  We consider a deep structured linear network under sparsity constraints. We
study sharp conditions guaranteeing the stability of the optimal parameters
defining the network. More precisely, we provide sharp conditions on the
network architecture and the sample under which the error on the parameters
defining the network scales linearly with the reconstruction error (i.e. the
risk). Therefore, under these conditions, the weights obtained with a
successful algorithms are well defined and only depend on the architecture of
the network and the sample. The features in the latent spaces are stably
defined. The stability property is required in order to interpret the features
defined in the latent spaces. It can also lead to a guarantee on the
statistical risk. This is what motivates this study. The analysis is based on
the recently proposed Tensorial Lifting. The particularity of this paper is to
consider a sparsity prior. This leads to a better stability constant. As an
illustration, we detail the analysis and provide sharp stability guarantees for
convolutional linear network under sparsity prior. In this analysis, we
distinguish the role of the network architecture and the sample input. This
highlights the requirements on the data in connection to parameter stability.
