# Simultaneous off-the-grid learning of mixtures issued from a continuous dictionary

arXiv: https://arxiv.org/abs/2210.16311 · HTML: https://mlsanity.com/paper/2210.16311/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 45 references that are in the map.
- First posted on arXiv 27 Oct 2022.

## Builds on (most cited first, 9 of 9)

- [Sparse Optimization on Measures with Over-parameterized Gradient Descent](https://mlsanity.com/paper/1907.10300/) (2019)
- [On Representer Theorems and Convex Regularization](https://mlsanity.com/paper/1806.09810/) (2018)
- [Exact Support Recovery for Sparse Spikes Deconvolution](https://mlsanity.com/paper/1306.6909/) (2013)
- [The Sliding Frank-Wolfe Algorithm and its Application to Super-Resolution Microscopy](https://mlsanity.com/paper/1811.06416/) (2018)
- [The Alternating Descent Conditional Gradient Method for Sparse Inverse Problems](https://mlsanity.com/paper/1507.01562/) (2015)
- [Adapting to unknown noise level in sparse deconvolution](https://mlsanity.com/paper/1606.04760/) (2016)
- [An off-the-grid approach to multi-compartment magnetic resonance fingerprinting](https://mlsanity.com/paper/2011.11193/) (2020)
- [Off-the-grid learning of mixtures from a continuous dictionary](https://mlsanity.com/paper/2207.00171/) (2022)
- [An Epigraphical Approach to the Representer Theorem](https://mlsanity.com/paper/1912.13224/) (2019)

## Abstract (arXiv)

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 (ℓ1,Lp)-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 chi2 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.