Analyzing and learning from high dimensional data have always been challenging in machine learning tasks, causing serious computational complexities and poor learning performances. Supervised dimensionality reduction is a popular technique to address such challenges in supervised learning tasks, where data are accompanied with labels. Traditionally, such techniques mostly learn one single transformation to project data into a low-dimensional discriminative subspace. However, learning only one transformation for the whole data could be dominated by one or several classes, and the rest of classes receive less discrimination in the reduced space. That is to say, learning one transformation is insufficient to properly discriminate classes of data in the reduced space because they may have complex and completely dissimilar distributions. This insufficiency becomes even more serious if the number of classes increases, leading to poor discrimination and lessening the learning performance in the reduced space. To overcome this limitation, we propose a novel supervised dimensionality reduction method, which learns per-class transformations by optimizing a newly designed and efficient objective function. The proposed method captures more discriminative information from each single class of data compared to the case of one single transformation. Moreover, the proposed objective function enjoys several desirable properties: (1) maximizing margins between the transformed classes of data, (2) having a closed-form solution, (3) being easily kernelized in the case of nonlinear data,(4) preventing overfitting, and (5) ensuring the transformations are sparse in rows so that discriminative features are learned in the reduced space. Experimental results verify that the proposed method is superior to the related state-of-the-art methods and promising in generating discriminative embeddings.

从高维数据进行分析和学习一直是机器学习任务中的挑战，导致严重的计算复杂性和差的学习性能。有监督的降维是一种流行的技术，可以解决有监督的学习任务中的此类挑战，即数据带有标签。传统上，此类技术通常会学习一次转换，以将数据投影到低维判别子空间中。但是，仅学习整个数据的一种转换可能会被一个或几个类所支配，而其余的类在缩小的空间中受到的歧视较少。也就是说，学习一种变换不足以正确地区分缩小空间中的数据类别，因为它们可能具有复杂且完全不同的分布。如果增加班级数量，这种不足会变得更加严重，导致辨别力差，并在缩小的空间内降低学习成绩。为了克服此限制，我们提出了一种新颖的监督降维方法，该方法通过优化新设计的高效目标函数来学习每类变换。与一次转换的情况相比，该方法从每一类数据中捕获了更多的判别信息。此外，提出的目标函数具有几个理想的属性：（1）最大化转换后的数据类别之间的边距；（2）具有封闭形式的解决方案；（3）在非线性数据的情况下容易被核化；（4）防止过度拟合 （5）确保变换的行稀疏，以便在缩小的空间中学习判别特征。实验结果证明，所提出的方法优于相关的最新技术，并有望产生判别性嵌入。