Capturing the Denoising Effect of PCA via Compression Ratio
This provides a novel way to measure PCA's impact and a competitive outlier detection method for high-dimensional noisy data, though it is incremental as it builds on existing PCA theory.
The paper tackles the problem of quantifying PCA's denoising effect on noisy data by introducing a compression ratio metric, showing it reduces intra-community distances more than inter-community distances, and uses this to design an outlier detection algorithm that improves clustering accuracy on real-world data like single-cell RNA-seq.
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 \emph{compression ratio} to capture the effect of PCA on high-dimensional noisy data. We show that, for data with \emph{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. Building on this new metric, we design a straightforward algorithm that could be used to detect outliers. Roughly speaking, we argue that points that have a \emph{lower variance of compression ratio} do not share a \emph{common signal} with others (hence could be considered outliers). We provide theoretical justification for this simple outlier detection algorithm and use simulations to demonstrate that our method is competitive with popular outlier detection tools. Finally, we run experiments on real-world high-dimension noisy data (single-cell RNA-seq) to show that removing points from these datasets via our outlier detection method improves the accuracy of clustering algorithms. Our method is very competitive with popular outlier detection tools in this task.