基于切空间判别学习的流形降维算法

在基于图像集的流形降维问题中，许多算法的核心思想都是把一个高维的流形直接降到一个维数相对较低、同时具有的判别信息更加充分的流形上.投影度量学习（projection metric learning，简称PML）是一种Grassmann流形降维算法.该算法是基于投影度量，并且使用RCG（Riemannian conjugate gradient）算法优化目标函数，其在多个数据集上都取得了较好的实验结果，但是对于复杂的人脸数据集，如YTC其实验结果相对较差，只取得了66.69%的正确率.同时，RCG算法的时间效率较差.基于上述原因，提出了基于切空间判别学习的流形降维算法.该算法首先对于PML中的投影矩阵添加扰动，使其成为对称正定（symmetric positive definite，简称SPD）矩阵；然后，使用LEM（log-euclidean metric）将其映射到切空间中；最后，利用基于特征值分解的迭代优化算法构造判别函数，得到变换矩阵.对提算法在多个标准数据集上进行了实验验证，并取得了较好的实验结果，从而验证了该算法的有效性.

Manifold Dimensional Reduction Algorithm Based on Tangent Space Discriminant Learning

Some good dimensional reduction algorithms based on image set have been developed. The core of these algorithms is performing a geometry-aware dimensionality reduction from the original manifold to a lower-dimensional, more discriminative manifold. Projection Metric Learning is a dimensional reduction algorithm that is based on Grassmann manifold. This algorithm, which is based on projection metric and RCG algorithm, has achieved better results on some benchmark datasets, but for some complicated face datasets, such as YTC, it has just obtained 66.69% classification accuracy. However, RCG algorithm has a poor performance of time efficiency. Based on the above reasons, a dimensional reduction algorithm based on the tangent space discriminant learning is presented. Firstly, perturbation is added to the projection matrix of PML to make it be a SPD matrix. Secondly LEM is adopted to map the element which lies on the SPD manifold to a tangent space, and then the iterative optimization algorithm based on eigen-decomposition is applied to find the discriminant function to obtain the transformation matrix. The experimental results on several standard datasets show the superiority of the proposed algorithm over other state-of-the-art algorithms.