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Regularised PCA to denoise and visualise data

机译：正则化PCA对数据​​进行消噪和可视化

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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.
机译：主成分分析（PCA）是一种建立完善的降维方法，通常用于对数据进行降噪和可视化。经典的PCA模型是固定效应模型，在该模型中，数据被生成为被噪声破坏的低秩的固定结构。在此模型下，就均方误差而言，PCA无法提供基础信号的最佳恢复。遵循与岭回归相同的原理，我们建议使用PCA的规范化版本，该版本本质上选择一定数量的维并缩小相应的奇异值。每个奇异值都乘以一个项，该项可以看作是信号方差与相关维度的总方差之比。该正则项是使用渐近结果解析得出的，也可以从该模型的贝叶斯处理中得到证明。与传统的PCA和软阈值估计策略相比，正规化的PCA在真实信号和图形输出的恢复方面提供了可喜的结果。在数据非常嘈杂的情况下，PCA和常规PCA之间的区别变得尤为重要。

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