🤖 AI Summary
This work addresses the limitations of conventional supervised dimensionality reduction methods when applied to high-cardinality, sparse, noisy, and class-imbalanced mixed categorical and discrete data. It introduces, for the first time, the density matrix formalism from quantum information theory into this domain, constructing a normalized Gram-type operator based on class-conditional frequencies that satisfies the axioms of a density matrix. This operator yields low-dimensional spectral embeddings in the one-hot encoded space, which are then combined with class-conditional kernel density estimation and a maximum likelihood decision rule for classification. Theoretical analysis reveals a structural property wherein the embedding rank is bounded by the number of classes, and establishes the method’s invariance and stability. Experiments demonstrate that the proposed approach significantly outperforms existing methods on both synthetic and real-world benchmarks, achieving superior robustness and classification performance.
📝 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.