A semi-group approach to Principal Component Analysis

12/07/2021
by   Martin Schlather, et al.
0

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.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/31/2018

The Stochastic Complexity of Principal Component Analysis

PCA (principal component analysis) and its variants are ubiquitous techn...
research
02/26/2023

Efficient fair PCA for fair representation learning

We revisit the problem of fair principal component analysis (PCA), where...
research
01/28/2019

Secure multi-party linear regression at plaintext speed

We detail a scheme for scalable, distributed, secure multiparty linear r...
research
11/10/2020

Supervised PCA: A Multiobjective Approach

Methods for supervised principal component analysis (SPCA) aim to incorp...
research
02/12/2020

Structure-Property Maps with Kernel Principal Covariates Regression

Data analysis based on linear methods, which look for correlations betwe...
research
09/04/2022

Orthogonal and Linear Regressions and Pencils of Confocal Quadrics

We develop further and enhance bridges between three disciplines: statis...
research
03/08/2017

Exact Dimensionality Selection for Bayesian PCA

We present a Bayesian model selection approach to estimate the intrinsic...

Please sign up or login with your details

Forgot password? Click here to reset