The goal of this paper is to develop novel tools for understanding the local structure of systems of functions, e.g. time-series data points, such as the total correlation function, the Cohen class of the data set, the data operator and the average lack of concentration. The Cohen class of the data operator gives a time-frequency representation of the data set. Furthermore, we show that the von Neumann entropy of the data operator captures local features of the data set and that it is related to the notion of the effective dimensionality. The accumulated Cohen class of the data operator gives us a low-dimensional representation of the data set and we quantify this in terms of the average lack of concentration and the von Neumann entropy of the data operator by an application of a Berezin-Lieb inequality. The framework for our approach is provided by quantum harmonic analysis.
翻译:----
目的是开发理解系统函数局部结构的新工具,例如时间序列数据点,如总相关函数,数据集的 Cohen 类,数据算子和平均缺乏浓度。数据算子的 Cohen 类给出数据集的时频表示。此外,我们展示了数据算子的 von Neumann 熵捕捉数据集的局部特征,并且它与有效维度的概念相关。通过使用 Berezin-Lieb 不等式的应用,我们将数据算子的积累 Cohen 类转化为数据集的低维表示,并以平均缺乏浓度和数据算子的 von Neumann 熵量化了这种表示。我们的方法框架由量子谐振分析提供。