# Off-the-grid learning of mixtures from a continuous dictionary

arXiv: https://arxiv.org/abs/2207.00171 · HTML: https://mlsanity.com/paper/2207.00171/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 54 references that are in the map.
- First posted on arXiv 29 Jun 2022.

## Builds on (most cited first, 10 of 10)

- [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)
- [Superresolution without Separation](https://mlsanity.com/paper/1506.03144/) (2015)
- [Adapting to unknown noise level in sparse deconvolution](https://mlsanity.com/paper/1606.04760/) (2016)
- [Deconvolution of Point Sources: A Sampling Theorem and Robustness Guarantees](https://mlsanity.com/paper/1707.00808/) (2017)
- [An off-the-grid approach to multi-compartment magnetic resonance fingerprinting](https://mlsanity.com/paper/2011.11193/) (2020)
- [Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems](https://mlsanity.com/paper/1905.04627/) (2019)

## Built on by (most cited first, 2 of 2)

- [Simultaneous off-the-grid learning of mixtures issued from a continuous dictionary](https://mlsanity.com/paper/2210.16311/) (2022)
- [Off-the-grid prediction and testing for linear combination of translated features](https://mlsanity.com/paper/2212.01169/) (2022)

## Abstract (arXiv)

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.