Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

A semi-group approach to Principal Component Analysis (2112.04026v1)

Published 7 Dec 2021 in stat.ME and math.RA

Abstract: Principal Component Analysis (PCA) is a well known procedure to reduce intrinsic complexity of a dataset, essentially through simplifying the covariance structure or the correlation structure. We introduce a novel algebraic, model-based point of view and provide in particular an extension of the PCA to distributions without second moments by formulating the PCA as a best low rank approximation problem. In contrast to hitherto existing approaches, the approximation is based on a kind of spectral representation, and not on the real space. Nonetheless, the prominent role of the eigenvectors is here reduced to define the approximating surface and its maximal dimension. In this perspective, our approach is close to the original idea of Pearson (1901) and hence to autoencoders. Since variable selection in linear regression can be seen as a special case of our extension, our approach gives some insight, why the various variable selection methods, such as forward selection and best subset selection, cannot be expected to coincide. The linear regression model itself and the PCA regression appear as limit cases.

Summary

We haven't generated a summary for this paper yet.