We propose a new supervised dimensionality reduction technique called Supervised Linear Centroid-Encoder (SLCE), a linear counterpart of the nonlinear Centroid-Encoder (CE) \citep{ghosh2022supervised}. SLCE works by mapping the samples of a class to its class centroid using a linear transformation. The transformation is a projection that reconstructs a point such that its distance from the corresponding class centroid, i.e., centroid-reconstruction loss, is minimized in the ambient space. We derive a closed-form solution using an eigendecomposition of a symmetric matrix. We did a detailed analysis and presented some crucial mathematical properties of the proposed approach. %We also provide an iterative solution approach based solving the optimization problem using a descent method. We establish a connection between the eigenvalues and the centroid-reconstruction loss. In contrast to Principal Component Analysis (PCA) which reconstructs a sample in the ambient space, the transformation of SLCE uses the instances of a class to rebuild the corresponding class centroid. Therefore the proposed method can be considered a form of supervised PCA. Experimental results show the performance advantage of SLCE over other supervised methods.

翻译：我们提出了一种新的监督降维技术，称为监督线性质心编码器（SLCE），它是非线性质心编码器（Centroid-Encoder, CE）\citep{ghosh2022supervised}的线性对应方法。SLCE通过线性变换将类别的样本映射到其类别质心。该变换是一个投影，它在原始空间中最小化点与对应类别质心之间的距离，即质心重建损失。我们利用对称矩阵的特征分解推导出了闭式解。我们对所提方法进行了详细分析，并给出了其若干关键数学性质。我们建立了特征值与质心重建损失之间的联系。与在主成分分析（PCA）中重建原始空间中的样本不同，SLCE的变换利用类别的实例来重建对应的类别质心。因此，所提方法可视为一种监督PCA的形式。实验结果表明，SLCE相比其他监督方法具有性能优势。