  Recently, convolutional auto-encoders (CAE) were introduced for image coding.
They achieved performance improvements over the state-of-the-art JPEG2000
method. However, these performances were obtained using massive CAEs featuring
a large number of parameters and whose training required heavy computational
power.\\ In this paper, we address the problem of lossy image compression using
a CAE with a small memory footprint and low computational power usage. In order
to overcome the computational cost issue, the majority of the literature uses
Lagrangian proximal regularization methods, which are time consuming
themselves.\\ In this work, we propose a constrained approach and a new
structured sparse learning method. We design an algorithm and test it on three
constraints: the classical $\ell_1$ constraint, the $\ell_{1,\infty}$ and the
new $\ell_{1,1}$ constraint. Experimental results show that the $\ell_{1,1}$
constraint provides the best structured sparsity, resulting in a high reduction
of memory and computational cost, with similar rate-distortion performance as
with dense networks.
