CIF: Small:Compressive-Projection Principal Component Analysis

这个奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。主成分分析(PCA)长期以来一直在降维和压缩方面发挥着核心作用。然而，主成分分析是一种依赖于数据的变换，传统上是通过计算昂贵的特征分解来确定的，这一事实经常阻碍其在严重资源受限的环境中的使用。高光谱图像就是一个很好的例子-尽管主成分分析在光谱应用于高光谱图像体积时提供了出色的去相关和降维，但许多高光谱传感器是机载或星载设备的事实限制了主成分分析的更广泛使用。在这类应用中，如果能够实现基于主元分析的降维和压缩，而不需要传统主元分析所带来的编码端开销，这将是非常有益的。研究了一种有效地将PCA的计算负担从资源受限的编码器转移到能力更强的基站解码器的过程。所研究的压缩投影PCA(CPPCA)是由传感器处随机选择的低维子空间上的投影驱动的，而CPPCA解码器仅在给定这些随机投影的情况下，不仅恢复与PCA变换相关的系数，而且恢复与PCA变换基本身的近似。通过使用编码侧随机投影，CPPCA允许降维与信号采集直接集成，从而消除了编码处的降维的显式计算。计算和存储负担被转移到CPPCA解码器，CPPCA解码器由一种新的特征向量重构过程组成，该过程基于投影子空间内Ritz向量驱动的凸集优化。研究活动的目的是从分析和实践两方面进一步了解CPPCA，包括探索在地理空间应用中产生的数据中的CPPCA，以及制定适应CPPCA的基本过程，以改进处理异常数据的性能。

"This award is funded under the American Recovery and Reinvestment Act of 2009(Public Law 111-5)."Principal component analysis (PCA) has long played a central role in dimensionality reduction and compression. However, the fact that PCA is a data-dependent transform that is traditionally determined via a computationally expensive eigendecomposition often hinders its use in severely resource-constrained settings. Hyperspectral imagery is a prime example---although PCA offers excellent decorrelation and dimensionality reduction when applied spectrally to hyperspectral image volumes, the fact that many hyperspectral sensors are airborne or spaceborne devices limits wider use of PCA. In such applications, it would be greatly beneficial if PCA-based dimensionality reduction and compression could be accomplished without the heavy encoder-side cost entailed by traditional PCA. This research investigates a process that effectively shifts the computational burden of PCA from the resource-constrained encoder to a more capable base-station decoder.The studied approach, compressive-projection PCA (CPPCA), is driven by projections at the sensor onto lower-dimensional subspaces chosen at random, while the CPPCA decoder, given only these random projections, recovers not only the coefficients associated with the PCA transform, but also an approximation to the PCA transform basis itself. By using encoder-side random projections, CPPCA permits dimensionality reduction to be integrated directly with signal acquisition such that explicit computation of dimensionality reduction at the encoder is eliminated. Computation and memory burdens are instead shifted to the CPPCA decoder which consists of a novel eigenvector-reconstruction process based on a convex-set optimization driven by Ritz vectors within the projected subspaces. Research activities are aimed at further understanding CPPCA both analytically and practically, including the exploration of CPPCA in data arising in geospatial applications, and the development of adaptations to the basic CPPCA process so as to improve performance on anomalous data.