MGCPL-Guided Categorical Data Clustering

• The paper introduces MCDC, a novel framework that iteratively optimizes weighted intra-cluster similarity to uncover nested clusters in categorical data.
• MCDC leverages stage-wise multi-granularity discovery with CAME to aggregate multi-level clusterings into high-quality partitional clusters.
• The method achieves linear time complexity and superior benchmark performance, offering scalable and robust clustering for large categorical datasets.

MGCPL-guided Categorical Data Clustering (MCDC) is a robust clustering methodology tailored for data sets composed exclusively of categorical features. The approach exploits the intrinsic nested granularity and overlap frequently observed in the discrete space of categorical data, where clusters manifest across multiple, hierarchically nested levels. By overcoming the challenge posed by the absence of well-defined distance metrics for categorical values, MCDC leverages Multi-Granular Competitive Penalization Learning (MGCPL) to interactively tune and converge clusters across successive granularities. This is followed by Cluster Aggregation based on MGCPL Encoding (CAME), which consolidates multi-granular labelings into high-quality partitional clusters. The pipeline combines theoretical convergence, per-cluster feature weighting, and linear time complexity, yielding superior performance across benchmark data sets and scalability for big data environments (Cai et al., 23 Jan 2026).

1. Categorical Data Clustering Challenges and Granularity Effects

Categorical data analysis is complicated by the absence of a natural Euclidean metric. Categorical features are defined by qualitative, discrete values with limited cardinality. The overlap among data points in feature space commonly results in the formation of compact clusters, while these small clusters themselves can aggregate into larger clusters, creating a nested multi-granular cluster structure. The implicit discrete distance space is not amenable to standard geometric clustering heuristics, prompting the need for algorithms capable of automatically discovering meaningful cluster numbers and granularities within such data (Cai et al., 23 Jan 2026).

2. Multi-Granular Competitive Penalization Learning (MGCPL)

MGCPL is an iterative clustering mechanism designed to capture categorical clustering structure at several nested levels. At a given granularity, with $k$ active cluster prototypes $C_1, ..., C_k$, MGCPL maximizes weighted intra-cluster similarity:

$$S(Q,\mathbf u)\;=\;\sum_{l=1}^k\;\sum_{i=1}^n u_l\,q_{il}\,s\bigl(x_i,C_l\bigr)$$

where $q_{il}$ denotes assignment, $u_l$ an adaptive cluster weight, and $s(x_i, C_l)$ the feature-wise similarity, adjusted via per-feature importance. Penalization at the prototype level is performed whenever the winning cluster for a data object is determined, with the closest rival penalized via negative update to its score, thus avoiding premature winner-takes-all collapse and facilitating the emergence of fine-grained clusters.

This iterative process involves:

1. Score computation and competitive assignment.
2. Per-iteration adjustment of cluster importance via score variable updates.
3. Per-cluster prototype update using the mode of categorical values.
4. Feature weighting per cluster, using the combination of inter-cluster difference ($\alpha_{rl}$) and intra-cluster similarity ($\beta_{rl}$).

Upon stabilization of assignments, clusters with near-zero adaptive weight are eliminated, yielding a reduced set of prototypes. The process is repeated sequentially, each time with the surviving clusters, until stabilization yields the finest granularity representing the multi-scale structure in the data (Cai et al., 23 Jan 2026).

3. Stage-wise Multi-Granularity Discovery and Encoding

MGCPL operates across $\sigma$ stages:

• Initialization with $k_0 \gg k^*$ clusters.
• Penalized competitive learning at each stage to obtain $k_1 < k_0$, followed by recursive reduction to $k_\sigma$ clusters, where convergence is achieved.
• Each stage produces a partition $Y_t$, and the sequence of granularities is collected as $\Gamma = \{Y_1, \dots, Y_\sigma\}$.

The multi-granular labelings from each stage ($Y_t \in \{1, ..., k_t\}^n$) enable the encoding of each data object into a categorical feature vector representing cluster assignments across all explored granularities (Cai et al., 23 Jan 2026). This forms the new feature space for aggregation.

4. Cluster Aggregation based on MGCPL Encoding (CAME)

CAME consolidates multi-granular clustering outcomes into a final partition. The set of assignment labels at each granularity level are treated as new categorical features. Given the desired final cluster number $k$, CAME optimizes a weighted $k$-modes objective:

$$\min_{Q,\Theta} P(Q, \Theta) = \sum_{l=1}^k \sum_{i=1}^n \sum_{r=1}^\sigma q_{il}\,\theta_r\,d(x_{ir}, Z_{lr})$$

where $d(\cdot, \cdot)$ is Hamming distance on the encoded categories, and $\theta_r$ are feature importances over granularities, recomputed based on intra-cluster similarity.

The algorithm alternates between updating assignments (mode-based label matching under $\Theta$) and updating feature weights ($\theta_r$), guaranteeing monotonic improvement and convergence. The result is a high-fidelity partitional clustering that integrates cluster information across all identified granularities (Cai et al., 23 Jan 2026).

5. Computational Complexity and Theoretical Properties

• MGCPL: Each granularity stage operates in $O(d n k_0)$ time, maintaining linear complexity in data dimensionality $d$, number of objects $n$, and initial clusters $k_0$.
• CAME: Aggregation phase proceeds in $O(\sigma n k)$.
• Full MCDC Pipeline: Linear time complexity overall.

Theoretical guarantees include monotonic increase and boundedness of penalized similarity during MGCPL updates, convergence of each stage, and classical convergence for the weighted $k$-modes in CAME. Per-cluster feature weighting ensures robustness to heterogeneous categorical distributions (Cai et al., 23 Jan 2026).

6. Empirical Performance and Comparative Analysis

Experimental evaluation on ten categorical data sets (eight UCI benchmarks, two synthetic) and comparison against nine baseline methods ($k$-modes, ROCK, WOCIL, FKMAWCW, GUDMM, ADC, two MCDC+ variants embedding GUDMM or FKMAWCW in CAME) demonstrates that MCDC—and especially MCDC+FKMAWCW—achieves top clustering metrics (ACC, ARI, AMI, FM). Superiority is statistically confirmed by Wilcoxon tests with 90% confidence.

Ablation studies highlight the importance of both CAME and per-feature weighting: omitting these components degrades performance, while reverting to single-granularity CPL diminishes clustering quality, substantiating the impact of the multi-granular approach.

The sequence $\{k_1, k_2, ..., k_\sigma\}$ reliably uncovers the true $k^*$ on all benchmarks, indicating effective automatic model selection. Scalability tests confirm linear growth in $n$, $d$, and $k$, with MCDC exhibiting substantially greater efficiency than hierarchical alternatives (Cai et al., 23 Jan 2026).

7. Concluding Principles and Prospective Applications

MGCPL-guided Categorical Data Clustering (MCDC) provides a systematic mechanism for uncovering the nested granular structure of categorical data. Through the interplay of competitive learning, rival penalization, and adaptive feature weighting, MGCPL extracts compact clusters across granularities. CAME consolidates these multi-granular encodings into high-accuracy partitions. The approach is computationally efficient, theoretically robust, and well-suited to large-scale, pure-categorical clustering scenarios, including pre-partitioning for distributed systems and boosting data analysis pipelines in big data contexts (Cai et al., 23 Jan 2026).