# On the stable recovery of deep structured linear networks under sparsity constraints

## TL;DR

This paper establishes sharp conditions for the stable recovery of deep structured linear networks with sparsity constraints, ensuring that parameter errors scale linearly with reconstruction errors, thus enabling reliable feature interpretation and statistical guarantees.

## Contribution

It introduces a novel stability analysis for deep linear networks with sparsity priors using Tensorial Lifting, providing sharp guarantees especially for convolutional networks.

## Key findings

- Stability error scales linearly with reconstruction error under certain conditions.
- Sparsity prior improves the stability constant.
- Analysis distinguishes effects of architecture and input data on stability.

## Abstract

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.