Regularized divergences and their gradient flows, generative modeling and structure-preserving learning.

生成建模算法是人工智能领域最近和正在取得的许多进展的基础，包括流行的图像和文本生成工具，以及材料设计、医学成像、药物发现和宇宙学等科学应用。这些算法的目标是从数据和可用知识开始学习和构建模型，然后部署学习到的模型来生成新的预测。这些预测以新数据的形式出现，比如新图像、新文本，甚至是药物设计的新候选分子。该项目涉及从信息论、深度学习、微分方程和概率论开发新的数学工具，以设计、改进、解释并最终信任这些学习算法。研究人员还将应用这些新算法来合并来自不同癌症研究的数据集，通过整合来自同一疾病但使用不同研究、技术和患者群体获得的数据来解决改进数据分析的关键需求。提出的研究的主要目标是在数据稀缺或获取昂贵时开发新的可靠的机器学习算法。作为本研究项目的一部分，研究生将接受这一领域的培训。概率散度和度量是用于度量不同概率模型之间或模型与数据之间的差异的数学对象，特别适用于非常高维的设置。需要仔细设计差异，以构建最能描述现有数据的模型。该项目将结合最优传输，信息论，偏微分方程和深度学习的工具来开发Lipschitz正则化散度，它在Wasserstein度量和信息论散度（例如Kullback-Leibler散度）之间进行插值，并提供灵活的损失函数家族来比较非绝对连续的概率度量。在机器学习应用中，人们经常需要构建算法来模拟奇异的目标分布，这些分布要么是由于其内在性质（如集中在低维结构上的概率），要么是因为它们通常只能通过数据来了解。这些新的散度将与深度学习相结合，在概率空间中构建梯度流，能够将任何初始分布传输到目标数据集。这些新方法也将适用于保存结构的学习，应用范围从医学到新分子的设计，其中数据可以显示对称性或物理限制。这些额外的知识将被考虑在概率散度中，以一种有效的方式构建保持结构的生成算法。每当数据稀缺和/或难以获得时，将研究和量化生成算法中结构的基本作用。这项研究的示范领域之一是生物信息学，在生物信息学中，由于预算限制或患者有限（例如罕见疾病），即使涉及同一种疾病，可用的真实数据集的样本量也很低。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Generative modeling algorithms underlie many recent and ongoing advances in artificial intelligence, both in popular image and text generation tools as well as scientific applications such as materials design, medical imaging, drug discovery, and cosmology, to name a few. The goal of such algorithms is to learn and construct a model starting from data and available knowledge, and then deploy the learned model to generate new predictions. These predictions are in the form of new data such as new images, new text or even new candidate molecules for drug design. This project involves the development of new mathematical tools from information theory, deep learning, differential equations and probability theory to design, improve, explain and ultimately trust such learning algorithms. The investigators will also apply these new algorithms to merge data sets from different cancer studies, addressing a critical need to improve data analysis by integrating data from the same disease but which are obtained using different studies, technologies, and patient groups. The primary goal of the proposed research is to develop new reliable machine learning algorithms when data is scarce or expensive to obtain. Graduate students will be trained in this field as part of this research project.Probability divergences and metrics are mathematical objects designed to measure discrepancies between different probabilistic models or between models and data and are especially adept in very high-dimensional settings. Divergences need to be carefully designed to construct models which best describe the available data. This project will combine tools from optimal transport, information theory, partial differential equations and deep learning to develop Lipschitz regularized divergences which interpolate between Wasserstein metrics and information-theoretic divergences (e.g. the Kullback-Leibler divergence) and which provide flexible families of loss functions to compare non-absolutely continuous probability measures. In machine learning applications one often needs to build algorithms to model target distributions which are singular, either by their intrinsic nature such as probabilities concentrated on low dimensional structures and/or because they are often only known through data. These new divergences will be combined with deep learning to build gradient flows in a probability space which are capable of transporting any initial distribution to a target data set. These new methods will also be adapted for structure-preserving learning, arising in applications ranging from medicine to the design of new molecules, where data can exhibit symmetries or physical constraints. This additional knowledge will be taken into account in the probability divergence to build structure-preserving generative algorithms in an efficient way. The essential role of structure in generative algorithms will be studied and quantified whenever data is scarce and/or expensive to obtain. One of the demonstration areas of this research is in bioinformatics, where available real datasets, even when they involve the same disease, have low sample size due to budgetary constraints or limited availability of patients e.g., in the case of rare diseases.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.