  Supersaturated designs investigate more factors than there are runs, and are
often constructed under a criterion measuring a design's proximity to an
unattainable orthogonal design. The most popular analysis identifies active
factors by inspecting the solution path of a penalized estimator, such as the
lasso. Recent criteria encouraging positive correlations between factors have
been shown to produce designs with more definitive solution paths so long as
the active factors have positive effects. Two open problems affecting the
understanding and practicality of supersaturated designs are: (1) do optimal
designs under existing criteria maximize support recovery probability across an
estimator's solution path, and (2) why do designs with positively correlated
columns produce more definitive solution paths when the active factors have
positive sign effects? To answer these questions, we develop criteria
maximizing the lasso's sign recovery probability. We prove that an orthogonal
design is an ideal structure when the signs of the active factors are unknown,
and a design constant small, positive correlations is ideal when the signs are
assumed known. A computationally-efficient design search algorithm is proposed
that first filters through optimal designs under new heuristic criteria to
select the one that maximizes the lasso sign recovery probability.
