RI: Small: Extracting and Understanding Sparse Structure in Spatiotemporal Data in Neuroscience and Other Applications

稀疏编码和流形学习是两种方法，每种方法都有自己的权利，对于理解复杂高维数据的结构至关重要。本项目的目标是将这两种方法联合收割机结合起来，以产生一种更强大的数据分析方法。研究人员将开发时空数据稀疏编码的数学，并将其与流形学习方法联合收割机相结合。从这项研究中出现的工具将为社会带来好处，因为它们适用于许多技术和医学领域，如信号处理，图像和视频编码，医学成像，神经数据分析，神经修复术，并有望对理解视觉皮层中的信息处理产生影响。稀疏编码是最初在神经科学中开发的一个概念，用于解释大脑中的感觉表征，现在在许多图像和信号处理以及数据分析任务中得到广泛应用。然而，当前的稀疏编码方法存在严重的局限性。一个主要问题是稀疏表示可能是脆弱的，随着时间的推移或响应于输入的微小变化而突然变化，并且它们对参数设置，初始条件和稀疏求解器的特定选择非常敏感。另一个限制是，如果数据位于低维流形中，例如声音波形或图像，则数据的稀疏代码与底层低维空间的几何形状之间的连接丢失。该小组认为，这两个限制应一并解决。在以前的工作和他们自己的初步研究的基础上，他们将开发一个稀疏编码的理论框架，以揭示稀疏编码的结果是独特的条件。基于这些理论见解，他们将设计新的算法来鲁棒地揭示时空数据中的持久稀疏结构。最后，他们将开发一种新的信号变换，称为稀疏流形变换，将传统的稀疏编码与流形学习相结合。

Sparse coding and manifold learning are two methods that, each in its own right, have proven essential for understanding the structure in complex high­ dimensional data. The goal of this project is to combine these two methods to yield a qualitatively more powerful approach to analyze data. The investigators will develop the mathematics of sparse coding of spatiotemporal data and combine it with approaches from manifold learning. The tools emerging from this research will bring benefits to society since they are applicable to many areas of technology and medicine, such as signal processing, image and video coding, medical imaging, neural data analysis, neuroprosthetics, and can be expected to have implications for understanding information processing in the visual cortex. Sparse coding is a concept originally developed in neuroscience to account for sensory representations in the brain, which now sees widespread use in many image and signal processing and data analysis tasks. However, there are critical limitations with current approaches to sparse coding. One major issue is that sparse representations can be brittle, changing abruptly over time or in response to small changes in the input, and they can be quite sensitive to parameter settings, initial conditions, and the particular choice of sparse solver. Another limitation is that if the data lie in a low­ dimensional manifold, such as sound waveforms or images, the connection between the sparse codes of the data and the geometry of the underlying low dimensional space is lost. The team conjectures that both of these limitations should be addressed together. Building on previous work and their own preliminary studies, they will develop a theoretical framework for sparse coding to reveal conditions under which the results of sparse coding are unique. Based on these theoretical insights, they will design novel algorithms for robustly revealing persistent sparse structure in spatio­temporal data. Finally they will develop a new signal transform, called sparse manifold transform, that combines traditional sparse coding with manifold learning.