用量子几何重构数据，揭示隐藏的全局结构。
Quantum Geometry of Data
- 将数据特征映射为希尔伯特空间中的量子态，以厄米矩阵表征。
- 直接从数据中提取内在维度、量子度量与贝里曲率等几何特性。
- 突破传统局部方法的维数诅咒，适用于认知建模与复杂数据分析。
我们展示了量子认知机器学习（QCML）如何将数据编码为量子几何。在QCML中，数据特征由学习得到的厄米矩阵表示，数据点被映射到希尔伯特空间中的状态。这种量子几何描述赋予数据集丰富的几何与拓扑结构——包括内在维度、量子度量和贝里曲率——均直接源自数据本身。该方法捕捉数据的全局特性，同时规避了局部方法固有的维数诅咒。我们在多个合成与真实世界数据集上进行了验证。QCML的量子几何表示有望推动量子认知框架下对认知现象的理解。
原文摘要 · Abstract (English)
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.
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