# Effective regions and kernels in continuous sparse regularisation, with application to sketched mixtures

arXiv: https://arxiv.org/abs/2507.08444 · HTML: https://mlsanity.com/paper/2507.08444/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 44 references that are in the map.
- First posted on arXiv 11 Jul 2025.

## Builds on (most cited first, 18 of 18)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [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)
- [Compressive Statistical Learning with Random Feature Moments](https://mlsanity.com/paper/1706.07180/) (2017)
- [Statistical Learning Guarantees for Compressive Clustering and Compressive Mixture Modeling](https://mlsanity.com/paper/2004.08085/) (2020)
- [Sketching Datasets for Large-Scale Learning (long version)](https://mlsanity.com/paper/2008.01839/) (2020)
- [Model Selection for Mixture Models - Perspectives and Strategies](https://mlsanity.com/paper/1812.09885/) (2018)
- [Adapting to unknown noise level in sparse deconvolution](https://mlsanity.com/paper/1606.04760/) (2016)
- [Mathematical Theory of Computational Resolution Limit in Multi-dimensions](https://mlsanity.com/paper/2103.11632/) (2021)
- [FastPart: Over-Parameterized Stochastic Gradient Descent for Sparse optimisation on Measures](https://mlsanity.com/paper/2312.05993/) (2023)
- [Model-based Clustering](https://mlsanity.com/paper/1807.01987/) (2018)
- [Revisiting RIP guarantees for sketching operators on mixture models](https://mlsanity.com/paper/2312.05573/) (2023)
- [Second Maximum of a Gaussian Random Field and Exact (t-)Spacing test](https://mlsanity.com/paper/2406.18397/) (2024)

## Abstract (arXiv)

This paper advances the general theory of continuous sparse regularisation on measures with the Beurling-LASSO (BLASSO). This TV-regularised convex program on the space of measures allows to recover a sparse measure using a noisy observation from a measurement operator. While previous works have uncovered the central role played by this operator and its associated kernel in order to get estimation error bounds, the latter requires a technical local positive curvature (LPC) assumption to be verified on a case-by-case basis. In practice, this yields only few LPC-kernels for which this condition is proved. In this paper, we prove that the ``sinc-4'' kernel, used for signal recovery and mixture problems, does satisfy the LPC assumption. Furthermore, we introduce the kernel switch analysis, which allows to leverage on a known LPC-kernel as a pivot kernel to prove error bounds. Together, these results provide easy-to-check conditions to get error bounds for a large family of translation-invariant model kernels. Besides, we also show that known BLASSO guarantees can be made adaptive to the noise level. This improves on known results where this error is fixed with some parameters depending on the model kernel. We illustrate the interest of our results in the case of mixture model estimation, using band-limiting smoothing and sketching techniques to reduce the computational burden of BLASSO.