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.

翻译：本文推进了关于测度空间上连续稀疏正则化的一般理论，重点研究Beurling-LASSO（BLASSO）方法。这种在测度空间上的全变差正则化凸优化程序，能够通过含噪声的观测数据从测量算子中恢复稀疏测度。尽管先前的研究已揭示该算子及其关联核函数在获得估计误差界中的核心作用，但相关结论需要依赖技术性的局部正曲率（LPC）假设，且该假设需针对具体案例逐一验证。实践中，仅有少数核函数被证明满足LPC条件。本文首先证明了用于信号恢复和混合问题的"sinc-4"核函数确实满足LPC假设。进一步，我们提出核函数切换分析法，该方法可利用已知的LPC核函数作为枢轴核，来推导误差界。这些结论共同为一大类平移不变模型核函数提供了易于验证的误差界条件。此外，我们还证明已知的BLASSO保证可以自适应于噪声水平，这改进了以往结果中误差固定且依赖模型核函数参数的情况。我们通过混合模型估计的案例说明本研究的价值，其中采用带限平滑技术和草图技术来降低BLASSO的计算负担。