding data patterns, wherein the intuitive Euclidean distance space is commonly adopted. However, this is not the case for implicit cluster distributions reflected by qualitative attribute values, e.g., the nominal values of attributes like symptoms, marital status, etc. This paper, therefore, discovered a tree-like distance structure to flexibly represent the local order relationship among intra-attribute qualitative values. That is, treating a value as the vertex of the tree allows to capture rich order relationships among the vertex value and the others. To obtain the trees in a clustering-friendly form, a joint learning mechanism is proposed to iteratively obtain more appropriate tree structures and clusters. It turns out that the latent distance space of the whole dataset can be well-represented by a forest consisting of the learned trees. Extensive experiments demonstrate that the joint learning adapts the forest to the clustering task to yield accurate results. Comparisons of 10 counterparts on 12 real benchmark datasets with significance tests verify the superiority of the proposed method.
1 Introduction
Qualitative attributes lack an inherent distance structure, while existing approaches either simplify value relationships or rely on prior semantic assumptions. COForest addresses this by jointly learning flexible graph structures and clusters.
- Qualitative values such as occupations lack a known Euclidean embedding, making their implicit distance structure difficult to represent for clustering.
- Hamming distance only tests whether qualitative values match, while other measures may consider values individually rather than their overall structure.
- Graph-based methods represent ordinal values with line graphs and nominal values with fully connected graphs, but depend on topology assumptions or domain knowledge.
- Random graph structures can outperform fully connected and semantic line graphs, indicating that restricting value relationships by prior knowledge may limit clustering accuracy.
- COForest jointly learns graph structures and clusters by alternating partition-based graph inference with clustering over the learned graph distances.
- Its order-forest formulation learns distance structures without prior knowledge bias and permits topology reconstruction beyond adjusting distances on a fixed graph.
2 Propose Method
COForest represents qualitative-value relationships with per-attribute minimum spanning trees and learns those structures jointly with clustering. The resulting trace-based distances form a valid forest-level distance metric while retaining local order relationships and linear complexity in samples and attributes when category cardinality is small.
- Motivation: Categorical values lack a known Euclidean distance structure, so clustering requires learning relationships among possible values rather than assuming numerical order.Prior line graphs impose semantic order, while fully connected graphs retain redundant paths.
- 2.2 Order Forest Construction: COForest constructs an order forest M whose tree M_r is a minimum spanning tree for attribute a_r.Each tree connects all possible values using o_r−1 minimum-sum edges.
- 2.2 Order Forest Construction: Each order tree provides a unique shortest-path trace between every pair of values, enabling local order relationships without forcing all values onto one line.Trace distance is obtained from the weights along that path.
- 2.3 Clustering-Friendly Trace Distance: Edge weights compare values through their cluster-wise probability distributions, and sample-cluster distance combines trace distances with within-cluster value distributions across attributes.The method uses p-norm differences between value distributions and aggregates attribute-level distances over the forest.
- 2.3 Clustering-Friendly Trace Distance: The learned trace distance and the overall sample-cluster distance are valid distance metrics.The trace metric satisfies non-negativity, symmetry, and triangle inequality; the paper states the forest-level metric is valid as well.
- 2.4 Joint Learning Algorithm: With maximum attribute cardinality treated as constant, COForest has overall time complexity O(nlkIE), linear in the number of samples n and attributes l.The unsimplified complexity is O(E(Inklς + nlς^2)).
3 Experiments
Experiments evaluate COForest against 10 counterparts on 12 real public datasets using three clustering metrics, ablations, convergence, efficiency, and learned-distance visualizations. COForest generally achieves superior clustering while its iterative learning converges efficiently and scales comparably with state-of-the-art methods.
- Experimental settings: COForest is compared with 10 counterparts on 12 real public datasets using clustering accuracy, ARI, and NMI.Each method is implemented 10 times, and average performance is reported.
- Clustering performance: COForest performs best on almost all datasets and significantly outperforms the compared methods in statistical tests.Friedman tests yield p-values of 0.00020 for CA and 0.00002 for ARI; post-hoc tests generally support COForest’s superiority.
- Ablation study: COForest outperforms its four ablated variants, supporting joint forest-and-clustering learning, the order-forest structure, and probability-distribution-based weights.COForest is not worse than COFI on all datasets, COFI is not worse than line-graph and fully connected variants on 10 of 12 datasets, and COFIII beats Hamming-distance COFIV on 10 datasets.
- Convergence and efficiency evaluation: After each order-forest update, the objective L decreases, and COForest converges within 15 iterations in most cases.The convergence curves track learning iterations and objective values across all 12 datasets.
- Convergence and efficiency evaluation: COForest’s execution time is lower than or similar to UDMC, DLC, and H2H, with an almost linear increase as sample size and attribute count grow.The efficiency study varies samples from 10k to 100k and attributes from 1k to 10k on synthetic datasets.
- Convergence and efficiency evaluation: t-SNE visualizations of the AC dataset use learned distances from COForest, CBDM, and ADC to illustrate cluster discrimination and distance-structure intuitiveness.The encoded data are reduced to two dimensions and marked with true labels.
4 Relate Work
Related work improves categorical-data clustering by learning distance representations, but remains constrained by assumptions about distance structure and prior knowledge.
- Distance measures and learning: Categorical-data clustering methods include distance measures and distance-learning approaches that incorporate distance definition into clustering.These approaches range from encoding and statistical measures to kernels, mixed representations, and other learned distance structures.
- Distance measures and learning: Hamming-based encoding captures only whether categorical values match, limiting its representation of dissimilarity.Statistical measures improve discrimination by using value frequencies and entropy-related information.
- Distance measures and learning: Distance-learning methods use kernels, mixed encodings, and other representations to model categorical data and attribute couplings.These methods often require careful hyperparameter tuning.
- Open challenge: Prior knowledge is both necessary for effective distance learning and difficult to obtain independently of predefined distance metrics.This creates a tension between assuming a suitable distance structure and learning data knowledge from that structure.
5 Concluding Remarks
COForest jointly learns tree-like distance structures and clusters, producing interpretable representations for qualitative data. Its reported performance is strong on static qualitative data when the true number of clusters is given, while broader settings remain future work.
- Concluding remarks: COForest jointly optimizes distance-structure construction and clustering through a parameter-free learning process given the number of clusters.The learned tree-like structures are described as concise and interpretable for representing implicit qualitative-data distributions.
- Concluding remarks: COForest demonstrates outstanding clustering performance on static qualitative data under the given true number of clusters.The paper identifies broader settings, including streaming mixed-type data and unknown imbalanced cluster counts, as future directions.