RI: Small: Making sense of incomplete sensor data

这个项目的重点是如何从部分测量中揭示整体。通常，物理变量只能不完全测量。例如，记录大脑活动的最新技术（局部场电位的多电极记录或功能性磁共振成像测量）只能部分解释大量神经元的活动模式。在这个项目的过程中，弗里茨·索默博士，和他在加州大学的合作者，研究并开发了从不完整测量中恢复复杂多维数据结构的方法。在一般情况下，当以低于称为奈奎斯特极限的数学极限的速率进行测量（或“二次采样”）时，原始信号不能完全重建。然而，最近的压缩感知（CS）理论表明，如果要压缩的信号具有稀疏结构，则低于奈奎斯特极限的子采样可以是无损的。CS理论的建立解释了何时可以从不完整的测量中完全恢复，并为完全数据重建提供了有效的算法。 这一结果是非常相关的，因为许多重要类别的传感器信号，如自然的图像和声音，有一个近似稀疏的structure.The目前的项目探讨CS的原则是否可以允许重建的数据结构，从测量，子采样一个未知的信号在一个未知的方式。在这种情况下，标准CS无法重建信号，因为该算法需要了解信号如何被二次采样以及其结构是什么。Sommer博士和他的团队计划开发可以单独使用子采样数据进行数据重建的方法。该思想是将联合收割机CS（一种测量具有稀疏结构的信号的原理）与稀疏编码（SC）（一种用于学习具有稀疏结构的信号的有效表示的原理）相结合。初步结果表明，这种称为自适应压缩感知（ACS）的方法组合确实可以“学习”地图，以便单独从子样本中恢复完整数据（Isely等人，2011）。该团队将研究在什么条件下，类似的结果适用于大类真实世界的传感器数据，这些数据并不完全稀疏，但可以通过稀疏表示很好地近似。此外，还在制定如何根据不完整的测量结果进行推断和决策的方法。特别是，与南加州大学Bosco Tjan博士实验室合作的一项试点调查将探索ACS是否可以应用于改善功能磁共振成像数据的解码。

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.