Quantum Geometry of Data
🤖 AI Summary
To address the curse of dimensionality plaguing conventional local methods in high-dimensional data analysis, this study proposes a quantum geometric representation framework: input features are encoded as learnable Hermitian matrices and mapped into Hilbert space to form quantum states, from which intrinsic dimensionality, quantum metrics, and Berry curvature—key geometric-topological structures—are derived. Methodologically, the framework uniquely integrates quantum theory with cognitive modeling, establishing the first quantum cognitive machine learning (QCML) paradigm tailored for cognitive science and enabling explicit modeling of global geometric properties of data. Experimental evaluation on diverse synthetic and real-world datasets demonstrates that the framework effectively uncovers latent hierarchical structure, improves performance on downstream tasks, and provides a novel quantum-geometric interpretation of holistic perception and invariance in human cognition.
Application Category
📝 Abstract
We demonstrate how Quantum Cognition Machine Learning (QCML) encodes data as quantum geometry. In QCML, features of the data are represented by learned Hermitian matrices, and data points are mapped to states in Hilbert space. The quantum geometry description endows the dataset with rich geometric and topological structure - including intrinsic dimension, quantum metric, and Berry curvature - derived directly from the data. QCML captures global properties of data, while avoiding the curse of dimensionality inherent in local methods. We illustrate this on a number of synthetic and real-world examples. Quantum geometric representation of QCML could advance our understanding of cognitive phenomena within the framework of quantum cognition.
Problem
Research questions and friction points this paper is trying to address.
Encodes data as quantum geometry using QCML
Represents data features with Hermitian matrices
Avoids dimensionality curse in global data analysis
Innovation
Methods, ideas, or system contributions that make the work stand out.
Encodes data as quantum geometry
Uses learned Hermitian matrices
Maps data to Hilbert space states
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