A novel noise-robust method for efficient online data decomposition

Given the critical role of data decomposition in managing complex datasets across various fields, addressing the challenge of noise is essential, as it can significantly degrade decomposition performance. This study emphasises the importance of noise-robust data decomposition and proposes a novel online method designed for efficient and reliable decomposition under noisy conditions. To address this challenge, we define an optimisation objective that simultaneously suppresses noise while satisfying physical constraints. To mitigate the sensitivity of gradient descent-based methods to initial conditions, which can often lead to local optima, we simplify the optimisation objective propose and advanced techniques. This approach eliminates irrelevant variables, fully exploits the efficiency of sparse matrices, and quickly solves for both spatial and temporal modes, significantly improving overall performance. Theoretical derivations and analyses are provided to elucidate the optimisation process and its impact on noise robustness. Numerical experiments demonstrate that the proposed method effectively avoids local optima and outperforms traditional methods in terms of noise robustness. Specifically, in real-world data experiments, the proposed method achieves an 89.6% reduction in the root mean square error of reconstructed data, compared to a 68.06% reduction achieved by existing methods, representing a 31.6% relative improvement. These results highlight the superior reliability and robustness of the proposed method in practical applications.