面向高维数据的稀疏与鲁棒线性判别分析模型与算法研究

In big data era, dealing with high dimensional data becomes more and more important. However, redundancy and low density of value information are usually difficult problems for high dimensional data mining. Dimensionality reduction is one of the most effective ways to cope with high dimensional data, and as a main dimensionality reduction method, linear discriminant analysis (LDA) has received a lot of attention. Nevertheless, traditional LDA is sensitive to outliers, and is lack of sparseness. Therefore, by combining our former work, this project aims at utilizing sparseness and robustness optimization methods to reduce the feature dimensions, and extracts valid information so as to perform subsequent mining. The research content of this project includes: (i)For high dimensional data, design new linear discriminant analysis models that can realize sparseness and robustness by using techniques such as sparse regularization and robust metrics, and explore linear discriminant analysis for matrices and tensors at the same time; (ii) Study contribution rate, orthogonality, and sparsity effectiveness of linear discriminant vectors, and construct corresponding theoretical guarantee; (iii) For the above models, propose effective and fast algorithms based on algorithms such as alternating direction method of multipliers and stochastic optimization technique, and apply them to detection of sick leaves’ images. The research content in this project is one of the key subjects in high dimensional data mining, which is of strong significance in both theory and reality.

在大数据时代，高维数据的处理越来越重要。然而，高维数据挖掘往往需要面对数据的大量冗余与价值密度低的难题。降维是处理高维数据最有效手段之一，而线性判别分析(LDA)作为主流的降维方法备受关注。但传统LDA易受离群点影响，且不具稀疏性。因此，本项目拟结合前期工作，利用稀疏优化及鲁棒学习研究高维数据的线性降维方法，并提取有效信息以进行后续挖掘，具体研究内容包括：(i) 针对高维数据，利用稀疏正则项和鲁棒度量等技术，设计具有稀疏性和鲁棒性的新型线性判别分析模型，同时探索矩阵和张量问题的线性判别分析；(ii) 研究线性判别向量的贡献率、正交性和稀疏有效性等理论，构建相应的理论保障；(iii) 对以上模型，提出可处理高维数据的交替方向乘子算法、随机优化等技术的有效快速求解算法，并将其应用于生病树叶图像检测。本项目研究内容是高维数据挖掘的关键课题之一，具有较强的理论和现实意义。

在基金支持下，本项目对高维数据的稀疏与鲁棒线性判别分析降维模型与算法及其应用进行了研究，具体包括稀疏与鲁棒模型的构建、有效算法的设计及应用。对此我们取得了若干有意义的成果。（1）针对高维数据，通过估计多类有标签数据的Bhattacharyya误差上界，构造几种基于L1-范数的稀疏鲁棒线性判别分析模型。（2）在此基础上，将所提基于L1-范数的稀疏鲁棒线性判别分析拓展到基于一般Lp-范数的线性判别分析模型以及2D数据输入的线性判别分析模型。（3）将以上线性判别分析中的稀疏鲁棒思想应用于更广泛的机器学习问题，包括分类、回归和聚类等，同时应用于实际图像数据。对以上研究内容，发表论文22篇，其中SCI论文21篇，CSCD期刊论文1篇。本研究为高维数据挖掘提供了新思路，具有一定的理论与现实意义。