In recent years, Large Language Models have garnered significant attention for their strong performance in various natural language tasks, such as machine translation and question answering. These models demonstrate an impressive ability to generalize across diverse tasks. However, their effectiveness in tackling domain-specific tasks, such as financial sentiment analysis and monetary policy understanding, remains a topic of debate, as these tasks often require specialized knowledge and precise reasoning. To address such challenges, researchers design various prompts to unlock the models' abilities. By carefully crafting input prompts, researchers can guide these models to produce more accurate responses. Consequently, prompt engineering has become a key focus of study. Despite the advancements in both models and prompt engineering, the relationship between the two-specifically, how prompt design impacts models' ability to perform domain-specific tasks-remains underexplored. This paper aims to bridge this research gap.

Recently, numerous tensor SVD (t-SVD)-based tensor recovery methods have emerged, showing promise in processing visual data. However, these methods often suffer from performance degradation when confronted with high-order tensor data exhibiting non-smooth changes, commonly observed in real-world scenarios but ignored by the traditional t-SVD-based methods. Our objective in this study is to provide an effective tensor recovery technique for handling non-smooth changes in tensor data and efficiently explore the correlations of high-order tensor data across its various dimensions without introducing numerous variables and weights. To this end, we introduce a new tensor decomposition and a new tensor norm called the Tensor $U_1$ norm. We utilize these novel techniques in solving the problem of high-order tensor completion problem and provide theoretical guarantees for the exact recovery of the resulting tensor completion models. An optimization algorithm is proposed to solve the resulting tensor completion model iteratively by combining the proximal algorithm with the Alternating Direction Method of Multipliers. Theoretical analysis showed the convergence of the algorithm to the Karush-Kuhn-Tucker (KKT) point of the optimization problem. Numerical experiments demonstrated the effectiveness of the proposed method in high-order tensor completion, especially for tensor data with non-smooth changes.

This study aims to solve the over-reliance on the rank estimation strategy in the standard tensor factorization-based tensor recovery and the problem of a large computational cost in the standard t-SVD-based tensor recovery. To this end, we proposes a new tensor norm with a dual low-rank constraint, which utilizes the low-rank prior and rank information at the same time. In the proposed tensor norm, a series of surrogate functions of the tensor tubal rank can be used to achieve better performance in harness low-rankness within tensor data. It is proven theoretically that the resulting tensor completion model can effectively avoid performance degradation caused by inaccurate rank estimation. Meanwhile, attributed to the proposed dual low-rank constraint, the t-SVD of a smaller tensor instead of the original big one is computed by using a sample trick. Based on this, the total cost at each iteration of the optimization algorithm is reduced to $\mathcal{O}(n^3\log n +kn^3)$ from $\mathcal{O}(n^4)$ achieved with standard methods, where $k$ is the estimation of the true tensor rank and far less than $n$. Our method was evaluated on synthetic and real-world data, and it demonstrated superior performance and efficiency over several existing state-of-the-art tensor completion methods.