CAREER: New Paradigms in Geometric Analysis of Data Sets and their Applications

The PI and his collaborators will develop algorithms for detecting and recovering underlying sparse geometric structures from massive high-dimensional data sets. In particular, they plan to explore the following frameworks: geometric optimization for the purpose of detecting low-dimensional geometric structures within point clouds; multiscale methods for the effective detection of local scales and their combination for capturing the most relevant local and global geometric information; online and adaptive algorithms for organizing data as mixtures of manifolds while separating outliers. The proposed methodologies will be justified by theoretical guarantees on performance, and these methodologies will be applied to a variety of data sets, many of which will be provided by industrial collaborators. The applications include: automatic detection of moving objects in video surveillance cameras; motion segmentation of video images, automatic segmentation of blood vessels in the brain taken via dynamic CT scans into arteries and veins.

Recently there has been a fundamental shift in the analysis and manipulation of certain types of data sets such as digital satellite images and magnetic resonance images (MRI). This revolution relies on the fact that while such images seemingly have a complex and high-dimensional structure, in fact they are relatively low-dimensional or 'sparse'. The basic observation was that this sparsity could be exploited to more rapidly acquire, transmit, reconstruct, and analyze such images. The PI and his collaborators are extending such 'dimensionality reduction' techniques to more general instances of data sets with the aim of identifying when seemingly high-dimensional collections of data are actually much more simple, and to then get a grip on what the simplified structure is. Such research has several important applications related to making computer aided decisions about data which has both security and medical significance. The hope is that the research yields speedy, efficient, and proven algorithms for separating various and important features of data which is changing in time. The practical benefits of the research would include reliable automation of security cameras. Many of the applications and themes suggested in this proposal are accessible to a broad community. The PI plans to take advantage of this accessibility in order to integrate the research effort with the education of younger researchers and students. In particular, the PI is committed to provide material to mathematics educators at all levels and involve undergraduate and graduate students in emerging industrial research. The PI will share his joint findings through publications and software, all available online to the scientific and engineering communities as well as the public at large.