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

2025-05-13

AI Summary

Global low-rank matrix approximation (LoRMA) neglects local structural variations, leading to critical detail loss in medical imaging. To address this, we propose an adaptive LoRMA method: overlapping image patches are clustered via k-means based on similarity, and singular value decomposition (SVD) is applied independently to each cluster—preserving global redundancy suppression while enabling local-aware compression. We establish, for the first time, a theoretical framework for adaptive low-rank approximation under overlapping patch clustering, quantifying the trade-off between patch size, compression ratio, and computational cost. This enables joint optimization: high-fidelity reconstruction in clinically critical regions and aggressive compression elsewhere. Evaluated on MRI, ultrasound, CT, and X-ray datasets, our method outperforms global SVD by +1.8 dB PSNR, +0.04 SSIM, −32% MSE, +6.5% IoU, and +11.2% EPI, while significantly suppressing block artifacts and enhancing diagnostic fidelity in lesion regions.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Computer Vision: Learning & Optimization for CV
• Data Mining & Knowledge Management: Data Compression

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

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.

Problem

• Adaptive low-rank matrix approximation for data compression
• Preserving local structural details in medical imaging
• Optimizing compression efficiency and diagnostic fidelity

Innovation

• Partitions data into overlapping patches
• Groups similar patches using k-means clustering
• Performs SVD within each cluster for compression