  Affine phase retrieval is the problem of recovering signals from the
magnitude-only measurements with a priori information. In this paper, we use
the $\ell_1$ minimization to exploit the sparsity of signals for affine phase
retrieval, showing that $O(k\log(en/k))$ Gaussian random measurements are
sufficient to recover all $k$-sparse signals by solving a natural $\ell_1$
minimization program, where $n$ is the dimension of signals. For the case where
measurements are corrupted by noises, the reconstruction error bounds are given
for both real-valued and complex-valued signals. Our results demonstrate that
the natural $\ell_1$ minimization program for affine phase retrieval is stable.
