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Data Analysis from Empirical Moments and the Christoffel Function
Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2020-03-16 , DOI: 10.1007/s10208-020-09451-2
Edouard Pauwels , Mihai Putinar , Jean-Bernard Lasserre

Spectral features of the empirical moment matrix constitute a resourceful tool for unveiling properties of a cloud of points, among which, density, support and latent structures. This matrix is readily computed from an input dataset, and its eigen decomposition can then be used to identify algebraic properties of the support or density/support estimates with the Christoffel function. It is already well known that the empirical moment matrix encodes a great deal of subtle attributes of the underlying measure. Starting from this object as base of observations, we combine ideas from statistics, real algebraic geometry, orthogonal polynomials and approximation theory for opening new insights relevant for machine learning problems with data supported on algebraic sets. Refined concepts and results from real algebraic geometry and approximation theory are empowering a simple tool (the empirical moment matrix) for the task of solving non-trivial questions in data analysis. We provide (1) theoretical support, (2) numerical experiments and (3) connections to real-world data as a validation of the stamina of the empirical moment matrix approach.



中文翻译:

根据经验矩和Christoffel函数进行数据分析

经验矩矩阵的谱特征构成了揭示点云特性的一种有用工具,其中包括密度,支撑和潜在结构。该矩阵很容易从输入数据集中计算出来,其特征分解可用于通过Christoffel函数识别支撑的代数性质或密度/支撑估计。众所周知,经验矩矩阵对基本度量的许多微妙属性进行了编码。从该对象作为观察的基础开始,我们将统计,真实代数几何,正交多项式和逼近理论的思想相结合,以利用与代数集支持的数据开辟与机器学习问题相关的新见解。解决数据分析中非平凡问题的简单工具(经验矩矩阵)。我们提供(1)理论支持,(2)数值实验和(3)与实际数据的连接,以验证经验矩矩阵方法的耐力。

更新日期:2020-04-21
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