标题： 具有分段常数特征值轮廓的简约高斯混合模型 显示英文标题

标题： Parsimonious Gaussian mixture models with piecewise-constant eigenvalue profiles

摘要： 高斯混合模型（GMMs）在统计学习中无处不在，尤其是在无监督问题中。 虽然完整的GMMs在高维空间中其协方差矩阵存在过度参数化的问题，而球形GMMs（具有各向同性协方差矩阵）显然缺乏灵活性，无法拟合某些各向异性分布。 连接这两个极端，我们引入了一种新的简洁GMM家族，其协方差特征值图是分段常数的。 这些模型扩展了多种低秩模型，如著名的概率主成分分析器混合模型（MPPCA），通过允许任何可能的特征值重数序列。 如果后者是预定义的，那么我们可以自然地推导出一个期望最大化（EM）算法来学习混合参数。 否则，为了解决联合学习混合参数和超参数这一众所周知的挑战性问题，我们提出了一种分组件惩罚的EM算法，其单调性得到了证明。 我们在各种无监督实验中展示了我们的模型所取得的优越似然-简洁性权衡：密度拟合、聚类和单图像去噪。 显示英文摘要

摘要： Gaussian mixture models (GMMs) are ubiquitous in statistical learning, particularly for unsupervised problems. While full GMMs suffer from the overparameterization of their covariance matrices in high-dimensional spaces, spherical GMMs (with isotropic covariance matrices) certainly lack flexibility to fit certain anisotropic distributions. Connecting these two extremes, we introduce a new family of parsimonious GMMs with piecewise-constant covariance eigenvalue profiles. These extend several low-rank models like the celebrated mixtures of probabilistic principal component analyzers (MPPCA), by enabling any possible sequence of eigenvalue multiplicities. If the latter are prespecified, then we can naturally derive an expectation-maximization (EM) algorithm to learn the mixture parameters. Otherwise, to address the notoriously-challenging issue of jointly learning the mixture parameters and hyperparameters, we propose a componentwise penalized EM algorithm, whose monotonicity is proven. We show the superior likelihood-parsimony tradeoffs achieved by our models on a variety of unsupervised experiments: density fitting, clustering and single-image denoising.