Collaborative Research: A Unifying Deep Learning Framework Using Cell Complex Neural Networks

深度学习促进了许多新技术的发展，从自动医学图像分析到人工智能支持的文档翻译。这些社会变革源于计算机视觉和自然语言处理领域的快速进步，也就是图像和文本的处理。然而，大范围的数据并不能最好地用像素网格或单词序列来表示。例如，(生物分子)形状或(社会)网络是显示局部和全局几何属性的数据类型，而现有的深度学习架构可能无法有效地利用这些属性。因此，有必要严格理解和扩展深度学习方法可以应用到的数据类型。本研究项目考虑更抽象的“单元格复杂”数据类型。这项工作引入并旨在量化“细胞复杂网络”在深度学习中的潜力。应用范围从计算生物学和医学、社会科学和艺术，到更好地理解深度学习本身。该项目将通过出版物和发布开放源码软件、演示材料和数据集来传播这些想法。这一结果有望加强深度学习与其他依赖几何、拓扑和组合对象的领域之间的现有联系。将通过仔细选择出版地点来满足不同机器学习社区的需求。将向本科生和研究生提供研究培训，招聘过程将鼓励代表人数不足的群体提出申请。结对编程课程，加上偶尔的黑客马拉松，将有助于培训下一代从业者在拓扑和几何方面的深度学习，以关键的软件工程实践补充他们的理论培训。该项目还将通过由来自不同背景的演讲者参加的公开讲座来促进推广。这项研究旨在开发一个统一的数学框架，其中深度学习模型和协议可以在细胞复杂领域中被普遍定义和执行。这些域概括了具有实际重要性的离散域，如图、点云、网格和单纯复形。该项目首先将现有的深度学习计算模块统一到细胞复杂神经网络(CXN)的框架中。研究人员计划在具有几何、拓扑和组合特征的域上严格构建神经网络计算基元的必要工具。他们将研究与这些模型相关的重要理论问题，例如根据专门为CXN定义的指标的概括性和表现性。其次，该项目将利用深度学习的力量来回答在研究具有这种拓扑和组合结构的数据时出现的问题。该项目将提供图表、点云和网格数据类型的基准，利用模拟和真实数据集。研究人员将开发一个开源的Python包，它收集了拓扑和组合深度学习原语，并提供了一个界面，可以研究它们的理论属性。第三，该项目将应用这些工具来理解深度学习本身。这项工作将利用CXN来提取在给定网络的训练期间产生的权重迭代序列的几何和拓扑摘要。调查人员将把这些摘要与手头的深度学习算法和体系结构的概括性联系起来。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Deep learning has fostered the development of many new technologies, ranging from automatic medical image analysis to document translation powered by artificial intelligence. These societal transformations originated with rapid advancements in the fields of computer vision and natural language processing, that is, the processing of images and texts. Yet, a wide range of data is not best represented by a grid of pixels or a sequence of words. For example, (biomolecular) shapes or (social) networks are data types exhibiting local and global geometric properties that might not be efficiently leveraged by existing deep learning architectures. Hence, there is a need to rigorously understand and expand the data types to which deep learning methods can be applied. This research project considers the more abstract "cell complex" data type. The work introduces and aims to quantify the potential of "cell complex networks" in deep learning. Applications range from computational biology and medicine, social science, and art, to a better understanding of deep learning itself. The project will disseminate these ideas through publications and the release of open-source software, demonstration material, and datasets. The results are expected to enhance existing ties between deep learning and other fields that rely on geometric, topological, and combinatorial objects. The needs of diverse machine learning communities will be addressed by carefully choosing publication venues. Research training will be provided to undergraduate and graduate students, where the recruiting process will encourage applications from underrepresented groups. Pair-programming sessions, together with occasional hackathons, will help train the next generation of practitioners in topological and geometric deep learning, complementing their theoretical training with pivotal software engineering practices. This project will also facilitate outreach through public lectures featuring speakers from diverse backgrounds.This research aims to develop a unifying mathematical framework where deep learning models and protocols can be universally defined and executed over cell complex domains. Such domains generalize discrete domains of practical importance such as graphs, point clouds, meshes, and simplicial complexes. The project first unifies existing deep learning computational blocks into the framework of cell complex neural networks (CXNs). The investigators plan to rigorously construct the necessary tools of neural network computational primitives over domains that have geometric, topological, and combinatorial characteristics. They will investigate important theoretical questions associated to these models, such as generalizability and expressiveness in the light of metrics specifically defined for CXNs. Second, the project will harness the power of deep learning in answering questions that arise when studying data with such topological and combinatorial structures. The project will provide benchmarks for graph, point cloud, and mesh data types, leveraging both simulated and real datasets. The investigators will develop an open-source Python package that gathers the topological and combinatorial deep learning primitives with an interface allowing study of their theoretical properties. Third, the project will apply these tools to the understanding of deep learning itself. The work will leverage CXNs to extract geometric and topological summaries of the sequence of weight iterates generated during the training of a given network. The investigators will connect these summaries to the generalizability of the deep learning algorithm and architecture at hand.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.