Learnable Weighting of Intra-Attribute Distances for Categorical Data Clustering with Nominal and Ordinal Attributes

Clustering mixed tabular data requires a unified metric space to bridge the inherent heterogeneity between continuous numerical measurements and discrete categorical symbols. Traditionally, algorithms rely entirely on dataset-internal statistics to estimate categorical relationships, which confines the learned metric to empirical co-occurrences and ignores conceptually obvious yet statistically unobserved affinities. Although LLMs offer external world knowledge, applying their text-centric reasoning to highly abstract tabular concepts presents significant challenges. Bridging this modality gap to construct a semantically complete metric typically requires embedding LLMs into iterative metric learning loops to dynamically optimize cross-modality representations. This incurs intractable computational overhead, forcing a compromise between semantic enrichment and scalability. Therefore, we propose GRACE, an LLM-grounded framework for scalable mixed-data clustering. GRACE shifts semantic acquisition to the attribute-value level via a multi-perspective LLM querying strategy, mapping heterogeneous values into knowledge-informed descriptions. Crucially, this one-shot grounding extracts general-purpose semantic representations that embed heterogeneous attributes into a unified space, decoupling expensive LLM invocation from iterative optimization. Furthermore, GRACE cross-validates these external semantics against dataset-internal statistical evidence to ensure alignment with the dataset-specific cluster structure. Ultimately, GRACE matches the scalability of conventional statistics-driven baselines while achieving superior clustering accuracy and conceptual interpretability over 11 competing methods. The source code is available at https://github.com/develop-yang/GRACE-GRACE-A

K-Means clustering algorithm is one of the most commonly used clustering algorithms because of its simplicity and efficiency. K-Means clustering algorithm based on Euclidean distance only pays attention to the linear distance between Euclidean distance is an efficient and interpretable similarity measurement, but its effectiveness may deteriorate in sample spaces with anisotropic structures, redundant features, or complex feature interactions. In this paper, we propose a novel distance metric called View distance. Inspired by orthographic projection, the proposed metric projects the sample space onto $n(n-1)/2$ two-dimensional planes and defines the final distance as the sum of the Euclidean distances across the projected planes. Theoretical derivations verify that the View distance strictly satisfies the metric axioms and norm constraints. Beyond that, the View distance achieves feature coupling through projection and not only enables constant features to indirectly participate in distance calculation, but also suppresses interference from redundant features while exhibiting anisotropic geometric properties. Furthermore, to address the high computational complexity and poor scalability of full-projection View distance, we propose a two-dimensional projection plane selection strategy based on iterative Maximum Weight Matching, which reduces the computational complexity of distance calculation from $\mathcal{O}(n^2)$ to $\mathcal{O}(k)$. Extensive experiments on 12 diverse datasets demonstrate that View distance and the deterministic selection strategy provide competitive or superior performance compared with Euclidean distance and other $L_{p}$ metrics while maintaining strong interpretability and computational efficiency. The View distance provides a new perspective and option for similarity measurements.

Ordinal Classification (OC) deals with classification tasks where the classes follow a natural order. Despite the progress in OC, many existing approaches fail to fully leverage the ordinal information, treating the problem as nominal classification and thereby losing performance potential. In this work, ADABORD, an AdaBoost framework specifically designed for ordinal classification problems, is introduced. The ordinal nature of the classes is incorporated into two key components of the well-known AdaBoost algorithm: 1) the base estimator, where decision trees with the ordinal Gini splitting criterion are proposed; 2) the error function used to update sample weights at each stage and the weights of the classifier in the final ensemble model, given by the absolute ranked probability score, a measure that accounts for both the ordering and the distance between classes. ADABORD is extensively compared against seven state-of-the-art methods on the TOC-UCO repository, the largest benchmark collection for OC to date. The experimental results, supported by statistical analysis, show that ADABORD significantly outperforms competing methods, particularly on datasets with five or more classes, where the ordinal structure becomes more pronounced. Source code, along with all experimental protocols, is publicly available to ensure reproducibility and facilitate future research in OC.