CIF: Small: Structured Signal Modeling via Nonconvex Optimization

现代信号处理和数据分析中的许多问题可以归结为为观测数据寻找结构化的低维模型。在适当的词典中，可以将图像、音频信号、视频序列等建模为稀疏。科学数据，从神经棘波分类到显微镜，可以被建模为特征模式的叠加，在空间上转换。学习的数据表示在图像处理和视觉中的各种经典问题上产生了出色的结果，并且在最近信号分类和计算科学发现的突破中也发挥了关键作用。相比之下，从理论上讲，人们对学习表示的性质以及学习它们的算法的了解要少得多。这个项目开发了理论和专门的优化算法，在适当的条件下，确保为观察到的信号提供良好的表示。研究人员研究了一类问题的理论和计算方面，包括稀疏字典学习、稀疏反卷积和卷积字典学习。这些问题的自然形式是非凸的。该项目的目标是发展对这些非凸问题的几何洞察力，并利用这些洞察力来开发恢复理论和可扩展的高效算法。关键挑战包括了解局部和全局极小值的性质，以及开发避免在鞍点附近停滞不前的高效算法。该算法在图像分类和科学成像中产生的真实信号数据上得到了验证。

Many problems in modern signal processing and data analysis can be cast as searching for a structured, low-dimensional model for observed data. Images, audio signals, video sequences, and more, can be modeled as sparse in an appropriate dictionary. Scientific data, from neural spike sorting to microscopy, can be modeled as superpositions of characteristic patterns, translated over space. Learned data representations yield excellent results on a variety of classical problems in image processing and vision, and are also play a crucial role in recent breakthroughs in signal classification and computational scientific discovery. In contrast, much less is known theoretically about the properties of learned representations, and of algorithms for learning them. This project develops theory and dedicated optimization algorithms, which are guaranteed, under appropriate conditions, to provide good representations for observed signals.The investigators study theoretical and computational aspects of a family of problems, including sparse dictionary learning, sparse deconvolution, and convolutional dictionary learning. The natural formulations of these problems are nonconvex. The goal of the project is to develop geometric insights into these nonconvex problems, and to use these insights to develop recovery theory and scalable, efficient algorithms. Key challenges include understanding the properties of local and global minima, and developing efficient algorithms that avoid stalling near saddle points. The algorithms are demonstrated on real signal data arising in image classification and scientific imaging.