Collaborative Research: CIF: Small: Deep Sparse Models: Analysis and Algorithms

深度卷积神经网络是一类数学模型，它为各种机器学习工具提供了令人印象深刻的成功，通常在不同领域获得最先进的结果。然而，他们的理论理解和这些算法背后的基本思想仍然难以捉摸。这些问题对于认识和描述其局限性，为其性能提供保证，甚至开发和设计改进的实用模型至关重要。获得这种理解的一种有希望的方法是对部署这些模型的样本类别进行假设（例如，因此这些是“足够简单”的），目的是提供关于它们的理论见解。该项目旨在进一步理解这种“多层卷积稀疏模型”，拓宽对其相关优化和学习问题的理解，并揭示深度学习方法。该项目旨在推进不同层数的广义稀疏模型的最新技术水平，重点关注推理和学习问题。可证明的和有效的优化方法将推导出与多层稀疏模型的反问题，依靠在最近的梯度和次梯度下降方法的新成果。这一建议将进一步扩大制定的追求其他设置，增加稳定性和鲁棒性的选择参数和离群值。此外，相应的无监督学习问题的有效算法将被提出和分析。样本复杂性和泛化界限的问题将在监督学习环境中进行研究。在整个项目中，将研究由此产生的算法与特定卷积网络架构的关系。该项目汇集了信号处理、字典学习、机器学习以及针对大规模问题的优化方法的设计、分析和实施方面的综合专业知识。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。

Deep convolutional neural networks are a class of mathematical models that provide a variety of machine learning tools with impressive success, often obtaining state-of-the-art results across different fields. Yet, their theoretical understanding and the fundamental ideas behind these algorithms have remained elusive. These questions are essential to recognize and characterize their limitations, to provide guarantees for their performance, and even to develop and engineer improved practical models. A promising approach to obtain this understanding is to make assumptions about the class of samples on which these models are deployed (e.g., so that these are "simple enough") with the intention of providing theoretical insights about them. Further understanding of this 'multi-layered convolutional sparse model' is what this project seeks accomplish, broadening the understanding of its related optimization and learning problems, and shedding light on deep learning methodologies.This project proposes to advance the state of the art in generalized sparse models of different numbers of layers, focusing on both inference and learning problems. Provable and efficient optimization methods will be derived for the inverse problems associated with multilayer sparse models by relying on new results in proximal gradient and subgradient descent methods. This proposal will further extend the formulation of the pursuit to other settings, increasing stability and robustness to the choice of parameters and to outliers. Furthermore, efficient algorithms for the corresponding unsupervised learning problem will be proposed and analyzed. Questions of sample complexity and generalization bounds will in turn be studied in supervised learning settings. Throughout this project, the resulting algorithms will be studied in terms of their relation to specific convolutional network architectures. The project brings together combined expertise in signal processing, dictionary learning, machine learning, and the design, analysis and implementation of optimization methods for large-scale problems.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.