CIF: Small:Compressive-Projection Principal Component Analysis

“这项奖励是根据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）在降维和压缩数据方面一直发挥着核心作用。然而，PCA是一种依赖于数据的变换，传统上是通过计算代价高昂的特征分解确定的，这一事实往往阻碍了它在资源严重受限的环境中的使用。高光谱图像是一个典型的例子——尽管PCA在光谱上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，这一事实限制了PCA的广泛使用。在这些应用中，如果基于PCA的降维和压缩可以在没有传统PCA所带来的沉重编码器侧成本的情况下完成，这将是非常有益的。本研究探讨了一种有效地将PCA的计算负担从资源受限的编码器转移到更有能力的基站解码器的过程。所研究的方法，压缩投影PCA (CPPCA)，是由传感器上随机选择的低维子空间的投影驱动的，而CPPCA解码器只给出这些随机投影，不仅恢复与PCA变换相关的系数，而且恢复近似于PCA变换基本身。通过使用编码器侧随机投影，CPPCA允许将降维直接集成到信号采集中，从而消除了编码器的显式降维计算。计算和内存负担被转移到CPPCA解码器，该解码器由基于凸集优化的特征向量重建过程组成，该过程由投影子空间中的里兹向量驱动。研究活动旨在从分析和实践两方面进一步理解CPPCA，包括探索在地理空间应用中产生的数据中的CPPCA，以及开发对CPPCA基本过程的适应，以提高对异常数据的处理性能。

"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.