Rethinking RoPE: A Mathematical Blueprint for N-dimensional Positional Encoding

Rotary Position Embedding (RoPE) is widely adopted in Transformers due to its ability to encode relative positions with high efficiency and extrapolation capability. However, existing RoPE variants lack a unified theoretical foundation, especially in higher dimensions. In this paper, we propose a systematic mathematical framework for RoPE grounded in Lie group and Lie algebra theory. We identify two core properties of RoPE, named relativity and reversibility, and derive general constraints and constructions for valid RoPE in 1D, 2D, and N-dimensional (ND). We prove that RoPE must lie in the basis of a maximal abelian subalgebra (MASA) of the special orthogonal Lie algebra, and show that standard RoPE corresponds to the maximal toral subalgebra. Furthermore, we propose to model inter-dimensional interactions by learning an orthogonal basis transformation. Our framework unifies and explains existing RoPE designs, while enabling principled extensions to new modalities and tasks.

Alleviating Hyperparameter-Tuning Burden in SVM Classifiers for Pulmonary Nodules Diagnosis with Multi-Task Bayesian Optimization

In the field of non-invasive medical imaging, radiomic features are utilized to measure tumor characteristics. However, these features can be affected by the techniques used to discretize the images, ultimately impacting the accuracy of diagnosis. To investigate the influence of various image discretization methods on diagnosis, it is common practice to evaluate multiple discretization strategies individually. This approach often leads to redundant and time-consuming tasks such as training predictive models and fine-tuning hyperparameters separately. This study examines the feasibility of employing multi-task Bayesian optimization to accelerate the hyperparameters search for classifying benign and malignant pulmonary nodules using RBF SVM. Our findings suggest that multi-task Bayesian optimization significantly accelerates the search for hyperparameters in comparison to a single-task approach. To the best of our knowledge, this is the first investigation to utilize multi-task Bayesian optimization in a critical medical context.

Evolving the pulmonary nodules diagnosis from classical approaches to deep learning aided decision support: three decades development course and future prospect

Lung cancer is the commonest cause of cancer deaths worldwide, and its mortality can be reduced significantly by performing early diagnosis and screening. Since the 1960s, driven by the pressing needs to accurately and effectively interpret the massive volume of chest images generated daily, computer-assisted diagnosis of pulmonary nodule has opened up new opportunities to relax the limitation from physicians subjectivity, experiences and fatigue. It has been witnessed that significant and remarkable advances have been achieved since the 1980s, and consistent endeavors have been exerted to deal with the grand challenges on how to accurately detect the pulmonary nodules with high sensitivity at low false-positives rate as well as on how to precisely differentiate between benign and malignant nodules. The main goal of this investigation is to provide a comprehensive state-of-the-art review of the computer-assisted nodules detection and benign-malignant classification techniques developed over three decades, which have evolved from the complicated ad hoc analysis pipeline of conventional approaches to the simplified seamlessly integrated deep learning techniques. This review also identifies challenges and highlights opportunities for future work in learning models, learning algorithms and enhancement schemes for bridging current state to future prospect and satisfying future demand. As far as the authors know, it is the first review of the literature of the past thirty years development in computer-assisted diagnosis of lung nodules. We acknowledge the value of potential multidisciplinary researches that will make the computer-assisted diagnosis of pulmonary nodules enter into the main stream of clinical medicine, and raise the state-of-the-art clinical applications as well as increase both welfares of physicians and patients.