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.
