CIF: Small: Recursive Robust Principal Components' Analyis (PCA)

CIF：小型：递归稳健主成分分析 (PCA)

• 批准号: 1117125
• 负责人: Namrata Vaswani
• 金额: $ 39.67万
• 依托单位: Iowa State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1117125&HistoricalAwards=false
• 关键词: CIF; Small; Recursive; Robust; Principal; CIF; Small; Recursive; Robust; Principal

We develop novel and provably stable polynomial time solutions for solving the recursive robust principal component analysis (PCA) problem. Here, "robust" refers to robustness to both independent and correlated sparse outliers. The goal of PCA is to find the principal component (PC) space, which is the minimum-dimension subspace that spans (or, in practice, approximately spans) a given dataset. Computing the PCs in the presence of outliers is called robust PCA. If the PC space changes over time, there is a need to update the PCs. Doing this recursively is referred to as recursive robust PCA. Key potential applications include automatic foreground extraction from similar-looking backgrounds in video; sensor-network-based detection and tracking of abnormal events such as forest fires; online detection of brain activation patterns from functional MRI sequences; and speech/audio extraction from large but correlated background noise.The key idea is to reformulate this as a problem of recursively recovering a time sequence of sparse signals in the presence of large but correlated noise. The noise must be correlated enough to have an approximately low rank covariance matrix that is either constant or changes slowly. The change in the support of the sparse signal sequences may or may not be slow, but it is highly correlated; e.g. the support can move, expand or deform over time. We ask the following practically relevant questions about performance guarantees of the proposed algorithms. (a) Under what conditions can we prove exact recovery? (b) When can be obtain time-invariant and small error bounds (i.e., show stability)? The research will be included in the curriculum at various levels and in undergraduate senior design and summer research projects.

我们开发了新颖且可证明的稳定的多项式时间解决方案，以解决递归鲁棒的主成分分析（PCA）问题。在这里，“鲁棒”是指对独立和相关的稀疏异常值的鲁棒性。 PCA的目的是找到主要组件（PC）空间，这是跨越（或实际上大约跨度）的最小维度子空间。在存在异常值的情况下计算PC被称为鲁棒PCA。如果PC空间随时间变化，则需要更新PC。递归进行此操作被称为递归鲁棒PCA。关键潜在应用包括从视频中相似背景的自动前景提取；基于传感器网络的检测和跟踪异常事件（例如森林火灾）；在线检测功能性MRI序列的大脑激活模式；以及从大而相关的背景噪声中提取语音/音频。关键思想是将其重新制定为在存在较大但相关的噪声的情况下递归恢复稀疏信号的时间顺序的问题。噪声必须足够相关，以具有恒定或缓慢变化的大约较低等级协方差矩阵。稀疏信号序列的支持的变化可能会或不慢，但高度相关。例如支持可以随着时间的流逝而移动，扩展或变形。我们询问有关拟议算法的性能保证的以下实际相关问题。 （a）在什么条件下，我们可以证明精确恢复？ （b）何时可以获取时间不变和小误差边界（即显示稳定性）？该研究将包括在各个层次的课程中，以及本科高级设计和夏季研究项目。