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SUPERMIX: SPARSE REGULARIZATION FOR MIXTURES
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DOI：10.1214/20-AOS2022.png)
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摘要
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This paper investigates the statistical estimation of a discrete mixing measure mu(0) involved in a kernel mixture model. Using some recent advances in l(1)-regularization over the space of measures, we introduce a data fitting and regularization convex program for estimating mu(0) in a grid-less manner from a sample of mixture law, this method is referred to as Beurling-LASSO. Our contribution is two-fold: we derive a lower bound on the bandwidth of our data fitting term depending only on the support of mu(0) and its socalled minimum separation to ensure quantitative support localization error bounds; and under a so-called nondegenerate source condition we derive a nonasymptotic support stability property. This latter shows that for a sufficiently large sample size n, our estimator has exactly as many weighted Dirac masses as the target mu(0), converging in amplitude and localization towards the true ones. Finally, we also introduce some tractable algorithms for solving this convex program based on Sliding Frank-Wolfe or Conic Particle Gradient Descent. Statistical performances of this estimator are investigated designing a socalled dual certificate, which is appropriate to our setting. Some classical situations as, for example, mixtures of super-smooth distributions (see, e.g., Gaussian distributions) or ordinary-smooth distributions (see, e.g., Laplace distributions), are discussed at the end of the paper.
Keyword:
Beurling Lasso
mixture recovery
dual certificate
kernel approach
super-resolution
期刊
IF:
3.7
论文数:
2.8K
被引数:
2.9W
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STRONG IDENTIFIABILITY AND OPTIMAL MINIMAX RATES FOR FINITE MIXTURE ESTIMATION.png)
ANNALS OF STATISTICS
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ANNALS OF STATISTICSIF3.7
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