Capturing the Denoising Effect of PCA via Compression Ratio

Principal component analysis (PCA) is one of the most fundamental tools in machine learning with broad use as a dimensionality reduction and denoising tool. In the later setting, while PCA is known to be effective at subspace recovery and is proven to aid clustering algorithms in some specific settings, its improvement of noisy data is still not well quantified in general. In this paper, we propose a novel metric called compression ratio to capture the effect of PCA on high-dimensional noisy data. We show that, for data with underlying community structure, PCA significantly reduces the distance of data points belonging to the same community while reducing inter-community distance relatively mildly. We explain this phenomenon through both theoretical proofs and experiments on real-world data.