Dimensionality Reduction Based on kCCC and Manifold Learning

JOURNAL OF MATHEMATICAL IMAGING AND VISION(2021)

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Abstract
This paper first proposes a statistic for measuring the correlation between two random variables. Because the data are usually polluted by noise and the feature of the data is usually task-relevant, this paper proposes to perform a transformation on the data before measuring their correlation. In addition, since the kernel method is a commonly used data transformation method in the field of machine learning, choosing different kernel functions is to choose different features, so we use kernel functions to perform this transformation. The random variable transformed by the kernel function becomes a kernelized random variable. Most importantly, the kernelized random variable is a random process, so we propose to use the norm of their cross-covariance function, which is called the kernelized cross-covariance criterion ( kCCC ), to measure the task-related correlation of two random variables. The kCCC criterion is a universal principle, based on which a variety of statistical machine learning algorithms can be constructed. This paper proposes to apply the kCCC to data dimensionality reduction, referred to as kCCC-DR for short. Further, we propose kCCC-DR in combination with the most widely studied and efficient local geometric property preservation method and manifold learning dimensionality reduction method, referred as kCCC-ML-DR for short. It is a dimensionality reduction method that maintains the global statistical characteristics and local geometric characteristics of the data at the same time. In the experiments presented in this paper, kCCC is combined with LLE, LE and LTSA. These algorithms are famous manifold learning algorithms, in which LLE is local linearity-preserving, LE is local similarity-preserving and LTSA is local homeomorphism-preserving. Experiments verify the effectiveness of our method.
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Key words
Dimensionality reduction, Statistical machine learning, Manifold learning
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