  We initiate the study of \emph{inverse} problems in approximate uniform
generation, focusing on uniform generation of satisfying assignments of various
types of Boolean functions. In such an inverse problem, the algorithm is given
uniform random satisfying assignments of an unknown function $f$ belonging to a
class $\C$ of Boolean functions, and the goal is to output a probability
distribution $D$ which is $\epsilon$-close, in total variation distance, to the
uniform distribution over $f^{-1}(1)$.
  Positive results: We prove a general positive result establishing sufficient
conditions for efficient inverse approximate uniform generation for a class
$\C$. We define a new type of algorithm called a \emph{densifier} for $\C$, and
show (roughly speaking) how to combine (i) a densifier, (ii) an approximate
counting / uniform generation algorithm, and (iii) a Statistical Query learning
algorithm, to obtain an inverse approximate uniform generation algorithm. We
apply this general result to obtain a poly$(n,1/\eps)$-time algorithm for the
class of halfspaces; and a quasipoly$(n,1/\eps)$-time algorithm for the class
of $\poly(n)$-size DNF formulas.
  Negative results: We prove a general negative result establishing that the
existence of certain types of signature schemes in cryptography implies the
hardness of certain inverse approximate uniform generation problems. This
implies that there are no {subexponential}-time inverse approximate uniform
generation algorithms for 3-CNF formulas; for intersections of two halfspaces;
for degree-2 polynomial threshold functions; and for monotone 2-CNF formulas.
  Finally, we show that there is no general relationship between the complexity
of the "forward" approximate uniform generation problem and the complexity of
the inverse problem for a class $\C$ -- it is possible for either one to be
easy while the other is hard.
