Self-supervised speech models (S3Ms) have become a common tool for the speech processing community, leveraging representations for downstream tasks. Clustering S3M representations yields discrete speech units (DSUs), which serve as compact representations for speech signals. DSUs are typically obtained by k-means clustering. Using DSUs often leads to strong performance in various tasks, including automatic speech recognition (ASR). However, even with the high dimensionality and redundancy of S3M representations, preprocessing S3M representations for better clustering remains unexplored, even though it can affect the quality of DSUs. In this paper, we investigate the potential of linear preprocessing methods for extracting DSUs. We evaluate standardization, principal component analysis, whitening, and independent component analysis (ICA) on DSU-based ASR benchmarks and demonstrate their effectiveness as preprocessing for k-means. We also conduct extensive analyses of their behavior, such as orthogonality or interpretability of individual components of ICA.