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，它提供了各种机器学习工具，取得了令人印象深刻的成功，经常在不同的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它们的局限性、为它们的性能提供保证、甚至开发和设计改进的实用模型都是至关重要的。要获得这种理解，一个有希望的方法是对部署这些模型的样本类别做出假设(例如，使这些模型“足够简单”)，目的是提供关于它们的理论见解。进一步理解这种“多层卷积稀疏模型”，拓宽了对其相关优化和学习问题的理解，并揭示了深度学习方法。本项目旨在促进不同层数的广义稀疏模型的发展，重点研究推理和学习问题。利用近邻梯度法和次梯度下降法的新结果，将得到与多层稀疏模型相关的反问题的可证明的、有效的优化方法。这一提议将进一步将追逐的提法扩展到其他环境，增加对参数选择和对异常值的稳定性和稳健性。此外，还将针对相应的无监督学习问题提出并分析有效的算法。样本复杂性和泛化范围的问题将依次在监督学习环境中进行研究。在整个项目中，将根据它们与特定卷积网络体系结构的关系来研究得到的算法。该项目汇集了信号处理、词典学习、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 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.