NSF-BSF: Computational Methods for Shape Space Analysis in Structural Biology

现代科学应用需要分析大量复杂的数据集。 一种普遍的情况是，每个数据点本身是几何的（例如，由于是图像），而整个数据集也携带形状结构（例如，由于潜在的运动）。 以结构生物学为例，冷冻电子显微镜和X射线自由电子激光等实验技术可以用来捕捉感兴趣的蛋白质的数十万张嘈杂图像。 然后，科学家们使用软件工具重建蛋白质的三维形状及其构象，这对基础科学和药物发现至关重要。 在这个项目中，将开发新的计算和数学方法来分析图像集和体积数据集，表现出连续的变化。 该项目将为研究生提供培训，并为美国和以色列的学生提供合作机会。研究的第一部分将整合来自最佳运输领域的指标与机器学习方法进行降维和聚类，以一种针对形状空间数据集分析的方式进行定制。 基于图形的方法将与Wasserstein度量相结合，重点是噪声鲁棒性，样本和计算效率以及几何变形方面的可解释性。 研究的第二部分将是开发一个代数框架，用于处理形状空间数据集中的对称性，例如旋转对称性。 组表示理论将用于主成分分析和基于图的学习方法，以实现比当前方法更好的效率。 该项目将产生具有广泛适用性的严格数学算法，以及用于紧迫科学应用的专业软件库。该奖项反映了NSF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。

Modern scientific applications require analyzing massive and complex datasets. A prevalent situation is that each data point is itself geometrical (due to being an image for example), while the totality of the dataset carries shape structure as well (due to underlying motion for instance). Consider structural biology, where experimental techniques known as cryo-electron microscopy and X-ray free electron lasers can be used to capture hundreds of thousands of noisy images of a protein of interest. Scientists then use software tools to reconstruct the three-dimensional shape of the protein and its conformations, which are vital to basic science and drug discovery. In this project new computational and mathematical methods will be developed for analyzing image sets and volumetric datasets which exhibit continuous variability. The project will provide graduate student training, and opportunities for students in the US and Israel to collaborate. The first part of the research will integrate metrics from the field of optimal transport with machine learning methods for dimensionality reduction and clustering, in a way that is tailored towards the analysis of shape space datasets. Graph-based methods will be combined with the Wasserstein metric with emphases on noise robustness, sample and computational efficiency, and interpretability in terms of geometric deformations. The second part of the research will be to develop an algebraic framework for dealing with symmetries in shape space datasets, such as rotational symmetries. Group representation theory will be used within principal component analysis and graph-based learning methods to achieve better efficiency than current approaches. This project will produce rigorous mathematical algorithms with broad applicability, and specialized software libraries for pressing scientific applications.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.