Quantifying the Informativeness of Similarity Measurements
Quantifying the Informativeness of Similarity Measurements
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发表时间：
2017-07
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通讯作者：
A. Brockmeier;Tingting Mu;S. Ananiadou;J. Y. Goulermas
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作者：
A. Brockmeier;Tingting Mu;S. Ananiadou;J. Y. Goulermas
在本文中，我们描述了一种无监督度量，用于量化由数据实例之间的成对相似性或关系形成的相关矩阵的“信息性”。该度量量化了相关性的异质性，并将其定义为相关矩阵与具有恒定非对角线条目的最近相关矩阵之间的距离。这种非参数概念通过允许可选的距离度量（如Bures和量子信息理论中的其他距离）来推广相关系数相等的现有检验统计量。对于一些距离和不相似度度量，我们推导了信息量的封闭形式表达式，可以作为机器学习应用的目标函数。经验表明，信息量是选择核参数、选择基于核的非线性降维维数以及识别结构图的有用标准。我们还考虑了在目标矩阵周围寻找最大信息量相关矩阵的问题，并探索了根据样本坐标或通过低维嵌入来参数化优化的问题。在后一种情况下，我们发现最大化基于bures的信息量度量（对于中心秩1相关矩阵是最大的）相当于最小化一个特定的矩阵范数，并提出了一种使用范数的近端算子来解决最小化问题的算法。提出的相关去噪算法持续改善了谱聚类。总的来说，我们发现信息量是一个新的和有用的标准来识别非琐碎的相关结构。
In this paper, we describe an unsupervised measure for quantifying the 'informativeness' of correlation matrices formed from the pairwise similarities or relationships among data instances. The measure quantifies the heterogeneity of the correlations and is defined as the distance between a correlation matrix and the nearest correlation matrix with constant off-diagonal entries. This non-parametric notion generalizes existing test statistics for equality of correlation coefficients by allowing for alternative distance metrics, such as the Bures and other distances from quantum information theory. For several distance and dissimilarity metrics, we derive closed-form expressions of informativeness, which can be applied as objective functions for machine learning applications. Empirically, we demonstrate that informativeness is a useful criterion for selecting kernel parameters, choosing the dimension for kernel-based nonlinear dimensionality reduction, and identifying structured graphs. We also consider the problem of finding a maximally informative correlation matrix around a target matrix, and explore parameterizing the optimization in terms of the coordinates of the sample or through a lower-dimensional embedding. In the latter case, we find that maximizing the Bures-based informativeness measure, which is maximal for centered rank-1 correlation matrices, is equivalent to minimizing a specific matrix norm, and present an algorithm to solve the minimization problem using the norm's proximal operator. The proposed correlation denoising algorithm consistently improves spectral clustering. Overall, we find informativeness to be a novel and useful criterion for identifying non-trivial correlation structure..