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

职业:数据集几何分析的新范式及其应用

基本信息

  • 批准号:
    0956072
  • 负责人:
  • 金额:
    $ 55.16万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2010
  • 资助国家:
    美国
  • 起止时间:
    2010-07-01 至 2016-06-30
  • 项目状态:
    已结题

项目摘要

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.
PI和他的合作者将开发用于从大量高维数据集中检测和恢复底层稀疏几何结构的算法。特别是,他们计划探索以下框架:几何优化,用于检测点云中的低维几何结构;多尺度方法,用于有效检测局部尺度及其组合,用于捕获最相关的局部和全局几何信息;在线和自适应算法,用于将数据组织为流形的混合物,同时分离离群值。所提出的方法将通过对性能的理论保证来证明,这些方法将应用于各种数据集,其中许多数据集将由工业合作者提供。这些应用包括:自动检测视频监控摄像头中的运动物体;视频图像的运动分割,通过动态CT扫描将大脑中的血管自动分割为动脉和静脉。 最近,在分析和处理某些类型的数据集,如数字卫星图像和磁共振图像(MRI)方面发生了根本性的转变。 这场革命依赖于这样一个事实,即虽然这些图像看起来具有复杂和高维的结构,但实际上它们是相对低维或“稀疏”的。基本的观察是,这种稀疏性可以被利用来更快地获取、传输、重建和分析这些图像。 PI和他的合作者正在将这种“降维”技术扩展到更一般的数据集实例,目的是识别看似高维的数据集合实际上更简单,然后掌握简化的结构是什么。 这种研究有几个重要的应用,涉及到对具有安全和医学意义的数据进行计算机辅助决策。 希望这项研究能够产生快速,高效和经过验证的算法,用于分离随时间变化的数据的各种重要特征。 这项研究的实际好处将包括安全摄像机的可靠自动化。 本提案中建议的许多应用程序和主题可供广泛的社区使用。PI计划利用这种可访问性,以便将研究工作与年轻研究人员和学生的教育相结合。特别是,PI致力于为各级数学教育工作者提供材料,并让本科生和研究生参与新兴工业研究。PI将通过出版物和软件分享他的联合发现,所有这些都可以在线提供给科学和工程界以及广大公众。

项目成果

期刊论文数量(0)
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Gilad Lerman其他文献

Estimation of Camera Locations in Highly Corrupted Scenarios: All About that Base, No Shape Trouble
高度损坏场景中摄像机位置的估计:一切都围绕该底座,没有形状问题
Phase transition in random tensors with multiple spikes
具有多个尖峰的随机张量的相变
  • DOI:
  • 发表时间:
    2018
  • 期刊:
  • 影响因子:
    0
  • 作者:
    Wei;Madeline Handschy;Gilad Lerman
  • 通讯作者:
    Gilad Lerman
$${l_p}$$ -Recovery of the Most Significant Subspace Among Multiple Subspaces with Outliers
  • DOI:
    10.1007/s00365-014-9242-6
  • 发表时间:
    2014-07-03
  • 期刊:
  • 影响因子:
    1.200
  • 作者:
    Gilad Lerman;Teng Zhang
  • 通讯作者:
    Teng Zhang
Analysis and algorithms for emℓ/emsubemp/em/sub-based semi-supervised learning on graphs
基于 emℓ/emsubemp/em/sub 的图上半监督学习的分析与算法
Topological Data Analysis and Machine Learning Theory
拓扑数据分析和机器学习理论
  • DOI:
  • 发表时间:
    2012
  • 期刊:
  • 影响因子:
    0
  • 作者:
    G. Carlsson;Rick Jardine;Dmitry Feichtner;D. Morozov;D. Attali;A. Bak;M. Belkin;Peter Bubenik;Brittany Terese Fasy;Jesse Johnson;Matthew Kahle;Gilad Lerman;Sayan Mukherjee;Monica Nicolau;A. Patel;Yusu Wang
  • 通讯作者:
    Yusu Wang

Gilad Lerman的其他文献

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{{ truncateString('Gilad Lerman', 18)}}的其他基金

Mathematically-Guaranteed Global Solutions to Structure-from-Motion
数学保证的运动结构全局解决方案
  • 批准号:
    2152766
  • 财政年份:
    2022
  • 资助金额:
    $ 55.16万
  • 项目类别:
    Continuing Grant
ATD: Robustness, Privacy, and Fairness in Threat Detection
ATD:威胁检测中的稳健性、隐私性和公平性
  • 批准号:
    2124913
  • 财政年份:
    2021
  • 资助金额:
    $ 55.16万
  • 项目类别:
    Standard Grant
ATD: Threat Detection Problems in Precision Agriculture and Satellite Imaging
ATD:精准农业和卫星成像中的威胁检测问题
  • 批准号:
    1830418
  • 财政年份:
    2018
  • 资助金额:
    $ 55.16万
  • 项目类别:
    Continuing Grant
Theory-Driven Solutions to Robust and Non-Convex Data Science Problems
稳健和非凸数据科学问题的理论驱动解决方案
  • 批准号:
    1821266
  • 财政年份:
    2018
  • 资助金额:
    $ 55.16万
  • 项目类别:
    Standard Grant
Novel Paradigms in Geometric Modeling of Large and High-Dimensional Data Sets
大型高维数据集几何建模的新范式
  • 批准号:
    1418386
  • 财政年份:
    2014
  • 资助金额:
    $ 55.16万
  • 项目类别:
    Standard Grant
Collaborative Research: Multi-manifold data modeling: theory, algorithms and applications
协作研究:多流形数据建模:理论、算法和应用
  • 批准号:
    0915064
  • 财政年份:
    2009
  • 资助金额:
    $ 55.16万
  • 项目类别:
    Continuing Grant
Computational Methods for Exploring the Geometry of Large Data Sets
探索大数据集几何的计算方法
  • 批准号:
    0612608
  • 财政年份:
    2006
  • 资助金额:
    $ 55.16万
  • 项目类别:
    Standard Grant

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