Density-Matrix Spectral Embeddings for Categorical Data: Operator Structure and Stability
Abstract
We introduce a supervised dimensionality reduction methodology for categorical (and discretized mixed-type) data based on a density-matrix construction induced by class-conditional frequencies. Given a labeled dataset encoded in a one-hot survey space, we assemble a frequency matrix whose columns aggregate feature occurrences within each class, and define a normalized Gram-type operator that satisfies the axioms of a density matrix. The resulting representation admits an intrinsic rank bound controlled by the number of classes, enabling low-dimensional spectral embeddings via dominant eigenmodes. Classification is performed in the reduced space through class-conditional kernel density estimation and a maximum-likelihood decision rule. We establish structural invariances, provide complexity estimates, and validate the approach on synthetic benchmarks probing high cardinality, sparsity, noise, and class imbalance.
Cite
@article{arxiv.2603.01975,
title = {Density-Matrix Spectral Embeddings for Categorical Data: Operator Structure and Stability},
author = {Raquel Bosch-Romeu and Antonio Falcó and osé-Antonio Rodríguez-Gallego},
journal= {arXiv preprint arXiv:2603.01975},
year = {2026}
}