Searcharxiv⌕ Search

arXiv subjects

Francois Malgouyres

Publications and source records attributed to Francois Malgouyres.

4 recordsLinked to original sources
arXiv2020

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

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.

math.OC↗
arXiv2013

Image restoration using sparse approximations of spatially varying blur operators in the wavelet domain

Restoration of images degraded by spatially varying blurs is an issue of increasing importance in the context of photography, satellite or microscopy imaging. One of the main difficulty to solve this problem comes from the huge dimensions of the blur matrix. It prevents the use of naive approaches for performing matrix-vector multiplications. In this paper, we propose to approximate the blur operator by a matrix sparse in the wavelet domain. We justify this approach from a mathematical point of view and investigate the approximation quality numerically. We finish by showing that the sparsity pattern of the matrix can be pre-defined, which is central in tasks such as blind deconvolution.

math.OC↗
arXiv2008

Average performance of the sparsest approximation using a general dictionary

We consider the minimization of the number of non-zero coefficients (the $\ell_0$ "norm") of the representation of a data set in terms of a dictionary under a fidelity constraint. (Both the dictionary and the norm defining the constraint are arbitrary.) This (nonconvex) optimization problem naturally leads to the sparsest representations, compared with other functionals instead of the $\ell_0$ "norm". Our goal is to measure the sets of data yielding a $K$-sparse solution--i.e. involving $K$ non-zero components. Data are assumed uniformly distributed on a domain defined by any norm--to be chosen by the user. A precise description of these sets of data is given and relevant bounds on the Lebesgue measure of these sets are derived. They naturally lead to bound the probability of getting a $K$-sparse solution. We also express the expectation of the number of non-zero components. We further specify these results in the case of the Euclidean norm, the dictionary being arbitrary.

math.OC↗