So, datasets are becoming so large that we now have investigations on the difference between Random Projections and Randomized PCA !
Projecting "better than randomly": How to reduce the dimensionality of very large datasets in a way that outperforms random projections by Michael Wojnowicz, Di Zhang, Glenn Chisholm, Xuan Zhao, Matt Wolff
For very large datasets, random projections (RP) have become the tool of choice for dimensionality reduction. This is due to the computational complexity of principal component analysis. However, the recent development of randomized principal component analysis (RPCA) has opened up the possibility of obtaining approximate principal components on very large datasets. In this paper, we compare the performance of RPCA and RP in dimensionality reduction for supervised learning. In Experiment 1, study a malware classification task on a dataset with over 10 million samples, almost 100,000 features, and over 25 billion non-zero values, with the goal of reducing the dimensionality to a compressed representation of 5,000 features. In order to apply RPCA to this dataset, we develop a new algorithm called large sample RPCA (LS-RPCA), which extends the RPCA algorithm to work on datasets with arbitrarily many samples. We find that classification performance is much higher when using LS-RPCA for dimensionality reduction than when using random projections. In particular, across a range of target dimensionalities, we find that using LS-RPCA reduces classification error by between 37% and 54%. Experiment 2 generalizes the phenomenon to multiple datasets, feature representations, and classifiers. These findings have implications for a large number of research projects in which random projections were used as a preprocessing step for dimensionality reduction. As long as accuracy is at a premium and the target dimensionality is sufficiently less than the numeric rank of the dataset, randomized PCA may be a superior choice. Moreover, if the dataset has a large number of samples, then LS-RPCA will provide a method for obtaining the approximate principal components.
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