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arXiv · 2603.01975

Density-Matrix Spectral Embeddings for Categorical Data: Operator Structure and Stability

Raquel Bosch-Romeu·Antonio Falcó·osé-Antonio Rodríguez-Gallego
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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.

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Raquel Bosch-Romeu, Antonio Falcó, osé-Antonio Rodríguez-Gallego. 2026-03-02. Density-Matrix Spectral Embeddings for Categorical Data: Operator Structure and Stability. https://arxiv.org/abs/2603.01975

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