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We study a deep linear network endowed with a structure. It takes the form ofa matrix $X$ obtained by multiplying $K$ matrices (called factors andcorresponding to the action of the layers). The action of each layer (i.e. afactor) is obtained by applying a fixed linear operator to a vector ofparameters satisfying a constraint. The number of layers is not limited.Assuming that $X$ is given and factors have been estimated, the error betweenthe product of the estimated factors and $X$ (i.e. the reconstruction error) iseither the statistical or the empirical risk. In this paper, we providenecessary and sufficient conditions on the network topology under which astability property holds. The stability property requires that the error on theparameters defining the factors (i.e. the stability of the recoveredparameters) scales linearly with the reconstruction error (i.e. the risk).Therefore, under these conditions on the network topology, any successfullearning task leads to stably defined features and therefore interpretablelayers/network.In order to do so, we first evaluate how the Segre embedding andits inverse distort distances. Then, we show that any deep structured linearnetwork can be cast as a generic multilinear problem (that uses the Segreembedding). This is the {\em tensorial lifting}. Using the tensorial lifting,we provide necessary and sufficient conditions for the identifiability of thefactors (up to a scale rearrangement). We finally provide the necessary andsufficient condition called \NSPlong~(because of the analogy with the usualNull Space Property in the compressed sensing framework) which guarantees thatthe stability property holds. We illustrate the theory with a practical examplewhere the deep structured linear network is a convolutional linear network. Asexpected, the conditions are rather strong but not empty. A simple test on thenetwork topology can be implemented to test if the condition holds.

Multilinear compressive sensing and an application to convolutional linear networks