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

TL;DR

This paper presents a novel spectral embedding technique for categorical data using a density-matrix approach, enabling low-dimensional representations for improved classification in high-cardinality, sparse, and noisy datasets.

Contribution

It introduces a density-matrix based spectral embedding method for categorical data, with theoretical analysis and validation on synthetic benchmarks.

Findings

Effective low-dimensional embeddings for high-cardinality data

Robust classification under noise and class imbalance

Structural invariances and complexity bounds established

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.