Regularised PCA to denoise and visualise data

Principal component analysis (PCA) is a well-established dimensionality reduction method commonly used to denoise and visualise data. A classical PCA model is the fixed effect model in which data are generated as a fixed structure of low rank corrupted by noise. Under this model, PCA does not provid...

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Bibliographic Details
Published in:Statistics and computing 2015-03, Vol.25 (2), p.471-486
Main Authors: Verbanck, Marie, Josse, Julie, Husson, François
Format: Article
Language:English
Subjects:
Artificial Intelligence
Asymptotic properties
Mathematical analysis
Mathematics and Statistics
Methodology
Probability and Statistics in Computer Science
Recovery
Regression
Ridges
Statistical Theory and Methods
Statistics
Statistics and Computing/Statistics Programs
Strategy
Variance
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Summary:Principal component analysis (PCA) is a well-established dimensionality reduction method commonly used to denoise and visualise data. A classical PCA model is the fixed effect model in which data are generated as a fixed structure of low rank corrupted by noise. Under this model, PCA does not provide the best recovery of the underlying signal in terms of mean squared error. Following the same principle as in ridge regression, we suggest a regularised version of PCA that essentially selects a certain number of dimensions and shrinks the corresponding singular values. Each singular value is multiplied by a term which can be seen as the ratio of the signal variance over the total variance of the associated dimension. The regularised term is analytically derived using asymptotic results and can also be justified from a Bayesian treatment of the model. Regularised PCA provides promising results in terms of the recovery of the true signal and the graphical outputs in comparison with classical PCA and with a soft thresholding estimation strategy. The distinction between PCA and regularised PCA becomes especially important in the case of very noisy data.
ISSN:0960-3174
1573-1375
DOI:10.1007/s11222-013-9444-y