# Low-rank approximation of sub-sampled matrix

Considering a large data matrix $X$ with zero-centered columns that is assumed to be approximately low-rank, it is common to do PCA and project the data onto the top few principal components, and use this representation for visualization or as input to other algorithms.

I have seen people perform PCA on a sub-matrix $X'$, consisting of a subset of the rows of $X$, and then project the full matrix onto the principal components of $X'$. More precisely, if $m,n$ are the number of rows and columns of $X$, and $m'< m$, then:

$$X' = X_{1,...m', 1,..n}\\ X' = USV^T\\ A = XV$$

And then, say, plot the first two columns of $A$.

My question: If there is "structure" in the data matrix $X$ (e.g. if rows were cells and columns were genes, and the PCA of the full matrix reveals two clearly distinguishable groups of cells in the the first two PCs), when do we lose that structure by sub-sampling? Are there theoretical bounds to "information lost" when we sub-sample the rows? Or perhaps better--how can I formulate this question more precisely?

In other words, if you are doing dimensionality reduction using PCA, how much information is lost by sub-sampling prior to PCA?