How to select a set of independent features predictive of all other features without a target variable

Question

Problem

I have a dataset NxD, where N - number of observation (~100k) and D - number of features (~10k) (More specifically it is a single-cell RNAseq data, so each observation is a single cell and each feature is a gene, values are counts of genes in each of the cells).

    *Cell* Cell1 Cell2 Cell3 ....
*Gene*
DPM1       .  . . . .  .  .  . .  .  . 1  . . .  .  .  .  .
SCYL3      .  . . . .  .  .  . .  .  . .  . . .  .  .  .  . 
C1orf112   .  . . . .  .  .  . .  .  . .  . . .  .  .  .  . 
FGR        .  . . . .  .  .  . .  1  . .  . . .  .  .  .  . 
CFH        .  . . . .  .  .  . .  .  . .  . . .  .  .  .  . 
FUCA2      .  . . . .  .  .  . .  .  . .  . . .  .  .  1  . 
GCLC       .  . . . .  .  .  . .  .  . .  . . .  .  .  .  . 

As you can see data is discrete and sparse (only a small set of genes have non-zero counts in each cell).

I would like to select a subset of features (genes) such that:

• The dimensionality of the subset d << D (~1k)
• Resulting features are not correlated
• Using this subset I can predict remaining features

Possible approaches

I have a few ideas in mind, but I would be very grateful for any comments or suggestions. Are there any standard ways of approaching this problem?

• Find clusters of genes in the space of cells (or reduced space) and select one gene per cluster, e.g. from the center. Presumably all genes in the cluster will be somewhat correlated with each other and therefore one gene should be enough to capture information about the other genes in the same cluster. Problems with this approach: clustering in high-dimensional space is hard, and this won't capture anticorrelated genes - they will belong to separate clusters, but still can be used to predict each others' expression.

• Same as above, but use hierarchical clustering on the distance matrix. Distance can be anything from simple covariance, to mutual information (this would take care of anticorrelated genes as well).

• Select most important features from multiple regularized regressions:

  $\forall i, \quad x_i \sim X_{-i}$, where $x_i$ is expression of gene $i$, $X_{-i}$ - expression of all other genes. The problem with this approach - the distribution of gene counts is negative binomial so both my target variable and predictor variables will be negative binomial distributed.

Answer 1

There is extensive literature on the application of unsupervised learning to gene-expression data, dating back to early microarray analysis of RNA. Back then, clustering was done for example to try to distinguish subclasses of tumors among individuals. The advent of RNA-seq moved the focus toward analysis of count data. The more recent extension to single cells (scRNA-seq), typically looking for different types of cells in a sample, requires additional attention to the sparsity of the data.

This recent review goes over ways to cluster scRNA-seq data in substantial detail, with over 100 references and with links to software for implementation. Clustering typically involve a combination of methods. For example, SC3 clustering combines principal components analysis (PCA*, as suggested in a comment) and parallel evaluation of multiple parameter values to provide a final k-means clustering. Other methods are hierarchically based, sometimes combined with other processing. There are tradeoffs among the approaches both in terms of practical implementation, depending on the scale of the data, and in terms of just what you are trying to accomplish with the clustering.

A frequently used method for clustering scRNA-seq data, not addressed well in that review, is t-distributed stochastic neighbor embedding (t-SNE), which attempts to reduce high-dimension data into a 2D representation of clusters. This article by Kobak and Berens shows how to use t-SNE on scRNA-seq data while avoiding problems with its "naive application."

Once you've identified the expression clusters in your data, you could proceed to find exemplar genes expressed highly within each cluster if your clustering is based on gene-expression values rather than on a transformation as in PCA. My guess, however, is that even then you will need to use more than 1 exemplar gene per cluster. That's true with clinical tests that try to distinguish classes of tumors from bulk-tissue RNA analysis, so it's even more likely to be the case with the low number of counts per gene typical with scRNA-seq for other than housekeeping genes. If your clusters are based on linear combinations of gene expression values as in PCA, you will have to find combinations of genes that can be used together to identify clusters.

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*PCA ensures that all components are orthogonal, which would match your desire that "resulting features are not correlated." That orthogonality, however, will not be seen on the single-gene level, it will represent the situation after any data pre-processing, and it might make it impossible to find individual genes that are exemplars of your clusters.