基于流形结构的原数据恢复与重构

With the development of information technology, the processing of the large high dimensional datasets becomes more and more popular, such as text, images, sounds,videoes and even gene data. To better handle these high dimensional datasets, their dimensionality usually needed to be reduced for compression storage, classification, clustering, and visualization etc. Recently, many dimensionality reduction (DR) algorithms have been developed, including linear methods such as PCA, MDS and nonlinear methods such as Isomap, LLE, LTSA, LLC etc., which are successfully applied for the feature extraction and representation in pattern classification. Many applications need re-projecting the features to the original space. Unfortunately, most of them cannot perform the reconstruction task.Though a few algorithms, such as LTSA, LLC, and deep learning can perform reconstruction.However , most of them, such Isomap, LLE, Laplacian Eigenmap cannot do this.In order to carry out the construction task, this proposal tries to investigate an approach for performing the manifold reconstruction based on the manifold assumption of the high dimensionality. We believe the algorithm could be widely applied in image recovery, information compression, patten recognition.

多数流形学习算法只是给出了如何将高维数据嵌入到低维空间中的方法，没有解决将嵌入数据映射回高维空间中的问题 ，即不能基于流形结构对原始数据进行恢复重建。如果能找到低维空间到高维空间的重构映射，那么有助于对非线性维数约简算法的嵌入质量评价，同时通过流形学习算法，我们发现数据潜在的结构，然后根据低维空间中点与点之间关系以及低维数据与高维数据点之间的对应关系来重构一些新的高维观测数据。重构新的高维观测数据是很有意义。它可以广泛应用在数据融合如图像插值、语音恢复与识别、数据压缩和数据的可视化等领域。鉴于问题的理论意义和应用价值，本项拟研究的问题是维数约简中的逆问题: 基于流形结构的数据恢复，即在给定约束条件，根据局部混合高斯模型，找出低维空间的流形结构倒高维空间的映射关系，从而实现从降维空间到高维数据空间的重构。

本项目研究的内容是维数约简中的逆问题，即在给定约束条件，找出低维空间的流形结构到高维空间的映射关系，从而实现从降维空间到高维数据空间的原数据重构。本项目中我们提出两个流形重构算法，一种是基于GMM模型的分段线性表达的思想,提出流形学习降维的框架，该框架不仅实现高维数据降维，并且可实现高维重构。另一种算法是基于字典学习理论提出了一种新的数据重构方法。通过建立低维字典和高维字典的联系从而进行重构，所提出的算法不仅计算效率高，而且占用存储少。本项目还开展了基于协同字典学习的人像识别应用研究。