Abstract
Ordinal data arise ubiquitously in survey research, psychology, medicine, economics, and recommender systems, yet kernel methods for such data typically rely on either nominal encodings or arbitrary numeric codings. The former discards order information; the lat- ter imposes a fictitious metric structure. This paper develops a principled framework for kernel design on ordinal scales and introduces a new class of Semantic–Aware Ordinal Ker- nels (SAOK) that simultaneously capture ordinal order and semantic proximity between categories. We begin by formalizing order–preserving embeddings of finite chains and characterizing a broad family of chain distances that are conditionally negative definite. Through Schoen- berg transforms, these distances generate valid positive definite ordinal kernels. Building on this foundation, SAOK augments order information with a semantic similarity matrix derived from label embeddings or response profiles, yielding kernels that respect both the latent order direction and the meanings of the categories. We analyze the induced RKHS geometry and show that SAOK promotes representa- tions in which ordinal levels are arranged along a low–dimensional, approximately monotone manifold. In synthetic latent–threshold models and real Likert–scale data, SAOK produces substantially stronger rank alignment, as quantified by Kendall’s τ , and clearer monotone gradients in kernel PCA embeddings, while maintaining competitive predictive accuracy and computational complexity comparable to standard Gaussian kernels. These results demonstrate that ordinal structure and semantic information can be in- tegrated at the kernel level in a theoretically grounded and computationally simple way, providing a geometric and interpretable approach to learning from ordinal data.
Publication Date
Spring 3-31-2026
Document Type
Technical Report
Recommended Citation
Ordinal data arise ubiquitously in survey research, psychology, medicine, economics, and recommender systems, yet kernel methods for such data typically rely on either nominal encodings or arbitrary numeric codings. The former discards order information; the latter imposes a fictitious metric structure. This paper develops a principled framework for kernel design on ordinal scales and introduces a new class of Semantic–Aware Ordinal Kernels (SAOK) that simultaneously capture ordinal order and semantic proximity between categories. We begin by formalizing order–preserving embeddings of finite chains and characterizing a broad family of chain distances that are conditionally negative definite. Through Schoenberg transforms, these distances generate valid positive definite ordinal kernels. Building on this foundation, SAOK augments order information with a semantic similarity matrix derived from label embeddings or response profiles, yielding kernels that respect both the latent order direction and the meanings of the categories. We analyze the induced RKHS geometry and show that SAOK promotes representations in which ordinal levels are arranged along a low–dimensional, approximately monotone manifold. In synthetic latent–threshold models and real Likert–scale data, SAOK produces substantially stronger rank alignment, as quantified by Kendall’s τ , and clearer monotone gradients in kernel PCA embeddings, while maintaining competitive predictive accuracy and computational complexity comparable to standard Gaussian kernels. These results demonstrate that ordinal structure and semantic information can be integrated at the kernel level in a theoretically grounded and computationally simple way, providing a geometric and interpretable approach to learning from ordinal data.
Campus
RIT – Main Campus
Included in
Applied Statistics Commons, Categorical Data Analysis Commons, Data Science Commons, Multivariate Analysis Commons, Statistical Methodology Commons, Statistical Models Commons
