This paper studies sparse density estimation via $\ell_1$ penalization (SPADES). We focus on estimation in high-dimensional mixture models and nonparametric adaptive density estimation. We show, respectively, that SPADES can recover, with high probability, the unknown components of a mixture of probability densities and that it yields minimax adaptive density estimates. These results are based on a general sparsity oracle inequality that the SPADES estimates satisfy. We offer a data driven method for the choice of the tuning parameter used in the construction of SPADES. The method uses the generalized bisection method first introduced in \citebb09. The suggested procedure bypasses the need for a grid search and offers substantial computational savings. We complement our theoretical results with a simulation study that employs this method for approximations of one and two-dimensional densities with mixtures. The numerical results strongly support our theoretical findings.
SPADE: Sequential-clustering Particle Annihilation via Discrepancy Estimation
对于经验有符号测度$μ= \frac{1}{N} \left(\sum_{i=1}^P δ_{x_i} - \sum_{i=1}^M δ_{y_i}\right)$，粒子湮灭（PA）同时从$\{x_i\}_{i=1}^P$和$\{y_i\}_{i=1}^M$两者中移除$N_A$粒子，从而产生另一经验有符号测度$ν$，使得$\int f d ν$在对于合适的测试函数$f$的可接受精度内近似于$\int f d μ$。这种携带相反重要性权重的粒子的湮灭已经被广泛用于减轻粒子模拟中的数值符号问题。在本文中，我们提出了一种算法PA在高维欧氏空间的基础上混合聚类和匹配，称为顺序聚类粒子湮灭通过离散估计（SPADE）。它包括两个步骤：通过控制粒子的数论差异来实现粒子的自适应聚类，并在每个簇中的正负粒子之间进行独立的随机匹配。证明了Koksma-Hlawka不等式的确定性误差界和浓度不等式的非渐近随机误差界都受两个因素的影响。一个因子度量点分布的不规则性并反映其离散性。另一种依赖于测试函数的变异性，受测试函数连续性的影响。只有后者隐含地依赖于维度$d$，这意味着SPADE可以免疫的灾难的维数为广泛的测试功能。直到$d=1080$的数值实验验证了我们的理论发现。
Regime-Adaptive Bayesian Optimization via Dirichlet Process Mixtures of Gaussian Processes
Department of Computer Science, Florida State University（佛罗里达州立大学计算机科学系）；Department of Computer Science, University of Chicago（芝加哥大学计算机科学系）；School of Mechanical Engineering, Purdue University（普渡大学机械工程学院）；College of Nanotechnology, Science, and Engineering, State University of New York at Albany（纽约州立大学阿尔巴尼分校纳米技术、科学与工程学院）
Booth School of Business University of Chicago（芝加哥大学商学院）；Volgenau School of Engineering, George Mason University（乔治·马歇尔大学工程学院）；Department of Systems Engineering and Operations Research George Mason University（乔治·马歇尔大学系统工程与运营管理系）
Scalable Dirichlet Process Mixture Models with Unknown Concentration and Adaptive Covariance for High-Dimensional Clustering Applied to Leukemia Transcriptomics
A discomfort-informed adaptive Gibbs sampler for finite mixture models
University of Florence（佛罗伦萨大学）；MRC Biostatistics Unit, University of Cambridge（剑桥大学医学研究委员会生物统计单元）；University of Warwick（华威大学）；Bocconi University（博科尼大学）；Integreat—Norwegian Centre for knowledge-driven machine learning, Department of Mathematics, University of Oslo（奥斯陆大学数学系 Integreat——挪威知识驱动机器学习中心）
