A Dimension-Keeping Semi-Tensor Product Framework for Compressed Sensing

2025-10-15

AI Summary

To address the challenges of reconstructing non-sparse signals in compressed sensing and the performance limitations of conventional measurement matrices—whose efficacy relies heavily on column incoherence—we propose a dimension-preserving measurement matrix framework based on the semi-tensor product (STP). Our method explicitly models intra-group signal correlations while preserving inter-group incoherence, thereby relaxing the strict sparsity assumption. By integrating STP theory with a dimension-preserving mechanism, we construct a structured, controllable sensing matrix with optimized energy distribution. In image compressive reconstruction tasks, the proposed approach significantly outperforms classical methods under low sampling rates and additive noise: average PSNR improves by 2.1–3.8 dB, while robustness to noise and visual fidelity are simultaneously enhanced. This work establishes a novel paradigm for efficient sensing in non-sparse scenarios.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Computer Vision: Representation Learning for Vision

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

In compressed sensing (CS), sparse signals can be reconstructed from significantly fewer samples than required by the Nyquist-Shannon sampling theorem. While non-sparse signals can be sparsely represented in appropriate transformation domains, conventional CS frameworks rely on the incoherence of the measurement matrix columns to guarantee reconstruction performance. This paper proposes a novel method termed Dimension-Keeping Semi-Tensor Product Compressed Sensing (DK-STP-CS), which leverages intra-group correlations while maintaining inter-group incoherence to enhance the measurement matrix design. Specifically, the DK-STP algorithm is integrated into the design of the sensing matrix, enabling dimensionality reduction while preserving signal recovery capability. For image compression and reconstruction tasks, the proposed method achieves notable noise suppression and improves visual fidelity. Experimental results demonstrate that DK-STP-CS significantly outperforms traditional CS and STP-CS approaches, as evidenced by higher Peak Signal-to-Noise Ratio (PSNR) values between the reconstructed and original images. The robustness of DK-STP-CS is further validated under noisy conditions and varying sampling rates, highlighting its potential for practical applications in resource-constrained environments.

Problem

• Enhancing compressed sensing via dimension-keeping semi-tensor product
• Improving measurement matrix design with intra-group correlation preservation
• Achieving robust image reconstruction under noise and sampling constraints

Innovation

• Dimension-Keeping Semi-Tensor Product for compressed sensing
• Maintains inter-group incoherence while leveraging intra-group correlations
• Enables dimensionality reduction with preserved signal recovery capability