半监督距离度量学习内蕴加速投影梯度算法

考虑求解一类半监督距离度量学习问题. 由于样本集(数据库)的规模与复杂性的激增, 在考虑距离度量学习问题时, 必须考虑学习来的距离度量矩阵具有稀疏性的特点. 因此, 在现有的距离度量学习模型中, 增加了学习矩阵的稀疏约束. 为了便于模型求解, 稀疏约束应用了Frobenius 范数约束. 进一步, 通过罚函数方法将Frobenius范数约束罚到目标函数, 使得具有稀疏约束的模型转化成无约束优化问题. 为了求解问题, 提出了正定矩阵群上加速投影梯度算法, 克服了矩阵群上不能直接进行线性组合的困难, 并分析了算法的收敛性. 最后通过UCI数据库的分类问题的例子, 进行了数值实验, 数值实验的结果说明了学习矩阵的稀疏性以及加速投影梯度算法的有效性.

An intrinsic accelerated projection gradient algorithm for semi-supervised metric learning

In this paper, we consider a class of semi-supervised metric learning problems. Due to the explosion in size and complexity of datasets, it is increasingly important to consider the sparse of metric learning. We add the constraint of sparse for the model of semi-supervised metric learning. To be easy to deal with the sparse constraint, we apply the Frobenius norm to define the sparse and transform it into the objective function of model by using the penalty parameter. Next we present an accelerated projection gradient algorithm, which is originally designed for convex smooth optimization in Euclidean space, over a positive definite matrix group.We analyze the convergence of our algorithm. Finally, we show the numericaltest to demonstrate the effectiveness of the proposed algorithm.