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III: Medium: Non-Convex Methods for Discovering High-Dimensional Structures in Big and Corrupted Data

III：媒介：在大数据和损坏数据中发现高维结构的非凸方法

基本信息

批准号：
1704458
负责人：
Rene Vidal
金额：
$ 115万
依托单位：
Johns Hopkins University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2023-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1704458&HistoricalAwards=false
关键词：
III Medium Non Convex Methods

项目摘要

Discovering structure in high-dimensional data, such as images, videos and 3D point clouds, has become an essential part of scientific discovery in many disciplines, including machine learning, computer vision, pattern recognition, and signal processing. This has motivated extraordinary advances in the past decade, including various sparse and low-rank modeling methods based on convex optimization with provable theoretical guarantees of correct recovery. However, existing theory and algorithms rely on the assumption that high-dimensional data can be well approximated by low-dimensional structures. While this assumption is adequate for some datasets, e.g., images of faces under varying illumination, it may be violated in many emerging datasets, e.g., 3D point clouds. The goal of this project is to develop a mathematical modeling framework and associated non-convex optimization tools for discovering high-dimensional structures in big and corrupted data.This project will develop provably correct and scalable optimization algorithms for learning a union of high-dimensional subspaces from big and corrupted data. The proposed algorithms will be based on a novel framework called Dual Principal Component Pursuit that, instead of learning a basis for each subspace, seeks to learn a basis for their orthogonal complements. In sharp contrast with existing sparse and low-rank methods, which require both the dimensions of the subspaces and the percentage of outliers to be sufficiently small, the proposed framework will lead to results where even subspaces of the highest possible dimension (i.e., hyperplanes) can be correctly recovered from highly corrupted data. This will be achieved by solving a family of non-convex sparse representation problems whose analysis will require the development of novel theoretical results to guarantee the correct recovery of the subspaces from corrupted data. The project will also develop scalable algorithms for solving these non-convex optimization problems and study conditions for their convergence to the global optimum. These algorithms will be evaluated in two major applications in computer vision: segmentation of point clouds and clustering of image categorization datasets.

在图像、视频和3D点云等高维数据中发现结构已成为许多学科科学发现的重要组成部分，包括机器学习、计算机视觉、模式识别和信号处理。这在过去的十年中推动了非凡的进步，包括基于凸优化的各种稀疏和低秩建模方法，这些方法具有正确恢复的可证明理论保证。然而，现有的理论和算法依赖于这样的假设，即高维数据可以很好地近似低维结构。虽然这种假设对于某些数据集是足够的，例如，在变化的照明下的面部图像，它可能在许多新兴的数据集中被违反，例如，三维点云。该项目的目标是开发一个数学建模框架和相关的非凸优化工具，用于发现大数据和损坏数据中的高维结构。该项目将开发可证明正确和可扩展的优化算法，用于从大数据和损坏数据中学习高维子空间的并集。所提出的算法将基于一种称为双主成分追踪的新框架，该框架不是学习每个子空间的基础，而是寻求学习其正交互补的基础。与现有的稀疏和低秩方法形成鲜明对比的是，现有的稀疏和低秩方法要求子空间的维度和离群值的百分比都足够小，所提出的框架将导致这样的结果，即使是最高可能维度的子空间（即，超平面）可以从高度损坏的数据中正确恢复。这将通过解决一系列非凸稀疏表示问题来实现，这些问题的分析将需要开发新的理论结果，以保证从损坏的数据中正确恢复子空间。该项目还将开发解决这些非凸优化问题的可扩展算法，并研究其收敛到全局最优值的条件。这些算法将在计算机视觉中的两个主要应用中进行评估：点云的分割和图像分类数据集的聚类。