This paper investigates the impact of alignment between the target function of interest and the kernel matrix on a variety of kernel-based methods based on a general loss belonging to a rich loss function family, which covers many commonly used methods in regression and classification problems. We consider the truncated kernel-based method (TKM) which is estimated within a reduced function space constructed by using the spectral truncation of the kernel matrix and compare its theoretical behavior to that of the standard kernel-based method (KM) under various settings. By using the kernel complexity function that quantifies the complexity of the induced function space, we derive the upper bounds for both TKM and KM, and further reveal their dependencies on the degree of target-kernel alignment. Specifically, for the alignment with polynomial decay, the established results indicate that under the just-aligned and weakly-aligned regimes, TKM and KM share the same learning rate. Yet, under the strongly-aligned regime, KM suffers the saturation effect, while TKM can be continuously improved as the alignment becomes stronger. This further implies that TKM has a strong ability to capture the strong alignment and provide a theoretically guaranteed solution to eliminate the phenomena of saturation effect. The minimax lower bound is also established for the squared loss to confirm the optimality of TKM.

This paper investigates the impact of alignment between the target function of interest and the kernel matrix on a variety of kernel-based methods based on a general loss belonging to a rich loss function family, which covers many commonly used methods in regression and classification problems. We consider the truncated kernel-based method (TKM) which is estimated within a reduced function space constructed by using the spectral truncation of the kernel matrix and compare its theoretical behavior to that of the standard kernel-based method (KM) under various settings. By using the kernel complexity function that quantifies the complexity of the induced function space, we derive the upper bounds for both TKM and KM, and further reveal their dependencies on the degree of target-kernel alignment. Specifically, for the alignment with polynomial decay, the established results indicate that under the just-aligned and weakly-aligned regimes, TKM and KM share the same learning rate. Yet, under the strongly-aligned regime, KM suffers the saturation effect, while TKM can be continuously improved as the alignment becomes stronger. This further implies that TKM has a strong ability to capture the strong alignment and provide a theoretically guaranteed solution to eliminate the phenomena of saturation effect.

In location-based resource allocation scenarios, the distances between each individual and the facility are desired to be approximately equal, thereby ensuring fairness. Individually fair clustering is often employed to achieve the principle of treating all points equally, which can be applied in these scenarios. This paper proposes a novel algorithm, tilted k-means (TKM), aiming to achieve individual fairness in clustering. We integrate the exponential tilting into the sum of squared errors (SSE) to formulate a novel objective function called tilted SSE. We demonstrate that the tilted SSE can generalize to SSE and employ the coordinate descent and first-order gradient method for optimization. We propose a novel fairness metric, the variance of the distances within each cluster, which can alleviate the Matthew Effect typically caused by existing fairness metrics. Our theoretical analysis demonstrates that the well-known k-means++ incurs a multiplicative error of O(k log k), and we establish the convergence of TKM under mild conditions. In terms of fairness, we prove that the variance generated by TKM decreases with a scaled hyperparameter. In terms of efficiency, we demonstrate the time complexity is linear with the dataset size. Our experiments demonstrate that TKM outperforms state-of-the-art methods in effectiveness, fairness, and efficiency.

Traditional Korean medicine (TKM) emphasizes individualized diagnosis and treatment. This uniqueness makes AI modeling difficult due to limited data and implicit processes. Large language models (LLMs) have demonstrated impressive medical inference, even without advanced training in medical texts. This study assessed the capabilities of GPT-4 in TKM, using the Korean National Licensing Examination for Korean Medicine Doctors (K-NLEKMD) as a benchmark. The K-NLEKMD, administered by a national organization, encompasses 12 major subjects in TKM. We optimized prompts with Chinese-term annotation, English translation for questions and instruction, exam-optimized instruction, and self-consistency. GPT-4 with optimized prompts achieved 66.18% accuracy, surpassing both the examination's average pass mark of 60% and the 40% minimum for each subject. The gradual introduction of language-related prompts and prompting techniques enhanced the accuracy from 51.82% to its maximum accuracy. GPT-4 showed low accuracy in subjects including public health & medicine-related law, internal medicine (2) which are localized in Korea and TKM. The model's accuracy was lower for questions requiring TKM-specialized knowledge. It exhibited higher accuracy in diagnosis-based and recall-based questions than in intervention-based questions. A positive correlation was observed between the consistency and accuracy of GPT-4's responses. This study unveils both the potential and challenges of applying LLMs to TKM. These findings underline the potential of LLMs like GPT-4 in culturally adapted medicine, especially TKM, for tasks such as clinical assistance, medical education, and research. But they also point towards the necessity for the development of methods to mitigate cultural bias inherent in large language models and validate their efficacy in real-world clinical settings.