This project focuses on how to reveal the whole from only partial measurements. Often, physical variables can only be incompletely measured. For example, the state-of-the-art techniques to record brain activity (multi-electrode recordings of local field potentials or fMRI measurements) give only a partial account of the activity patterns of large numbers of neurons. In the course of this project, Fritz Sommer, Ph.D., and his collaborators from the University of California, investigate and develop methods for recovering complex multi-dimensional data structure from incomplete measurements. In the general case, when measurements are made (or "subsampled") at a rate lower than a mathematical limit called the Nyquist limit, the original signal cannot be fully reconstructed. However, the recent theory of compressed sensing (CS) demonstrated that subsampling below the Nyquist limit can be lossless if the signal to be compressed has sparse structure. The established theory of CS explains when full recovery from incomplete measurements is possible and provides efficient algorithms for full data reconstruction. This result is extremely relevant because many important classes of sensor signals, such as natural images and sounds, have an approximately sparse structure.
The current project explores whether the principle of CS can allow reconstruction of data structure from measurements that subsample an unknown signal in an unknown fashion. Standard CS cannot reconstruct the signal in such situations because the algorithm requires knowledge of how the signal was subsampled and what its structure is. Dr. Sommer and his team plan to develop methods for data reconstruction that can be performed with the subsampled data alone. The idea is to combine CS, a principle about measuring a signal with sparse structure, with sparse coding (SC), a principle for learning efficient representations of signals with sparse structure. Preliminary results suggest that this combination of methods, called adaptive compressed sensing (ACS), can indeed "learn" the map for recovering the full data from the subsamples alone (Isely et al. 2011). The team will investigate under what conditions a similar result holds for the large class of real-world sensor data that are not exactly sparse but can be well-approximated by sparse representations. Also being developed are methods on how to draw inferences and make decisions based on incomplete measurements. In particular, a pilot investigation in collaboration with Dr. Bosco Tjan's lab at the University of Southern California will explore whether ACS can be applied to improve the decoding of fMRI data.
