Clustering-based Low-Rank Matrix Approximation: An Adaptive Theoretical Analysis with Application to Data Compression

Low-rank matrix approximation (LoRMA) is a fundamental tool for compressing high-resolution data matrices by extracting important features while suppressing redundancy. Low-rank methods, such as global singular value decomposition (SVD), apply uniform compression across the entire data matrix, often ignoring important local variations and leading to the loss of fine structural details. To address these limitations, we introduce an adaptive LoRMA, which partitions data matrix into overlapping patches, groups structurally similar patches into several clusters using k-means, and performs SVD within each cluster. We derive the overall compression factor accounting for patch overlap and analyze how patch size influences compression efficiency and computational cost. While the proposed adaptive LoRMA method is applicable to any data exhibiting high local variation, we focus on medical imaging due to its pronounced local variability. We evaluate and compare our adaptive LoRMA against global SVD across four imaging modalities: MRI, ultrasound, CT scan, and chest X-ray. Results demonstrate that adaptive LoRMA effectively preserves structural integrity, edge details, and diagnostic relevance, as measured by peak signal-to-noise ratio (PSNR), structural similarity index (SSIM), mean squared error (MSE), intersection over union (IoU), and edge preservation index (EPI). Adaptive LoRMA significantly minimizes block artifacts and residual errors, particularly in pathological regions, consistently outperforming global SVD in terms of PSNR, SSIM, IoU, EPI, and achieving lower MSE. Adaptive LoRMA prioritizes clinically salient regions while allowing aggressive compression in non-critical regions, optimizing storage efficiency. Although adaptive LoRMA requires higher processing time, its diagnostic fidelity justifies the overhead for high-compression applications.