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.

Visualizing high-dimensional data is an essential task in Data Science and Machine Learning. The Centroid-Encoder (CE) method is similar to the autoencoder but incorporates label information to keep objects of a class close together in the reduced visualization space. CE exploits nonlinearity and labels to encode high variance in low dimensions while capturing the global structure of the data. We present a detailed analysis of the method using a wide variety of data sets and compare it with other supervised dimension reduction techniques, including NCA, nonlinear NCA, t-distributed NCA, t-distributed MCML, supervised UMAP, supervised PCA, Colored Maximum Variance Unfolding, supervised Isomap, Parametric Embedding, supervised Neighbor Retrieval Visualizer, and Multiple Relational Embedding. We empirically show that centroid-encoder outperforms most of these techniques. We also show that when the data variance is spread across multiple modalities, centroid-encoder extracts a significant amount of information from the data in low dimensional space. This key feature establishes its value to use it as a tool for data visualization.