  In this paper, we study a general framework for compressive sensing assuming
the existence of the prior knowledge that $\boldsymbol{x}^*$ belongs to the
union of multiple convex sets, $\boldsymbol{x}^{*} \in \bigcup_i
\mathcal{C}_i$. In fact, by proper choices of these convex sets in the above
framework, the problem can be transformed to well known CS problems such as the
phase retrieval, quantized compressive sensing, and model-based CS. First we
analyze the impact of this prior knowledge on the minimum number of
measurements $M$ to guarantee the uniqueness of the solution. Then we formulate
a universal objective function for signal recovery, which is both
computationally inexpensive and flexible. Then, an algorithm based on
\emph{multiplicative weight update} and \emph{proximal gradient descent} is
proposed and analyzed for computation and its properties are analyzed for
signal reconstruction. Finally, we investigate as to how we can improve the
signal recovery by introducing regularizers into the objective function.
