  We show that the signal-processing paradigm known as compressed sensing (CS)
is applicable to genome-wide association studies (GWAS) and genomic selection
(GS). The aim of GWAS is to isolate trait-associated loci, whereas GS attempts
to predict the phenotypic values of new individuals on the basis of training
data. CS addresses a problem common to both endeavors, namely that the number
of genotyped markers often greatly exceeds the sample size. We show using CS
methods and theory that all loci of nonzero effect can be identified (selected)
using an efficient algorithm, provided that they are sufficiently few in number
(sparse) relative to sample size. For heritability h2 = 1, there is a sharp
phase transition to complete selection as the sample size is increased. For
heritability values less than one, complete selection can still occur although
the transition is smoothed. The transition boundary is only weakly dependent on
the total number of genotyped markers. The crossing of a transition boundary
provides an objective means to determine when true effects are being recovered;
we discuss practical methods for detecting the boundary. For h2 = 0.5, we find
that a sample size that is thirty times the number of nonzero loci is
sufficient for good recovery.
