  A first analysis applying compressive sensing to a quantitative biological
trait and its compressible "frequency domain" is presented. Consider an $n$-bit
genetic sequence and suppose we want to discover a function that maps
participating alleles (or even environmental influences) to a particular trait.
Under plausible assumptions of how they evolved, certain traits can be viewed
as "smooth" functions on the $n$-dimensional Boolean lattice of possible
genomes. This allows approximation of their Fourier transforms, i.e., their
gene networks, as sparse, dominated by "low-frequency" components. In turn, we
can apply compressive sensing methods to collect relatively few samples, yet
achieve accurate recovery.
  For an arbitrary quantitative trait affected by $n=26$ genes and with $812$
meaningful gene interactions, our simulations show noisy trait measurements
($\mathrm{SNR}=20\,\mathrm{dB}$) from just $M=44,\!336$ genomes in a population
of size $N = 2^{26}$ (undersample ratio $M/N\approx0.00066$) permit discovering
its gene network and predicting trait values, both with about $97.6\%$
accuracy. More dramatic undersample ratios are possible for traits affected by
more genes. Work is currently underway to see if empirical data fit the
proposed model. If so, it could offer a radical reduction in the number of
measurements -- from exponential to polynomial in some cases -- necessary to
quantify the relationship between genomes and certain traits.
