Clément Hardy

Cristina Butucea, Jean-François Delmas, Anne Dutfoy et al.

We consider a general non-linear model where the signal is a finite mixture of an unknown, possibly increasing, number of features issued from a continuous dictionary parameterized by a real non-linear parameter. The signal is observed with Gaussian (possibly correlated) noise in either a continuous or a discrete setup. We propose an off-the-grid optimization method, that is, a method which does not use any discretization scheme on the parameter space, to estimate both the non-linear parameters of the features and the linear parameters of the mixture. We use recent results on the geometry of off-the-grid methods to give minimal separation on the true underlying non-linear parameters such that interpolating certificate functions can be constructed. Using also tail bounds for suprema of Gaussian processes we bound the prediction error with high probability. Assuming that the certificate functions can be constructed, our prediction error bound is up to $\log$-factors similar to the rates attained by the Lasso predictor in the linear regression model. We also establish convergence rates that quantify with high probability the quality of estimation for both the linear and the non-linear parameters. We develop in full details our main results for two applications: the Gaussian spike deconvolution and the scaled exponential model.

Cristina Butucea, Jean-François Delmas, Anne Dutfoy et al.

In this paper we observe a set, possibly a continuum, of signals corrupted by noise. Each signal is a finite mixture of an unknown number of features belonging to a continuous dictionary. The continuous dictionary is parametrized by a real non-linear parameter. We shall assume that the signals share an underlying structure by assuming that each signal has its active features included in a finite and sparse set. We formulate regularized optimization problem to estimate simultaneously the linear coefficients in the mixtures and the non-linear parameters of the features. The optimization problem is composed of a data fidelity term and a $(\ell_1,L^p)$-penalty. We call its solution the Group-Nonlinear-Lasso and provide high probability bounds on the prediction error using certificate functions. Following recent works on the geometry of off-the-grid methods, we show that such functions can be constructed provided the parameters of the active features are pairwise separated by a constant with respect to a Riemannian metric.When the number of signals is finite and the noise is assumed Gaussian, we give refinements of our results for $p=1$ and $p=2$ using tail bounds on suprema of Gaussian and $χ^2$ random processes. When $p=2$, our prediction error reaches the rates obtained by the Group-Lasso estimator in the multi-task linear regression model. Furthermore, for $p=2$ these prediction rates are faster than for $p=1$ when all signals share most of the non-linear parameters.

Clément Hardy, Yvain Quéau, David Tschumperlé

The photometric stereo (PS) problem consists in reconstructing the 3D-surface of an object, thanks to a set of photographs taken under different lighting directions. In this paper, we propose a multi-scale architecture for PS which, combined with a new dataset, yields state-of-the-art results. Our proposed architecture is flexible: it permits to consider a variable number of images as well as variable image size without loss of performance. In addition, we define a set of constraints to allow the generation of a relevant synthetic dataset to train convolutional neural networks for the PS problem. Our proposed dataset is much larger than pre-existing ones, and contains many objects with challenging materials having anisotropic reflectance (e.g. metals, glass). We show on publicly available benchmarks that the combination of both these contributions drastically improves the accuracy of the estimated normal field, in comparison with previous state-of-the-art methods.