  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.
