CDS&E: Structure-Aware Representation Learning Using Deep Networks

近年来，深度神经网络被广泛应用于分析科学、工业和社会各个领域产生的大数据。在信号处理领域，深度卷积神经网络（CNN）是最成功的计算模型之一，在视觉目标识别、目标检测和定位等领域有着广泛的应用。尽管深度网络取得了广泛的成功，但这些模型计算的数据表示不一定以一种可理解的方式反映输入数据中的几何信息；深度模型的卓越表现与这种表现的可解释性之间仍然存在差距。特别是，从大量数据中训练的深度网络通常在模型参数中没有特定的结构，这也导致模型中存在显着的冗余。该项目将研究在深度网络中施加适当结构的数学基础，旨在建立更具可分析性和效率的网络模型，并在理论上保证性能。研究结果将直接应用于各种机器学习任务，提高现有模型的准确性、计算效率和可解释性。即将开发的理论分析将加深对深度网络的数学理解，这对下一代机器学习的计算工具非常重要。参与该项目的学生将在数学和电气工程的跨学科环境中接受培训，培养分析和软件实现的技能，这对他们未来在学术界或工业界的职业生涯有益。该项目还将通过课程开发培训未来的数学家和电气工程师，特别是通过公共在线存储库的机器学习和图像科学课程。该项目探索了使用深度网络进行表征学习的新可能性，深度网络是一种具有巨大潜力的工具，可以解决当今大数据分析和人工智能中的关键挑战。该项目的目标是开发新的深度卷积神经网络（CNN）模型的数学分析，以及基于分析的具有适当结构的CNN的创新设计。具体来说，pi将研究：(1)卷积层通道中的几何结构，通过卷积层通道中的几何结构，CNN表示可以协同并显式地编码数据中的几何信息，从而提高可解释性和鲁棒性；(2)卷积滤波器的空间结构，通过它可以解析地施加滤波器的正则性，从而使CNN表示可以证明对输入变化是稳定的；(3)高效的软件实现，将理论结果转化为计算机视觉中的目标检测和图像分割等应用。pi将使用谐波分析和近似理论的工具来解决该领域的以下开放问题：以原则性的方式去除训练过的CNN滤波器中的冗余，同时避免欠拟合，更有效地学习数据中几何变换的不变表示，以及从数据中训练的自适应网络学习深度表示的理论保证。新的数学理解将指导深度网络的设计，以实现更好的性能，准确性和计算速度，以及更好的学习数据表示的可解释性。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。

In recent years, deep neural networks have been widely applied to analyze Big Data produced in various fields of science, industry and society. In the area of signal processing, deep Convolutional Neural Network (CNN) is one of the most successful computational models, with numerous applications including visual object recognition, object detection and localization. Despite the wide success of deep networks, the data representation computed by these models does not necessarily reflect the geometric information in the input data in an understandable way; a gap remains between the remarkable performance of deep models and the interpretability of such performance. In particular, deep networks trained from enormous amounts of data typically have no specific structures in the model parameters, which also leads to significant redundancy in the model. This project will investigate the mathematical foundation for imposing appropriate structures in deep networks, aiming at more analyzable and efficient network models with theoretically guaranteed performance. The results will have direct applications in various machine learning tasks, improving the accuracy, computational efficiency and interpretability of existing models. The theoretical analysis to be developed will deepen the mathematical understanding of deep networks, which is important for the next generation of computational tools for machine learning. Students engaged in the project will be trained in an interdisciplinary environment of mathematics and electrical engineering, developing skills in both analysis and software implementation, which benefits their future careers in academia or industry. The project will also train future mathematicians and electrical engineers through course development, especially courses on machine learning and image sciences with public online repositories. The project explores the new possibility of representation learning using deep networks, a tool with immense potential to address key challenges in today's Big Data analysis and artificial intelligence. The goal of the project is to develop novel mathematical analysis of the deep Convolutional Neural Network (CNN) model, as well as innovative designs of CNNs with appropriate structures based on the analysis. Specifically, the PIs will study: (1) the geometric structures in the channels of the convolutional layers, with which the CNN representations can collaboratively and explicitly encode geometric information in the data with improved interpretability and robustness; (2) the spatial structures of the convolutional filters, by which the filter regularity can be analytically imposed so that the CNN representations can be provably stable to input variations; and (3) efficient software implementation, which transfers the theoretical results into applications such as object detection and image segmentation in computer vision. The PIs will use tools from harmonic analysis and approximation theory to address the following open problems in the field: the removal of redundancy in trained CNN filters in a principled way while avoiding under-fitting, the more efficient learning of invariant representations with respect to geometrical transforms in the data, and the theoretical guarantees of the deep representations learned by an adaptive network that is trained from data. The new mathematical understanding will guide the design of deep networks to achieve better performance, in accuracy and computational speed, and better interpretability of the learned data representation.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.