We consider a model where a signal (discrete or continuous) is observed with an additive Gaussian noise process. The signal is issued from a linear combination of a finite but increasing number of translated features. The features are continuously parameterized by their location and depend on some scale parameter. The general model considered in this paper is a non-linear extension of the classical high-dimensional regression model. 

First, we build off-the-grid estimators of both the linear parameters and the location parameters. The prediction bounds are analogous to those obtained for sparse linear regression. Next, we propose a goodness-of-fit test for the model and give non-asymptotic upper bounds of the testing risk and of the minimax separation rate between two distinguishable signals. In particular, our test encompasses the signal detection framework. We deduce upper bounds on the minimal energy, expressed as the 2-norm of the linear coefficients, to successfully detect a signal in presence of noise. It turns out that, our upper bound on the minimax separation rate matches (up to a logarithmic factor) the lower bound on the minimax separation rate for signal detection in the high dimensional linear model associated to a fixed dictionary of features.

Underlying paper: https://arxiv.org/abs/2212.01169

Personal website of Cristina Butucea

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