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Medical images are inherently high-resolution and contain locally varying structures that are crucial for diagnosis. Efficient compression of such data must therefore preserve diagnostic fidelity while minimizing redundancy. Low-rank matrix approximation (LoRMA) techniques have shown strong potential for image compression by capturing global correlations; however, they often fail to adapt to local structural variations across regions of interest, i.e., tumor areas or other diagnostically significant regions. To address these limitations, we introduce an adaptive LoRMA, which partitions a medical image 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. An ablation study against block-based SVD is further conducted to isolate the contribution of similarity-based clustering, and a contextual comparison with JPEG2000 is included to situate the proposed approach within established medical image compression frameworks. Results demonstrate that adaptive LoRMA effectively preserves structural integrity, edge details, and diagnostic relevance, as measured by average peak signal-to-noise ratio (PSNR), average structural similarity index (SSIM), average mean squared error (MSE), average 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.