发表机构
Centrale Lyon, INSA Lyon, Université Lyon 1, Université Jean Monnet, CNRS, ICJ UMR5208; Institut Universitaire de France (IUF)（里昂中央理工学院、里昂国立应用科学学院、里昂第一大学、让·莫内大学、法国国家科学研究中心、ICJ UMR5208; 法兰西学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对未知分量数与对角协方差的高斯混合模型，提出结合锥粒子梯度下降与黎曼梯度下降的求解方法，理论证明指数收敛，实验表明其比EM算法对分量数过指定更鲁棒。
AI 中文摘要
本文研究了Beurling-LASSO（BLASSO）的数值求解方法，这是一种在测度空间中促进稀疏性的凸优化框架。我们考虑将其应用于估计具有未知分量数量和未知对角协方差矩阵的高斯混合模型（GMMs）。我们的方法将锥粒子梯度下降（CPGD）原理与黎曼梯度下降相结合，以考虑高斯分布的底层Fisher-Rao几何结构。我们的贡献有两方面。首先，我们为算法的收敛性提供了理论保证。特别地，我们在解的非退化条件下建立了指数局部收敛性，并将这一假设与底层统计目标的分离条件联系起来。其次，我们解决了CPGD的实际实现方面，并展示了说明其性能的数值实验。在所考虑的测试案例中，这些实验表明CPGD对分量数量过度指定的鲁棒性优于EM算法。我们还研究了分量分离对恢复精度的影响。
英文摘要
This paper investigates the numerical resolution of the Beurling-LASSO (BLASSO), a convex optimization framework that promotes sparsity in the space of measures. We consider its application to the estimation of Gaussian mixture models (GMMs) with an unknown number of components and unknown diagonal covariance matrices. Our approach combines the Conic Particle Gradient Descent (CPGD) principle with Riemannian gradient descent, to account for the underlying Fisher-Rao geometry of Gaussian distributions. Our contributions are twofold. First, we provide theoretical guarantees for the convergence of our algorithm. In particular, we establish exponential local convergence under a non-degeneracy condition on the solution and relate this assumption to a separation condition on the underlying statistical target. Second, we address practical implementation aspects of CPGD and present numerical experiments illustrating its performance. On the test cases considered, these experiments suggest that CPGD is more robust to overspecification of the number of components than the EM algorithm. We also investigate the impact of component separation on recovery accuracy.
