Developments in Gaussian Processes and Beyond: Applications in Geostatistics and Deep Learning

高斯过程及其他过程的发展：地统计学和深度学习中的应用

• 批准号: 2014371
• 负责人: Anindya Bhadra
• 金额: $ 12万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-08-01 至 2023-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2014371&HistoricalAwards=false
• 关键词: Developments; Gaussian; Processes; Beyond; Applications

Gaussian processes have diverse applications in statistics and machine learning and are of great contemporary interest. To give a few examples, they arise in the modeling of spatial data, computer experiments, and in studying the limits of deep neural networks. Key reasons for the appeal of Gaussian processes include their simplicity and wide tractability: the entire process is characterized by just the mean and the covariance functions. Yet, although Gaussian processes are popular with well-developed theoretical and computational properties, there are some distinct limitations in using them. Moreover, there are several situations where Gaussian processes are inappropriate as a modeling choice. New methodology will be developed to address some of these limitations, with wide-ranging implications from spatial statistics to deep learning. Publicly available software development, student mentoring, and broad dissemination of research will have impacts beyond the particular research problems at hand.Key areas of the technical investigation are as follows. The first issue concerns the use of the ubiquitous Matern covariance function. A key benefit of the Matern family is the precise control over the smoothness of the resultant Gaussian processes (GP) realizations. However, the tails of the Matern covariance decay exponentially fast, which is inappropriate in the presence of polynomial dependence. Polynomial covariances such as Cauchy remedy this issue, but at the expense of a loss of control over smoothness, in that, GP realizations using Cauchy covariances are either infinitely differentiable or not at all. The PI will develop a new covariance function that combines the flexibility of the Matern and polynomial covariances. Next, the PI will study the limiting behavior of deep neural networks under global-local horseshoe regularization priors on the weights. The lack of bounded moments necessitates the construction of a new Levy process that can be used to study the limits of neural networks under such priors, thereby aiding uncertainty quantification. The PI will study the theoretical and computational properties of the resultant process. Finally, the PI will use recently developed global-local shrinkage approaches for Bayesian regularization in GP regression, with distinct improvements upon existing methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

高斯流程在统计和机器学习中具有不同的应用，并且具有极大的兴趣。为了举一些例子，它们出现在空间数据，计算机实验的建模以及研究深神经网络的限制时。高斯流程吸引力的关键原因包括它们的简单性和广泛的障碍：整个过程的特征是平均值和协方差功能。然而，尽管高斯流程在发达的理论和计算属性中很受欢迎，但使用它们存在一些明显的局限性。此外，在几种情况下，高斯流程不合适作为建模选择。将开发新的方法来解决其中一些局限性，并具有从空间统计到深度学习的广泛含义。公开可用的软件开发，学生指导和广泛的研究将影响超出特定研究问题的影响。技术调查的关键领域如下。第一个问题涉及使用无处不在的母乳协方差函数。 Matern家族的关键优势是对所得高斯工艺（GP）实现的平稳性的精确控制。然而，在多项式依赖性的情况下，母乳协方差衰减的尾巴呈指数速度，这是不合适的。多项式协方差，例如凯奇（Cauchy）的补救措施，但以失去对平滑度的控制为代价，使用凯奇（Cauchy）协方差实现的GP实现是无限的，要么根本没有。 PI将开发出一种新的协方差函数，结合了Matern和多项式协方差的灵活性。接下来，PI将研究重量的全球局部马蹄正规化先验下深层神经网络的限制行为。缺乏有限的力矩需要建立一个新的征费过程，该过程可用于研究此类先验的神经网络的限制，从而有助于不确定性量化。 PI将研究所得过程的理论和计算特性。最后，PI将在GP回归中使用最近开发的全球本地收缩方法来进行贝叶斯正规化，对现有方法进行了明显的改进。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的审查标准通过评估来支持的。

项目成果

A Laplace Mixture Representation of the Horseshoe and Some Implications

马蹄形的拉普拉斯混合表示及其一些含义

• DOI: 10.1109/lsp.2022.3228491
• 发表时间: 2022
• 期刊: IEEE Signal Processing Letters
• 影响因子: 3.9
• 作者: Sagar, Ksheera;Bhadra, Anindya
• 通讯作者: Bhadra, Anindya

Joint mean–covariance estimation via the horseshoe

通过马蹄形进行联合均值协方差估计

• DOI: 10.1016/j.jmva.2020.104716
• 发表时间: 2021
• 期刊: Journal of Multivariate Analysis
• 影响因子: 1.6
• 作者: Li, Yunfan;Datta, Jyotishka;Craig, Bruce A.;Bhadra, Anindya
• 通讯作者: Bhadra, Anindya

SURE-tuned bridge regression

• DOI: 10.1007/s11222-023-10350-z
• 发表时间: 2024-02-01
• 期刊: STATISTICS AND COMPUTING
• 影响因子: 2.2
• 作者: Loria，Jorge;Bhadra，Anindya
• 通讯作者: Bhadra，Anindya

Beyond Matérn: On A Class of Interpretable Confluent Hypergeometric Covariance Functions

超越马特恩：关于一类可解释的汇合超几何协方差函数

• DOI: 10.1080/01621459.2022.2027775
• 发表时间: 2022
• 期刊: Journal of the American Statistical Association
• 影响因子: 3.7
• 作者: Ma, Pulong;Bhadra, Anindya
• 通讯作者: Bhadra, Anindya

Discussion to: Bayesian graphical models for modern biological applications by Y. Ni, V. Baladandayuthapani, M. Vannucci and F.C. Stingo

• DOI: 10.1007/s10260-021-00600-7
• 发表时间: 2021-11
• 期刊: Statistical Methods & Applications
• 影响因子: 1
• 作者: M. Schweinberger