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 spatioÂtemporal data. Finally they will develop a new signal transform, called sparse manifold transform, that combines traditional sparse coding with manifold learning.
